1.
Distance
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Distance is a numerical description of how far apart objects are. In physics or everyday usage, distance may refer to a physical length, in most cases, distance from A to B is interchangeable with distance from B to A. In mathematics, a function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a set of rules. The circumference of the wheel is 2π × radius, and assuming the radius to be 1, in engineering ω = 2πƒ is often used, where ƒ is the frequency. Chessboard distance, formalized as Chebyshev distance, is the number of moves a king must make on a chessboard to travel between two squares. Distance measures in cosmology are complicated by the expansion of the universe, the term distance is also used by analogy to measure non-physical entities in certain ways. In computer science, there is the notion of the distance between two strings. For example, the dog and dot, which vary by only one letter, are closer than dog and cat. In this way, many different types of distances can be calculated, such as for traversal of graphs, comparison of distributions and curves, distance cannot be negative, and distance travelled never decreases. Distance is a quantity or a magnitude, whereas displacement is a vector quantity with both magnitude and direction. Directed distance is a positive, zero, or negative scalar quantity, the distance covered by a vehicle, person, animal, or object along a curved path from a point A to a point B should be distinguished from the straight-line distance from A to B. For example, whatever the distance covered during a trip from A to B and back to A. In general the straight-line distance does not equal distance travelled, except for journeys in a straight line, directed distances are distances with a directional sense. They can be determined along straight lines and along curved lines, for instance, just labelling the two endpoints as A and B can indicate the sense, if the ordered sequence is assumed, which implies that A is the starting point. A displacement is a kind of directed distance defined in mechanics. A directed distance is called displacement when it is the distance along a line from A and B. This implies motion of the particle, the distance traveled by a particle must always be greater than or equal to its displacement, with equality occurring only when the particle moves along a straight path
Distance
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d (A, B) > d (A, C) + d (C, B)
2.
Classical mechanics
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In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. Classical mechanics describes the motion of objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases, Classical mechanics also provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When both quantum and classical mechanics cannot apply, such as at the level with high speeds. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, however, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and accurate form. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and these advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newtons work, particularly through their use of analytical mechanics. The following introduces the concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, the motion of a point particle is characterized by a small number of parameters, its position, mass, and the forces applied to it. Each of these parameters is discussed in turn, in reality, the kind of objects that classical mechanics can describe always have a non-zero size. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the degrees of freedom. However, the results for point particles can be used to such objects by treating them as composite objects. The center of mass of a composite object behaves like a point particle, Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space, non-relativistic mechanics also assumes that forces act instantaneously. The position of a point particle is defined with respect to a fixed reference point in space called the origin O, in space. A simple coordinate system might describe the position of a point P by means of a designated as r. In general, the point particle need not be stationary relative to O, such that r is a function of t, the time
Classical mechanics
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Sir Isaac Newton (1643–1727), an influential figure in the history of physics and whose three laws of motion form the basis of classical mechanics
Classical mechanics
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Diagram of orbital motion of a satellite around the earth, showing perpendicular velocity and acceleration (force) vectors.
Classical mechanics
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Hamilton 's greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics.
3.
Second law of motion
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Newtons laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a measure of the force. These three laws have been expressed in different ways, over nearly three centuries, and can be summarised as follows. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation. Newtons laws are applied to objects which are idealised as single point masses, in the sense that the size and this can be done when the object is small compared to the distances involved in its analysis, or the deformation and rotation of the body are of no importance. In this way, even a planet can be idealised as a particle for analysis of its orbital motion around a star, in their original form, Newtons laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newtons laws of motion for rigid bodies called Eulers laws of motion, if a body is represented as an assemblage of discrete particles, each governed by Newtons laws of motion, then Eulers laws can be derived from Newtons laws. Eulers laws can, however, be taken as axioms describing the laws of motion for extended bodies, Newtons laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Other authors do treat the first law as a corollary of the second, the explicit concept of an inertial frame of reference was not developed until long after Newtons death. In the given mass, acceleration, momentum, and force are assumed to be externally defined quantities. This is the most common, but not the interpretation of the way one can consider the laws to be a definition of these quantities. Newtonian mechanics has been superseded by special relativity, but it is useful as an approximation when the speeds involved are much slower than the speed of light. The first law states that if the net force is zero, the first law can be stated mathematically when the mass is a non-zero constant, as, ∑ F =0 ⇔ d v d t =0. Consequently, An object that is at rest will stay at rest unless a force acts upon it, an object that is in motion will not change its velocity unless a force acts upon it. This is known as uniform motion, an object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest, if an object is moving, it continues to move without turning or changing its speed
Second law of motion
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Newton's First and Second laws, in Latin, from the original 1687 Principia Mathematica.
Second law of motion
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Isaac Newton (1643–1727), the physicist who formulated the laws
4.
Continuum mechanics
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Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century, research in the area continues till today. Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies, Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical properties are represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience, Materials, such as solids, liquids and gases, are composed of molecules separated by space. On a microscopic scale, materials have cracks and discontinuities, a continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material. More specifically, the continuum hypothesis/assumption hinges on the concepts of an elementary volume. This condition provides a link between an experimentalists and a viewpoint on constitutive equations as well as a way of spatial and statistical averaging of the microstructure. The latter then provide a basis for stochastic finite elements. The levels of SVE and RVE link continuum mechanics to statistical mechanics, the RVE may be assessed only in a limited way via experimental testing, when the constitutive response becomes spatially homogeneous. Specifically for fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made, consider car traffic on a highway---with just one lane for simplicity. Somewhat surprisingly, and in a tribute to its effectiveness, continuum mechanics effectively models the movement of cars via a differential equation for the density of cars. The familiarity of this situation empowers us to understand a little of the continuum-discrete dichotomy underlying continuum modelling in general. To start modelling define that, x measure distance along the highway, t is time, ρ is the density of cars on the highway, cars do not appear and disappear. Consider any group of cars, from the car at the back of the group located at x = a to the particular car at the front located at x = b. The total number of cars in this group N = ∫ a b ρ d x, since cars are conserved d N / d t =0. The only way an integral can be zero for all intervals is if the integrand is zero for all x, consequently, conservation derives the first order nonlinear conservation PDE ∂ ρ ∂ t + ∂ ∂ x =0 for all positions on the highway. This conservation PDE applies not only to car traffic but also to fluids, solids, crowds, animals, plants, bushfires, financial traders and this PDE is one equation with two unknowns, so another equation is needed to form a well posed problem
Continuum mechanics
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Figure 1. Configuration of a continuum body
5.
Statics
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When in static equilibrium, the acceleration of the system is zero and the system is either at rest, or its center of mass moves at constant velocity. The application of Newtons second law to a system gives, F = m a, where bold font indicates a vector that has magnitude and direction. F is the total of the acting on the system, m is the mass of the system. The summation of forces will give the direction and the magnitude of the acceleration will be proportional to the mass. The assumption of static equilibrium of a =0 leads to, the summation of forces, one of which might be unknown, allows that unknown to be found. Likewise the application of the assumption of zero acceleration to the summation of moments acting on the leads to. The summation of moments, one of which might be unknown and these two equations together, can be applied to solve for as many as two loads acting on the system. From Newtons first law, this implies that the net force, the net forces equaling zero is known as the first condition for equilibrium, and the net torque equaling zero is known as the second condition for equilibrium. A scalar is a quantity which only has a magnitude, such as mass or temperature, a vector has a magnitude and a direction. There are several notations to identify a vector, including, A bold faced character V An underlined character V A character with an arrow over it V →, vectors are added using the parallelogram law or the triangle law. Vectors contain components in orthogonal bases, unit vectors i, j, and k are, by convention, along the x, y, and z axes, respectively. Force is the action of one body on another, a force is either a push or a pull. A force tends to move a body in the direction of its action, the action of a force is characterized by its magnitude, by the direction of its action, and by its point of application. Thus, force is a quantity, because its effect depends on the direction as well as on the magnitude of the action. Forces are classified as either contact or body forces, a contact force is produced by direct physical contact, an example is the force exerted on a body by a supporting surface. A body force is generated by virtue of the position of a body within a field such as a gravitational, electric. An example of a force is the weight of a body in the Earths gravitational field. In addition to the tendency to move a body in the direction of its application, the axis may be any line which neither intersects nor is parallel to the line of action of the force
Statics
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Example of a beam in static equilibrium. The sum of force and moment is zero.
6.
Acceleration
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Acceleration, in physics, is the rate of change of velocity of an object with respect to time. An objects acceleration is the net result of any and all forces acting on the object, the SI unit for acceleration is metre per second squared. Accelerations are vector quantities and add according to the parallelogram law, as a vector, the calculated net force is equal to the product of the objects mass and its acceleration. For example, when a car starts from a standstill and travels in a line at increasing speeds. If the car turns, there is an acceleration toward the new direction, in this example, we can call the forward acceleration of the car a linear acceleration, which passengers in the car might experience as a force pushing them back into their seats. When changing direction, we call this non-linear acceleration, which passengers might experience as a sideways force. If the speed of the car decreases, this is an acceleration in the direction from the direction of the vehicle. Passengers may experience deceleration as a force lifting them forwards, mathematically, there is no separate formula for deceleration, both are changes in velocity. Each of these accelerations might be felt by passengers until their velocity matches that of the car, an objects average acceleration over a period of time is its change in velocity divided by the duration of the period. Mathematically, a ¯ = Δ v Δ t, instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimal interval of time. The SI unit of acceleration is the metre per second squared, or metre per second per second, as the velocity in metres per second changes by the acceleration value, every second. An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, in this case it is said to be undergoing centripetal acceleration. Proper acceleration, the acceleration of a relative to a free-fall condition, is measured by an instrument called an accelerometer. As speeds approach the speed of light, relativistic effects become increasingly large and these components are called the tangential acceleration and the normal or radial acceleration. Geometrical analysis of space curves, which explains tangent, normal and binormal, is described by the Frenet–Serret formulas. Uniform or constant acceleration is a type of motion in which the velocity of an object changes by an amount in every equal time period. A frequently cited example of uniform acceleration is that of an object in free fall in a gravitational field. The acceleration of a body in the absence of resistances to motion is dependent only on the gravitational field strength g
Acceleration
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Components of acceleration for a curved motion. The tangential component a t is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector (or in the opposite direction). The normal component (also called centripetal component for circular motion) a c is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.
Acceleration
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Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as time interval Δt → 0 of Δ v / Δt
7.
Angular momentum
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In physics, angular momentum is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque. The definition of momentum for a point particle is a pseudovector r×p. This definition can be applied to each point in continua like solids or fluids, unlike momentum, angular momentum does depend on where the origin is chosen, since the particles position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object via the moment of inertia I. However, while ω always points in the direction of the rotation axis, Angular momentum is additive, the total angular momentum of a system is the vector sum of the angular momenta. For continua or fields one uses integration, torque can be defined as the rate of change of angular momentum, analogous to force. Applications include the gyrocompass, control moment gyroscope, inertial systems, reaction wheels, flying discs or Frisbees. In general, conservation does limit the motion of a system. In quantum mechanics, angular momentum is an operator with quantized eigenvalues, Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the spin of elementary particles does not correspond to literal spinning motion, Angular momentum is a vector quantity that represents the product of a bodys rotational inertia and rotational velocity about a particular axis. Angular momentum can be considered an analog of linear momentum. Thus, where momentum is proportional to mass m and linear speed v, p = m v, angular momentum is proportional to moment of inertia I. Unlike mass, which only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation. Unlike linear speed, which occurs in a line, angular speed occurs about a center of rotation. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center and this simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case, L = r m v ⊥, where v ⊥ = v sin θ is the component of the motion. It is this definition, × to which the moment of momentum refers
Angular momentum
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This gyroscope remains upright while spinning due to the conservation of its angular momentum.
Angular momentum
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An ice skater conserves angular momentum – her rotational speed increases as her moment of inertia decreases by drawing in her arms and legs.
8.
Couple (mechanics)
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In mechanics, a couple is a system of forces with a resultant moment but no resultant force. A better term is force couple or pure moment and its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass. In rigid body mechanics, force couples are free vectors, meaning their effects on a body are independent of the point of application, the resultant moment of a couple is called a torque. This is not to be confused with the term torque as it is used in physics, instead, torque is a special case of moment. Torque has special properties that moment does not have, in particular the property of being independent of reference point, definition A couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance or moment. The simplest kind of couple consists of two equal and opposite forces whose lines of action do not coincide and this is called a simple couple. The forces have an effect or moment called a torque about an axis which is normal to the plane of the forces. The SI unit for the torque of the couple is newton metre. When d is taken as a vector between the points of action of the forces, then the couple is the product of d and F, i. e. τ = | d × F |. The moment of a force is defined with respect to a certain point P, and in general when P is changed. However, the moment of a couple is independent of the reference point P, in other words, a torque vector, unlike any other moment vector, is a free vector. The proof of claim is as follows, Suppose there are a set of force vectors F1, F2, etc. that form a couple, with position vectors r1, r2. The moment about P is M = r 1 × F1 + r 2 × F2 + ⋯ Now we pick a new reference point P that differs from P by the vector r. The new moment is M ′ = × F1 + × F2 + ⋯ Now the distributive property of the cross product implies M ′ = + r ×, however, the definition of a force couple means that F1 + F2 + ⋯ =0. Therefore, M ′ = r 1 × F1 + r 2 × F2 + ⋯ = M This proves that the moment is independent of reference point, which is proof that a couple is a free vector. A force F applied to a body at a distance d from the center of mass has the same effect as the same force applied directly to the center of mass. The couple produces an acceleration of the rigid body at right angles to the plane of the couple. The force at the center of mass accelerates the body in the direction of the force without change in orientation, conversely, a couple and a force in the plane of the couple can be replaced by a single force, appropriately located
Couple (mechanics)
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Classical mechanics
9.
D'Alembert's principle
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DAlemberts principle, also known as the Lagrange–dAlembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond dAlembert and it is the dynamic analogue to the principle of virtual work for applied forces in a static system and in fact is more general than Hamiltons principle, avoiding restriction to holonomic systems. A holonomic constraint depends only on the coordinates and time and it does not depend on the velocities. The principle does not apply for irreversible displacements, such as sliding friction, DAlemberts contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish. That is to say that the generalized forces Q j need not include constraint forces and it is equivalent to the somewhat more cumbersome Gausss principle of least constraint. The general statement of dAlemberts principle mentions the time derivatives of the momenta of the system. The momentum of the mass is the product of its mass and velocity, p i = m i v i. In many applications, the masses are constant and this reduces to p i ˙ = m i v ˙ i = m i a i. However, some applications involve changing masses and in those cases both terms m ˙ i v i and m i v ˙ i have to remain present, to date, nobody has shown that DAlemberts principle is equivalent to Newtons Second Law. This is true only for very special cases e. g. rigid body constraints. However, a solution to this problem does exist. Consider Newtons law for a system of particles, i, if arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces, the constraint forces do no work. Such displacements are said to be consistent with the constraints and this leads to the formulation of dAlemberts principle, which states that the difference of applied forces and inertial forces for a dynamic system does no virtual work. There is also a principle for static systems called the principle of virtual work for applied forces. DAlembert showed that one can transform an accelerating rigid body into an equivalent static system by adding the so-called inertial force, the inertial force must act through the center of mass and the inertial torque can act anywhere. The system can then be analyzed exactly as a static system subjected to this force and moment. The advantage is that, in the equivalent static system one can take moments about any point and this often leads to simpler calculations because any force can be eliminated from the moment equations by choosing the appropriate point about which to apply the moment equation. Even in the course of Fundamentals of Dynamics and Kinematics of machines, in textbooks of engineering dynamics this is sometimes referred to as dAlemberts principle
D'Alembert's principle
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Jean d'Alembert (1717—1783)
D'Alembert's principle
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Free body diagram of a wire pulling on a mass with weight W, showing the d’Alembert inertia “force” ma.
D'Alembert's principle
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Free body diagram depicting an inertia moment and an inertia force on a rigid body in free fall with an angular velocity.
10.
Potential energy
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In physics, potential energy is energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors. The unit for energy in the International System of Units is the joule, the term potential energy was introduced by the 19th century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotles concept of potentiality. Potential energy is associated with forces that act on a body in a way that the work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, that are called potential forces, can be represented at every point in space by vectors expressed as gradients of a scalar function called potential. Potential energy is the energy of an object. It is the energy by virtue of a position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass is performed by a force that works against the force field of the potential. This work is stored in the field, which is said to be stored as potential energy. If the external force is removed the field acts on the body to perform the work as it moves the body back to the initial position. Suppose a ball which mass is m, and it is in h position in height, if the acceleration of free fall is g, the weight of the ball is mg. There are various types of energy, each associated with a particular type of force. Chemical potential energy, such as the energy stored in fossil fuels, is the work of the Coulomb force during rearrangement of mutual positions of electrons and nuclei in atoms and molecules. Thermal energy usually has two components, the energy of random motions of particles and the potential energy of their mutual positions. Forces derivable from a potential are also called conservative forces, the work done by a conservative force is W = − Δ U where Δ U is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, common notations for potential energy are U, V, also Ep. Potential energy is closely linked with forces, in this case, the force can be defined as the negative of the vector gradient of the potential field. If the work for a force is independent of the path, then the work done by the force is evaluated at the start
Potential energy
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In the case of a bow and arrow, when the archer does work on the bow, drawing the string back, some of the chemical energy of the archer's body is transformed into elastic potential-energy in the bent limbs of the bow. When the string is released, the force between the string and the arrow does work on the arrow. Thus, the potential energy in the bow limbs is transformed into the kinetic energy of the arrow as it takes flight.
Potential energy
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A trebuchet uses the gravitational potential energy of the counterweight to throw projectiles over two hundred meters
Potential energy
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Springs are used for storing elastic potential energy
Potential energy
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Archery is one of humankind's oldest applications of elastic potential energy
11.
Frame of reference
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In physics, a frame of reference consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements. In n dimensions, n+1 reference points are sufficient to define a reference frame. Using rectangular coordinates, a frame may be defined with a reference point at the origin. In Einsteinian relativity, reference frames are used to specify the relationship between an observer and the phenomenon or phenomena under observation. In this context, the phrase often becomes observational frame of reference, a relativistic reference frame includes the coordinate time, which does not correspond across different frames moving relatively to each other. The situation thus differs from Galilean relativity, where all possible coordinate times are essentially equivalent, the need to distinguish between the various meanings of frame of reference has led to a variety of terms. For example, sometimes the type of system is attached as a modifier. Sometimes the state of motion is emphasized, as in rotating frame of reference, sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference, in this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a system may be employed for many purposes where the state of motion is not the primary concern. For example, a system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors and it seems useful to divorce the various aspects of a reference frame for the discussion below. A coordinate system is a concept, amounting to a choice of language used to describe observations. Consequently, an observer in a frame of reference can choose to employ any coordinate system to describe observations made from that frame of reference. A change in the choice of coordinate system does not change an observers state of motion. This viewpoint can be found elsewhere as well, which is not to dispute that some coordinate systems may be a better choice for some observations than are others. Choice of what to measure and with what observational apparatus is a separate from the observers state of motion. D. Norton, The discussion is taken beyond simple space-time coordinate systems by Brading, extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations of quantum field theory, classical relativistic mechanics, and quantum gravity
Frame of reference
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An observer O, situated at the origin of a local set of coordinates – a frame of reference F. The observer in this frame uses the coordinates (x, y, z, t) to describe a spacetime event, shown as a star.
12.
Impulse (physics)
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In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Since force is a quantity, impulse is also a vector in the same direction. Impulse applied to an object produces an equivalent vector change in its linear momentum, the SI unit of impulse is the newton second, and the dimensionally equivalent unit of momentum is the kilogram meter per second. The corresponding English engineering units are the pound-second and the slug-foot per second, a resultant force causes acceleration and a change in the velocity of the body for as long as it acts. Conversely, a force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly. This is often called the impulse-momentum theorem, as a result, an impulse may also be regarded as the change in momentum of an object to which a resultant force is applied. Impulse has the units and dimensions as momentum. In the International System of Units, these are kg·m/s = N·s, in English engineering units, they are slug·ft/s = lbf·s. The term impulse is also used to refer to a force or impact. This type of impulse is often idealized so that the change in momentum produced by the force happens with no change in time and this sort of change is a step change, and is not physically possible. However, this is a model for computing the effects of ideal collisions. The application of Newtons second law for variable mass allows impulse, in the case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse. This fact can be used to derive the Tsiolkovsky rocket equation, which relates the vehicles propulsive change in velocity to the specific impulse. Wave–particle duality defines the impulse of a wave collision, the preservation of momentum in the collision is then called phase matching. Applications include, Compton effect nonlinear optics Acousto-optic modulator Electron phonon scattering Serway, Raymond A. Jewett, John W. Physics for Scientists, Physics for Scientists and Engineers, Mechanics, Oscillations and Waves, Thermodynamics
Impulse (physics)
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A large force applied for a very short duration, such as a golf shot, is often described as the club giving the ball an impulse.
Impulse (physics)
13.
Inertia
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Inertia is the resistance of any physical object to any change in its state of motion, this includes changes to its speed, direction, or state of rest. It is the tendency of objects to keep moving in a line at constant velocity. The principle of inertia is one of the principles of classical physics that are used to describe the motion of objects. Inertia comes from the Latin word, iners, meaning idle, Inertia is one of the primary manifestations of mass, which is a quantitative property of physical systems. In common usage, the inertia may refer to an objects amount of resistance to change in velocity, or sometimes to its momentum. Thus, an object will continue moving at its current velocity until some force causes its speed or direction to change. On the surface of the Earth, inertia is often masked by the effects of friction and air resistance, both of which tend to decrease the speed of moving objects, and gravity. Aristotle explained the continued motion of projectiles, which are separated from their projector, by the action of the surrounding medium, Aristotle concluded that such violent motion in a void was impossible. Despite its general acceptance, Aristotles concept of motion was disputed on several occasions by notable philosophers over nearly two millennia, for example, Lucretius stated that the default state of matter was motion, not stasis. Philoponus proposed that motion was not maintained by the action of a surrounding medium, although this was not the modern concept of inertia, for there was still the need for a power to keep a body in motion, it proved a fundamental step in that direction. This view was opposed by Averroes and by many scholastic philosophers who supported Aristotle. However, this view did not go unchallenged in the Islamic world, in the 14th century, Jean Buridan rejected the notion that a motion-generating property, which he named impetus, dissipated spontaneously. Buridans position was that an object would be arrested by the resistance of the air. Buridan also maintained that impetus increased with speed, thus, his idea of impetus was similar in many ways to the modern concept of momentum. Buridan also believed that impetus could be not only linear, but also circular in nature, buridans thought was followed up by his pupil Albert of Saxony and the Oxford Calculators, who performed various experiments that further undermined the classical, Aristotelian view. Their work in turn was elaborated by Nicole Oresme who pioneered the practice of demonstrating laws of motion in the form of graphs, benedetti cites the motion of a rock in a sling as an example of the inherent linear motion of objects, forced into circular motion. The law of inertia states that it is the tendency of an object to resist a change in motion. According to Newton, an object will stay at rest or stay in motion unless acted on by a net force, whether it results from gravity, friction, contact
Inertia
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Galileo Galilei
14.
Moment of inertia
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It depends on the bodys mass distribution and the axis chosen, with larger moments requiring more torque to change the bodys rotation. It is a property, the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems. One of its definitions is the moment of mass with respect to distance from an axis r, I = ∫ Q r 2 d m. For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about a perpendicular to the plane. When a body is rotating, or free to rotate, around an axis, the amount of torque needed to cause any given angular acceleration is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram metre squared in SI units, moment of inertia plays the role in rotational kinetics that mass plays in linear kinetics - both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, for a point-like mass, the moment of inertia about some axis is given by mr2, where r is the distance to the axis, and m is the mass. For an extended body, the moment of inertia is just the sum of all the pieces of mass multiplied by the square of their distances from the axis in question. For an extended body of a shape and uniform density. In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, the term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Eulers second law. Comparison of this frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. Moment of inertia appears in momentum, kinetic energy, and in Newtons laws of motion for a rigid body as a physical parameter that combines its shape. There is a difference in the way moment of inertia appears in planar. The moment of inertia of a flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis, if the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their arms or divers curl their bodies into a tuck position during a dive. For a simple pendulum, this yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as. Thus, moment of inertia depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation
Moment of inertia
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Tightrope walker Samuel Dixon using the long rod's moment of inertia for balance while crossing the Niagara River in 1890.
Moment of inertia
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Flywheels have large moments of inertia to smooth out mechanical motion. This example is in a Russian museum.
Moment of inertia
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Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to conservation of angular momentum.
Moment of inertia
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Pendulums used in Mendenhall gravimeter apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth.
15.
Mass
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In physics, mass is a property of a physical body. It is the measure of a resistance to acceleration when a net force is applied. It also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram, Mass is not the same as weight, even though mass is often determined by measuring the objects weight using a spring scale, rather than comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity and this is because weight is a force, while mass is the property that determines the strength of this force. In Newtonian physics, mass can be generalized as the amount of matter in an object, however, at very high speeds, special relativity postulates that energy is an additional source of mass. Thus, any body having mass has an equivalent amount of energy. In addition, matter is a defined term in science. There are several distinct phenomena which can be used to measure mass, active gravitational mass measures the gravitational force exerted by an object. Passive gravitational mass measures the force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force, according to Newtons second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A bodys mass also determines the degree to which it generates or is affected by a gravitational field and this is sometimes referred to as gravitational mass. The standard International System of Units unit of mass is the kilogram, the kilogram is 1000 grams, first defined in 1795 as one cubic decimeter of water at the melting point of ice. Then in 1889, the kilogram was redefined as the mass of the prototype kilogram. As of January 2013, there are proposals for redefining the kilogram yet again. In this context, the mass has units of eV/c2, the electronvolt and its multiples, such as the MeV, are commonly used in particle physics. The atomic mass unit is 1/12 of the mass of a carbon-12 atom, the atomic mass unit is convenient for expressing the masses of atoms and molecules. Outside the SI system, other units of mass include, the slug is an Imperial unit of mass, the pound is a unit of both mass and force, used mainly in the United States
Mass
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Depiction of early balance scales in the Papyrus of Hunefer (dated to the 19th dynasty, ca. 1285 BC). The scene shows Anubis weighing the heart of Hunefer.
Mass
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The kilogram is one of the seven SI base units and one of three which is defined ad hoc (i.e. without reference to another base unit).
Mass
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Galileo Galilei (1636)
Mass
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Distance traveled by a freely falling ball is proportional to the square of the elapsed time
16.
Power (physics)
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In physics, power is the rate of doing work. It is the amount of energy consumed per unit time, having no direction, it is a scalar quantity. In the SI system, the unit of power is the joule per second, known as the watt in honour of James Watt, another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written, because this integral depends on the trajectory of the point of application of the force and torque, this calculation of work is said to be path dependent. As a physical concept, power requires both a change in the universe and a specified time in which the change occurs. This is distinct from the concept of work, which is measured in terms of a net change in the state of the physical universe. The output power of a motor is the product of the torque that the motor generates. The power involved in moving a vehicle is the product of the force of the wheels. The dimension of power is divided by time. The SI unit of power is the watt, which is equal to one joule per second, other units of power include ergs per second, horsepower, metric horsepower, and foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the required to lift 550 pounds by one foot in one second. Other units include dBm, a logarithmic measure with 1 milliwatt as reference, food calories per hour, Btu per hour. This shows how power is an amount of energy consumed per unit time. If ΔW is the amount of work performed during a period of time of duration Δt and it is the average amount of work done or energy converted per unit of time. The average power is simply called power when the context makes it clear. The instantaneous power is then the value of the average power as the time interval Δt approaches zero. P = lim Δ t →0 P a v g = lim Δ t →0 Δ W Δ t = d W d t. In the case of constant power P, the amount of work performed during a period of duration T is given by, W = P t
Power (physics)
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Ansel Adams photograph of electrical wires of the Boulder Dam Power Units, 1941–1942
17.
Work (physics)
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In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and then dropped, the SI unit of work is the joule. The SI unit of work is the joule, which is defined as the work expended by a force of one newton through a distance of one metre. The dimensionally equivalent newton-metre is sometimes used as the unit for work, but this can be confused with the unit newton-metre. Usage of N⋅m is discouraged by the SI authority, since it can lead to confusion as to whether the quantity expressed in newton metres is a torque measurement, or a measurement of energy. Non-SI units of work include the erg, the foot-pound, the foot-poundal, the hour, the litre-atmosphere. Due to work having the physical dimension as heat, occasionally measurement units typically reserved for heat or energy content, such as therm, BTU. The work done by a constant force of magnitude F on a point that moves a distance s in a line in the direction of the force is the product W = F s. For example, if a force of 10 newtons acts along a point that travels 2 meters and this is approximately the work done lifting a 1 kg weight from ground level to over a persons head against the force of gravity. Notice that the work is doubled either by lifting twice the weight the distance or by lifting the same weight twice the distance. Work is closely related to energy, the work-energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in energy is caused by an equal amount of negative work done by the resultant force. From Newtons second law, it can be shown that work on a free, rigid body, is equal to the change in energy of the velocity and rotation of that body. The work of forces generated by a function is known as potential energy. These formulas demonstrate that work is the associated with the action of a force, so work subsequently possesses the physical dimensions. The work/energy principles discussed here are identical to Electric work/energy principles, constraint forces determine the movement of components in a system, constraining the object within a boundary. Constraint forces ensure the velocity in the direction of the constraint is zero and this only applies for a single particle system. For example, in an Atwood machine, the rope does work on each body, there are, however, cases where this is not true
Work (physics)
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A baseball pitcher does positive work on the ball by applying a force to it over the distance it moves while in his grip.
Work (physics)
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A force of constant magnitude and perpendicular to the lever arm
Work (physics)
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Gravity F = mg does work W = mgh along any descending path
Work (physics)
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Lotus type 119B gravity racer at Lotus 60th celebration.
18.
Space
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Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, in Isaac Newtons view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the visibility of spatial depth in his Essay Towards a New Theory of Vision. Kant referred to the experience of space in his Critique of Pure Reason as being a pure a priori form of intuition. In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in space is conceived as curved. According to Albert Einsteins theory of relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a model for the shape of space. In the seventeenth century, the philosophy of space and time emerged as an issue in epistemology. At its heart, Gottfried Leibniz, the German philosopher-mathematician, and Isaac Newton, unoccupied regions are those that could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete. Space could be thought of in a way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people, but since there would be no observational way of telling these universes apart then, according to the identity of indiscernibles, there would be no real difference between them. According to the principle of sufficient reason, any theory of space that implied that there could be two possible universes must therefore be wrong. Newton took space to be more than relations between objects and based his position on observation and experimentation
Space
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Gottfried Leibniz
Space
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A right-handed three-dimensional Cartesian coordinate system used to indicate positions in space.
Space
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Isaac Newton
Space
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Immanuel Kant
19.
Time
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Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future. Time is often referred to as the dimension, along with the three spatial dimensions. Time has long been an important subject of study in religion, philosophy, and science, nevertheless, diverse fields such as business, industry, sports, the sciences, and the performing arts all incorporate some notion of time into their respective measuring systems. Two contrasting viewpoints on time divide prominent philosophers, one view is that time is part of the fundamental structure of the universe—a dimension independent of events, in which events occur in sequence. Isaac Newton subscribed to this realist view, and hence it is referred to as Newtonian time. This second view, in the tradition of Gottfried Leibniz and Immanuel Kant, holds that time is neither an event nor a thing, Time in physics is unambiguously operationally defined as what a clock reads. Time is one of the seven fundamental physical quantities in both the International System of Units and International System of Quantities, Time is used to define other quantities—such as velocity—so defining time in terms of such quantities would result in circularity of definition. The operational definition leaves aside the question there is something called time, apart from the counting activity just mentioned, that flows. Investigations of a single continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy. Furthermore, it may be there is a subjective component to time. Temporal measurement has occupied scientists and technologists, and was a motivation in navigation. Periodic events and periodic motion have long served as standards for units of time, examples include the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum, and the beat of a heart. Currently, the unit of time, the second, is defined by measuring the electronic transition frequency of caesium atoms. Time is also of significant social importance, having economic value as well as value, due to an awareness of the limited time in each day. In day-to-day life, the clock is consulted for periods less than a day whereas the calendar is consulted for periods longer than a day, increasingly, personal electronic devices display both calendars and clocks simultaneously. The number that marks the occurrence of an event as to hour or date is obtained by counting from a fiducial epoch—a central reference point. Artifacts from the Paleolithic suggest that the moon was used to time as early as 6,000 years ago. Lunar calendars were among the first to appear, either 12 or 13 lunar months, without intercalation to add days or months to some years, seasons quickly drift in a calendar based solely on twelve lunar months
Time
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The flow of sand in an hourglass can be used to keep track of elapsed time. It also concretely represents the present as being between the past and the future.
Time
Time
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Horizontal sundial in Taganrog
Time
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A contemporary quartz watch
20.
Torque
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Torque, moment, or moment of force is rotational force. Just as a force is a push or a pull. Loosely speaking, torque is a measure of the force on an object such as a bolt or a flywheel. For example, pushing or pulling the handle of a wrench connected to a nut or bolt produces a torque that loosens or tightens the nut or bolt, the symbol for torque is typically τ, the lowercase Greek letter tau. When it is called moment of force, it is denoted by M. The SI unit for torque is the newton metre, for more on the units of torque, see Units. This article follows US physics terminology in its use of the word torque, in the UK and in US mechanical engineering, this is called moment of force, usually shortened to moment. In US physics and UK physics terminology these terms are interchangeable, unlike in US mechanical engineering, Torque is defined mathematically as the rate of change of angular momentum of an object. The definition of states that one or both of the angular velocity or the moment of inertia of an object are changing. Moment is the term used for the tendency of one or more applied forces to rotate an object about an axis. For example, a force applied to a shaft causing acceleration, such as a drill bit accelerating from rest. By contrast, a force on a beam produces a moment, but since the angular momentum of the beam is not changing. Similarly with any force couple on an object that has no change to its angular momentum and this article follows the US physics terminology by calling all moments by the term torque, whether or not they cause the angular momentum of an object to change. The concept of torque, also called moment or couple, originated with the studies of Archimedes on levers, the term torque was apparently introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884. A force applied at an angle to a lever multiplied by its distance from the levers fulcrum is its torque. A force of three newtons applied two metres from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum. More generally, the torque on a particle can be defined as the product, τ = r × F, where r is the particles position vector relative to the fulcrum. Alternatively, τ = r F ⊥, where F⊥ is the amount of force directed perpendicularly to the position of the particle, any force directed parallel to the particles position vector does not produce a torque
Torque
21.
Velocity
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The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of its speed and direction of motion, Velocity is an important concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a vector quantity, both magnitude and direction are needed to define it. The scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI system as metres per second or as the SI base unit of. For example,5 metres per second is a scalar, whereas 5 metres per second east is a vector, if there is a change in speed, direction or both, then the object has a changing velocity and is said to be undergoing an acceleration. To have a constant velocity, an object must have a constant speed in a constant direction, constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at a constant 20 kilometres per hour in a path has a constant speed. Hence, the car is considered to be undergoing an acceleration, Speed describes only how fast an object is moving, whereas velocity gives both how fast and in what direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified, however, if the car is said to move at 60 km/h to the north, its velocity has now been specified. The big difference can be noticed when we consider movement around a circle and this is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Velocity is defined as the rate of change of position with respect to time, average velocity can be calculated as, v ¯ = Δ x Δ t. The average velocity is less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, from this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time is the displacement, x. In calculus terms, the integral of the velocity v is the displacement function x. In the figure, this corresponds to the area under the curve labeled s. Since the derivative of the position with respect to time gives the change in position divided by the change in time, although velocity is defined as the rate of change of position, it is often common to start with an expression for an objects acceleration. As seen by the three green tangent lines in the figure, an objects instantaneous acceleration at a point in time is the slope of the tangent to the curve of a v graph at that point. In other words, acceleration is defined as the derivative of velocity with respect to time, from there, we can obtain an expression for velocity as the area under an a acceleration vs. time graph
Velocity
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As a change of direction occurs while the cars turn on the curved track, their velocity is not constant.
22.
Newton's laws of motion
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Newtons laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a measure of the force. These three laws have been expressed in different ways, over nearly three centuries, and can be summarised as follows. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation. Newtons laws are applied to objects which are idealised as single point masses, in the sense that the size and this can be done when the object is small compared to the distances involved in its analysis, or the deformation and rotation of the body are of no importance. In this way, even a planet can be idealised as a particle for analysis of its orbital motion around a star, in their original form, Newtons laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newtons laws of motion for rigid bodies called Eulers laws of motion, if a body is represented as an assemblage of discrete particles, each governed by Newtons laws of motion, then Eulers laws can be derived from Newtons laws. Eulers laws can, however, be taken as axioms describing the laws of motion for extended bodies, Newtons laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Other authors do treat the first law as a corollary of the second, the explicit concept of an inertial frame of reference was not developed until long after Newtons death. In the given mass, acceleration, momentum, and force are assumed to be externally defined quantities. This is the most common, but not the interpretation of the way one can consider the laws to be a definition of these quantities. Newtonian mechanics has been superseded by special relativity, but it is useful as an approximation when the speeds involved are much slower than the speed of light. The first law states that if the net force is zero, the first law can be stated mathematically when the mass is a non-zero constant, as, ∑ F =0 ⇔ d v d t =0. Consequently, An object that is at rest will stay at rest unless a force acts upon it, an object that is in motion will not change its velocity unless a force acts upon it. This is known as uniform motion, an object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest, if an object is moving, it continues to move without turning or changing its speed
Newton's laws of motion
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Newton's First and Second laws, in Latin, from the original 1687 Principia Mathematica.
Newton's laws of motion
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Isaac Newton (1643–1727), the physicist who formulated the laws
23.
Analytical mechanics
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In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by scientists and mathematicians during the 18th century and onward. A scalar is a quantity, whereas a vector is represented by quantity, the equations of motion are derived from the scalar quantity by some underlying principle about the scalars variation. Analytical mechanics takes advantage of a systems constraints to solve problems, the constraints limit the degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates and it does not always work for non-conservative forces or dissipative forces like friction, in which case one may revert to Newtonian mechanics or use the Udwadia–Kalaba equation. Two dominant branches of mechanics are Lagrangian mechanics and Hamiltonian mechanics. There are other such as Hamilton–Jacobi theory, Routhian mechanics. All equations of motion for particles and fields, in any formalism, one result is Noethers theorem, a statement which connects conservation laws to their associated symmetries. Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics, rather it is a collection of equivalent formalisms which have broad application. In fact the principles and formalisms can be used in relativistic mechanics and general relativity. Analytical mechanics is used widely, from physics to applied mathematics. The methods of analytical mechanics apply to particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom, the definitions and equations have a close analogy with those of mechanics. Generalized coordinates and constraints In Newtonian mechanics, one customarily uses all three Cartesian coordinates, or other 3D coordinate system, to refer to a position during its motion. In physical systems, however, some structure or other system usually constrains the motion from taking certain directions. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motions geometry and these are known as generalized coordinates, denoted qi. Difference between curvillinear and generalized coordinates Generalized coordinates incorporate constraints on the system, there is one generalized coordinate qi for each degree of freedom, i. e. each way the system can change its configuration, as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates, DAlemberts principle The foundation which the subject is built on is DAlemberts principle
Analytical mechanics
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As the system evolves, q traces a path through configuration space (only some are shown). The path taken by the system (red) has a stationary action (δ S = 0) under small changes in the configuration of the system (δ q).
24.
Routhian mechanics
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In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions, the Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, and has a similar mathematical form to the Hamiltonian, but is not exactly the same. The difference between the Lagrangian, Hamiltonian, and Routhian functions are their variables, the Routhian differs from these functions in that some coordinates are chosen to have corresponding generalized velocities, the rest to have corresponding generalized momenta. This choice is arbitrary, and can be done to simplify the problem, in each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian. The full set thus has the advantages of both sets of equations, with the convenience of splitting one set of coordinates to the Hamilton equations, and the rest to the Lagrangian equations. The Lagrangian equations are powerful results, used frequently in theory, however, if cyclic coordinates occur there will still be equations to solve for all the coordinates, including the cyclic coordinates despite their absence in the Lagrangian. Overall fewer equations need to be solved compared to the Lagrangian approach, as with the rest of analytical mechanics, Routhian mechanics is completely equivalent to Newtonian mechanics, all other formulations of classical mechanics, and introduces no new physics. It offers a way to solve mechanical problems. The velocities dqi/dt are expressed as functions of their corresponding momenta by inverting their defining relation, in this context, pi is said to be the momentum canonically conjugate to qi. The Routhian is intermediate between L and H, some coordinates q1, q2, qn are chosen to have corresponding generalized momenta p1, p2. Pn, the rest of the coordinates ζ1, ζ2, ζs to have generalized velocities dζ1/dt, dζ2/dt. Dζs/dt, and time may appear explicitly, where again the generalized velocity dqi/dt is to be expressed as a function of generalized momentum pi via its defining relation. The choice of n coordinates are to have corresponding momenta. The above is used by Landau and Lifshitz, and Goldstein, some authors may define the Routhian to be the negative of the above definition. Below, the Routhian equations of motion are obtained in two ways, in the other useful derivatives are found that can be used elsewhere. Consider the case of a system with two degrees of freedom, q and ζ, with generalized velocities dq/dt and dζ/dt, now change variables, from the set to, simply switching the velocity dq/dt to the momentum p. This change of variables in the differentials is the Legendre transformation, the differential of the new function to replace L will be a sum of differentials in dq, dζ, dp, d, and dt. Notice the Routhian replaces the Hamiltonian and Lagrangian functions in all the equations of motion, the remaining equation states the partial time derivatives of L and R are negatives ∂ L ∂ t = − ∂ R ∂ t. n, and j =1,2
Routhian mechanics
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Edward John Routh, 1831–1907.
25.
Damping ratio
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In engineering, the damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium, a mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system is trying to return to its equilibrium position, sometimes losses damp the system and can cause the oscillations to gradually decay in amplitude towards zero or attenuate. The damping ratio is a measure of describing how rapidly the oscillations decay from one bounce to the next, where the spring–mass system is completely lossless, the mass would oscillate indefinitely, with each bounce of equal height to the last. This hypothetical case is called undamped, If the system contained high losses, for example if the spring–mass experiment were conducted in a viscous fluid, the mass could slowly return to its rest position without ever overshooting. Commonly, the mass tends to overshoot its starting position, and then return, with each overshoot, some energy in the system is dissipated, and the oscillations die towards zero. Between the overdamped and underdamped cases, there exists a level of damping at which the system will just fail to overshoot. This case is called critical damping, the key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time. The damping ratio is a parameter, usually denoted by ζ and it is particularly important in the study of control theory. It is also important in the harmonic oscillator, the damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. This equation can be solved with the approach, X = C e s t, where C and s are both complex constants. That approach assumes a solution that is oscillatory and/or decaying exponentially, using it in the ODE gives a condition on the frequency of the damped oscillations, s = − ω n. Undamped, Is the case where ζ →0 corresponds to the simple harmonic oscillator. Underdamped, If s is a number, then the solution is a decaying exponential combined with an oscillatory portion that looks like exp . This case occurs for ζ <1, and is referred to as underdamped, overdamped, If s is a real number, then the solution is simply a decaying exponential with no oscillation. This case occurs for ζ >1, and is referred to as overdamped, critically damped, The case where ζ =1 is the border between the overdamped and underdamped cases, and is referred to as critically damped. This turns out to be an outcome in many cases where engineering design of a damped oscillator is required. The factors Q, damping ratio ζ, and exponential decay rate α are related such that ζ =12 Q = α ω0, a lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times
Damping ratio
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The effect of varying damping ratio on a second-order system.
26.
Inertial frame of reference
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In classical physics and special relativity, an inertial frame of reference is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner. The physics of a system in an inertial frame have no causes external to the system, all inertial frames are in a state of constant, rectilinear motion with respect to one another, an accelerometer moving with any of them would detect zero acceleration. Measurements in one frame can be converted to measurements in another by a simple transformation. In general relativity, in any region small enough for the curvature of spacetime and tidal forces to be negligible, systems in non-inertial frames in general relativity dont have external causes because of the principle of geodesic motion. Physical laws take the form in all inertial frames. For example, a ball dropped towards the ground does not go straight down because the Earth is rotating. Someone rotating with the Earth must account for the Coriolis effect—in this case thought of as a force—to predict the horizontal motion, another example of such a fictitious force associated with rotating reference frames is the centrifugal effect, or centrifugal force. The motion of a body can only be described relative to something else—other bodies, observers and these are called frames of reference. If the coordinates are chosen badly, the laws of motion may be more complex than necessary, for example, suppose a free body that has no external forces on it is at rest at some instant. In many coordinate systems, it would begin to move at the next instant, however, a frame of reference can always be chosen in which it remains stationary. Similarly, if space is not described uniformly or time independently, indeed, an intuitive summary of inertial frames can be given as, In an inertial reference frame, the laws of mechanics take their simplest form. In an inertial frame, Newtons first law, the law of inertia, is satisfied, Any free motion has a constant magnitude, the force F is the vector sum of all real forces on the particle, such as electromagnetic, gravitational, nuclear and so forth. The extra terms in the force F′ are the forces for this frame. The first extra term is the Coriolis force, the second the centrifugal force, also, fictitious forces do not drop off with distance. For example, the force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis. All observers agree on the forces, F, only non-inertial observers need fictitious forces. The laws of physics in the frame are simpler because unnecessary forces are not present. In Newtons time the stars were invoked as a reference frame
Inertial frame of reference
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Figure 1: Two frames of reference moving with relative velocity. Frame S' has an arbitrary but fixed rotation with respect to frame S. They are both inertial frames provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.
27.
Mechanics of planar particle motion
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This article describes a particle in planar motion when observed from non-inertial reference frames. The most famous examples of motion are related to the motion of two spheres that are gravitationally attracted to one another, and the generalization of this problem to planetary motion. See centrifugal force, two-body problem, orbit and Keplers laws of planetary motion and those problems fall in the general field of analytical dynamics, the determination of orbits from given laws of force. This article is focused more on the issues surrounding planar motion. The Lagrangian approach to fictitious forces is introduced, unlike real forces such as electromagnetic forces, fictitious forces do not originate from physical interactions between objects. The appearance of fictitious forces normally is associated with use of a frame of reference. For solving problems of mechanics in non-inertial reference frames, the advice given in textbooks is to treat the fictitious forces like real forces, elaboration of this point and some citations on the subject follow. Examples are Cartesian coordinates, polar coordinates and curvilinear coordinates, or as seen from a rotating frame. A time-dependent description of observations does not change the frame of reference in which the observations are made, in discussion of a particle moving in a circular orbit, in an inertial frame of reference one can identify the centripetal and tangential forces. It then seems to be no problem to switch hats, change perspective and that switch is unconscious, but real. Suppose we sit on a particle in planar motion. What analysis underlies a switch of hats to introduce fictitious centrifugal, to explore that question, begin in an inertial frame of reference. In Figure 1, the arc length s is the distance the particle has traveled along its path in time t, the path r with components x, y in Cartesian coordinates is described using arc length s as, r =. One way to look at the use of s is to think of the path of the particle as sitting in space, like the left by a skywriter. Any position on this path is described by stating its distance s from some starting point on the path, then an incremental displacement along the path ds is described by, d r = = d s, where primes are introduced to denote derivatives with respect to s. The magnitude of this displacement is ds, showing that, =1, the unit magnitude of these vectors is a consequence of Eq.1. As an aside, notice that the use of vectors that are not aligned along the Cartesian xy-axes does not mean we are no longer in an inertial frame. All it means is that we are using unit vectors that vary with s to describe the path, the radius of curvature is introduced completely formally as,1 ρ = d θ d s
Mechanics of planar particle motion
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The arc length s(t) measures distance along the skywriter's trail. Image from NASA ASRS
Mechanics of planar particle motion
Mechanics of planar particle motion
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Figure 2: Two coordinate systems differing by a displacement of origin. Radial motion with constant velocity v in one frame is not radial in the other frame. Angular rate, but
28.
Rigid body dynamics
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Rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. This excludes bodies that display fluid highly elastic, and plastic behavior, the dynamics of a rigid body system is described by the laws of kinematics and by the application of Newtons second law or their derivative form Lagrangian mechanics. The formulation and solution of rigid body dynamics is an important tool in the simulation of mechanical systems. If a system of particles moves parallel to a fixed plane, in this case, Newtons laws for a rigid system of N particles, Pi, i=1. N, simplify because there is no movement in the k direction. Determine the resultant force and torque at a reference point R, to obtain F = ∑ i =1 N m i A i, T = ∑ i =1 N ×, where ri denotes the planar trajectory of each particle. In this case, the vectors can be simplified by introducing the unit vectors ei from the reference point R to a point ri. Several methods to describe orientations of a body in three dimensions have been developed. They are summarized in the following sections, the first attempt to represent an orientation is attributed to Leonhard Euler. The values of three rotations are called Euler angles. These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles, in aerospace engineering they are usually referred to as Euler angles. Euler also realized that the composition of two rotations is equivalent to a rotation about a different fixed axis. Therefore, the composition of the three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed. Based on this fact he introduced a way to describe any rotation, with a vector on the rotation axis. Therefore, any orientation can be represented by a vector that leads to it from the reference frame. When used to represent an orientation, the vector is commonly called orientation vector, or attitude vector. A similar method, called axis-angle representation, describes a rotation or orientation using a unit vector aligned with the axis. With the introduction of matrices the Euler theorems were rewritten, the rotations were described by orthogonal matrices referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a matrix is commonly called orientation matrix
Rigid body dynamics
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Human body modelled as a system of rigid bodies of geometrical solids. Representative bones were added for better visualization of the walking person.
Rigid body dynamics
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Movement of each of the components of the Boulton & Watt Steam Engine (1784) is modeled by a continuous set of rigid displacements
29.
Centripetal force
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A centripetal force is a force that makes a body follow a curved path. Its direction is orthogonal to the motion of the body. Isaac Newton described it as a force by which bodies are drawn or impelled, or in any way tend, in Newtonian mechanics, gravity provides the centripetal force responsible for astronomical orbits. One common example involving centripetal force is the case in which a body moves with uniform speed along a circular path, the centripetal force is directed at right angles to the motion and also along the radius towards the centre of the circular path. The mathematical description was derived in 1659 by the Dutch physicist Christiaan Huygens, the direction of the force is toward the center of the circle in which the object is moving, or the osculating circle. The speed in the formula is squared, so twice the speed needs four times the force, the inverse relationship with the radius of curvature shows that half the radial distance requires twice the force. Expressed using the orbital period T for one revolution of the circle, the rope example is an example involving a pull force. The centripetal force can also be supplied as a push force, newtons idea of a centripetal force corresponds to what is nowadays referred to as a central force. Another example of centripetal force arises in the helix that is traced out when a particle moves in a uniform magnetic field in the absence of other external forces. In this case, the force is the centripetal force that acts towards the helix axis. Below are three examples of increasing complexity, with derivations of the formulas governing velocity and acceleration, uniform circular motion refers to the case of constant rate of rotation. Here are two approaches to describing this case, assume uniform circular motion, which requires three things. The object moves only on a circle, the radius of the circle r does not change in time. The object moves with constant angular velocity ω around the circle, therefore, θ = ω t where t is time. Now find the velocity v and acceleration a of the motion by taking derivatives of position with respect to time, consequently, a = − ω2 r. negative shows that the acceleration is pointed towards the center of the circle, hence it is called centripetal. While objects naturally follow a path, this centripetal acceleration describes the circular motion path caused by a centripetal force. The image at right shows the relationships for uniform circular motion. In this subsection, dθ/dt is assumed constant, independent of time, consequently, d r d t = lim Δ t →0 r − r Δ t = d ℓ d t
Centripetal force
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A body experiencing uniform circular motion requires a centripetal force, towards the axis as shown, to maintain its circular path.
30.
Reactive centrifugal force
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In classical mechanics, a reactive centrifugal force forms part of an action–reaction pair with a centripetal force. In accordance with Newtons first law of motion, an object moves in a line in the absence of any external forces acting on the object. A curved path may however ensue when a physical acts on it, the two forces will only have the same magnitude in the special cases where circular motion arises and where the axis of rotation is the origin of the rotating frame of reference. It is the force that is the subject of this article. Any force directed away from a center can be called centrifugal, centrifugal simply means directed outward from the center. Similarly, centripetal means directed toward the center, the reactive centrifugal force discussed in this article is not the same thing as the centrifugal pseudoforce, which is usually whats meant by the term centrifugal force. The figure at right shows a ball in circular motion held to its path by a massless string tied to an immovable post. The figure is an example of a real force. In this system a centripetal force upon the ball provided by the string maintains the motion. In this model, the string is assumed massless and the rotational motion frictionless, the string transmits the reactive centrifugal force from the ball to the fixed post, pulling upon the post. Again according to Newtons third law, the post exerts a reaction upon the string, labeled the post reaction, the two forces upon the string are equal and opposite, exerting no net force upon the string, but placing the string under tension. It should be noted, however, that the reason the post appears to be immovable is because it is fixed to the earth. If the rotating ball was tethered to the mast of a boat, for example, even though the reactive centrifugal is rarely used in analyses in the physics literature, the concept is applied within some mechanical engineering concepts. An example of this kind of engineering concept is an analysis of the stresses within a rapidly rotating turbine blade, the blade can be treated as a stack of layers going from the axis out to the edge of the blade. Each layer exerts a force on the immediately adjacent, radially inward layer. At the same time the inner layer exerts a centripetal force on the middle layer, while and the outer layer exerts an elastic centrifugal force. It is the stresses in the blade and their causes that mainly interest mechanical engineers in this situation, another example of a rotating device in which a reactive centrifugal force can be identified used to describe the system behavior is the centrifugal clutch. A centrifugal clutch is used in small engine-powered devices such as saws, go-karts
Reactive centrifugal force
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A two-shoe centrifugal clutch. The motor spins the input shaft that makes the shoes go around, and the outer drum (removed) turns the output power shaft.
31.
Pendulum (mathematics)
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The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allows the equations of motion to be solved analytically for small-angle oscillations. e, the bob does not trace an ellipse but an arc. The motion does not lose energy to friction or air resistance, the differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However adding a restriction to the size of the oscillations amplitude gives a form whose solution can be easily obtained, the error due to the approximation is of order θ3. Given the initial conditions θ = θ0 and dθ/dt =0, the period of the motion, the time for a complete oscillation is which is known as Christiaan Huygenss law for the period. Note that under the small-angle approximation, the period is independent of the amplitude θ0, T0 =2 π ℓ g can be expressed as ℓ = g π2 T024. If SI units are used, and assuming the measurement is taking place on the Earths surface, then g ≈9.81 m/s2, and g/π2 ≈1. Therefore, a reasonable approximation for the length and period are. Note that this integral diverges as θ0 approaches the vertical lim θ0 → π T = ∞, so that a pendulum with just the right energy to go vertical will never actually get there. For comparison of the approximation to the solution, consider the period of a pendulum of length 1 m on Earth at initial angle 10 degrees is 41 m g K ≈2.0102 s. The linear approximation gives 2 π1 m g ≈2.0064 s, the difference between the two values, less than 0. 2%, is much less than that caused by the variation of g with geographical location. From here there are ways to proceed to calculate the elliptic integral. Figure 4 shows the relative errors using the power series, T0 is the linear approximation, and T2 to T10 include respectively the terms up to the 2nd to the 10th powers. The resulting power series is, T =2 π ℓ g, given Eq.3 and the arithmetic–geometric mean solution of the elliptic integral, K = π2 M, where M is the arithmetic-geometric mean of x and y. This yields an alternative and faster-converging formula for the period, T =2 π M ℓ g, the animations below depict the motion of a simple pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity. The small graph above each pendulum is the phase plane diagram, the horizontal axis is displacement. With a large enough initial velocity the pendulum does not oscillate back and forth, a compound pendulum is one where the rod is not massless, and may have extended size, that is, an arbitrarily shaped rigid body swinging by a pivot. In this case the period depends on its moment of inertia I around the pivot point
Pendulum (mathematics)
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Figure 1. Force diagram of a simple gravity pendulum.
Pendulum (mathematics)
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Animation of a pendulum showing the velocity and acceleration vectors.
32.
Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. Newtons Principia formulated the laws of motion and universal gravitation that dominated scientists view of the universe for the next three centuries. Newtons work on light was collected in his influential book Opticks. He also formulated a law of cooling, made the first theoretical calculation of the speed of sound. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He was knighted by Queen Anne in 1705 and he spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint and his father, also named Isaac Newton, had died three months before. Born prematurely, he was a child, his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Newtons mother had three children from her second marriage. From the age of twelve until he was seventeen, Newton was educated at The Kings School, Grantham which taught Latin and Greek. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, Henry Stokes, master at the Kings School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a bully, he became the top-ranked student. In June 1661, he was admitted to Trinity College, Cambridge and he started as a subsizar—paying his way by performing valets duties—until he was awarded a scholarship in 1664, which guaranteed him four more years until he would get his M. A. He set down in his notebook a series of Quaestiones about mechanical philosophy as he found it, in 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his B. A. degree in August 1665, in April 1667, he returned to Cambridge and in October was elected as a fellow of Trinity. Fellows were required to become ordained priests, although this was not enforced in the restoration years, however, by 1675 the issue could not be avoided and by then his unconventional views stood in the way. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II. A and he was elected a Fellow of the Royal Society in 1672. Newtons work has been said to distinctly advance every branch of mathematics then studied and his work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers
Isaac Newton
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Portrait of Isaac Newton in 1689 (age 46) by Godfrey Kneller
Isaac Newton
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Newton in a 1702 portrait by Godfrey Kneller
Isaac Newton
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Isaac Newton (Bolton, Sarah K. Famous Men of Science. NY: Thomas Y. Crowell & Co., 1889)
Isaac Newton
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Replica of Newton's second Reflecting telescope that he presented to the Royal Society in 1672
33.
Jeremiah Horrocks
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Jeremiah Horrocks, sometimes given as Jeremiah Horrox, was an English astronomer. Jeremiah Horrocks was born at Lower Lodge Farm in Toxteth Park and his father James had moved to Toxteth Park to be apprenticed to Thomas Aspinwall, a watchmaker, and subsequently married his masters daughter Mary. Both families were well educated Puritans, the Horrocks sent their sons to the University of Cambridge. For their unorthodox beliefs the Puritans were excluded from public office, in 1632 Horrocks matriculated at Emmanuel College at the University of Cambridge as a sizar. At Cambridge he associated with the mathematician John Wallis and the platonist John Worthington, at that time he was one of only a few at Cambridge to accept Copernicuss revolutionary heliocentric theory, and he studied the works of Johannes Kepler, Tycho Brahe and others. In 1635 for reasons not clear Horrocks left Cambridge without graduating, now committed to the study of astronomy, Horrocks began to collect astronomical books and equipment, by 1638 he owned the best telescope he could find. Liverpool was a town so navigational instruments such as the astrolabe. But there was no market for the very specialised astronomical instruments he needed and he was well placed to do this, his father and uncles were watchmakers with expertise in creating precise instruments. While a youth he read most of the treatises of his day and marked their weaknesses. Tradition has it that after he left home he supported himself by holding a curacy in Much Hoole, near Preston in Lancashire, according to local tradition in Much Hoole, he lived at Carr House, within the Bank Hall Estate, Bretherton. Carr House was a property owned by the Stones family who were prosperous farmers and merchants. Horrocks was the first to demonstrate that the Moon moved in a path around the Earth. He anticipated Isaac Newton in suggesting the influence of the Sun as well as the Earth on the moons orbit, in the Principia Newton acknowledged Horrockss work in relation to his theory of lunar motion. In the final months of his life Horrocks made detailed studies of tides in attempting to explain the nature of causation of tidal movements. Keplers tables had predicted a near-miss of a transit of Venus in 1639 but, having made his own observations of Venus for years, Horrocks predicted a transit would indeed occur. Horrocks made a simple helioscope by focusing the image of the Sun through a telescope onto a plane surface, from his location in Much Hoole he calculated the transit would begin at approximately 3,00 pm on 24 November 1639, Julian calendar. The weather was cloudy but he first observed the tiny black shadow of Venus crossing the Sun at about 3,15 pm, the 1639 transit was also observed by William Crabtree from his home in Broughton near Manchester. His figure of 95 million kilometres was far from the 150 million kilometres known today and it presented Horrocks enthusiastic and romantic nature, including humorous comments and passages of original poetry
Jeremiah Horrocks
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Making the first observation of the transit of Venus in 1639
Jeremiah Horrocks
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A representation of Horrocks' recording of the transit published in 1662 by Johannes Hevelius
Jeremiah Horrocks
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The title page of Jeremiah Horrocks' Opera Posthuma, published by the Royal Society in 1672.
Jeremiah Horrocks
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Jeremiah Horrocks Observatory on Moor Park, Preston
34.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
Leonhard Euler
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Portrait by Jakob Emanuel Handmann (1756)
Leonhard Euler
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1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Leonhard Euler
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Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as " v − e + f = 2".
Leonhard Euler
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Euler's grave at the Alexander Nevsky Monastery
35.
Jean le Rond d'Alembert
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Jean-Baptiste le Rond dAlembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was also co-editor with Denis Diderot of the Encyclopédie, DAlemberts formula for obtaining solutions to the wave equation is named after him. The wave equation is referred to as dAlemberts equation. Born in Paris, dAlembert was the son of the writer Claudine Guérin de Tencin and the chevalier Louis-Camus Destouches. Destouches was abroad at the time of dAlemberts birth, days after birth his mother left him on the steps of the Saint-Jean-le-Rond de Paris church. According to custom, he was named after the saint of the church. DAlembert was placed in an orphanage for foundling children, but his father found him and placed him with the wife of a glazier, Madame Rousseau, Destouches secretly paid for the education of Jean le Rond, but did not want his paternity officially recognized. DAlembert first attended a private school, the chevalier Destouches left dAlembert an annuity of 1200 livres on his death in 1726. Under the influence of the Destouches family, at the age of twelve entered the Jansenist Collège des Quatre-Nations. Here he studied philosophy, law, and the arts, graduating as baccalauréat en arts in 1735, in his later life, DAlembert scorned the Cartesian principles he had been taught by the Jansenists, physical promotion, innate ideas and the vortices. The Jansenists steered DAlembert toward a career, attempting to deter him from pursuits such as poetry. Theology was, however, rather unsubstantial fodder for dAlembert and he entered law school for two years, and was nominated avocat in 1738. He was also interested in medicine and mathematics, Jean was first registered under the name Daremberg, but later changed it to dAlembert. The name dAlembert was proposed by Johann Heinrich Lambert for a moon of Venus. In July 1739 he made his first contribution to the field of mathematics, at the time Lanalyse démontrée was a standard work, which dAlembert himself had used to study the foundations of mathematics. DAlembert was also a Latin scholar of note and worked in the latter part of his life on a superb translation of Tacitus. In 1740, he submitted his second scientific work from the field of fluid mechanics Mémoire sur la réfraction des corps solides, in this work dAlembert theoretically explained refraction. In 1741, after failed attempts, dAlembert was elected into the Académie des Sciences
Jean le Rond d'Alembert
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Jean-Baptiste le Rond d'Alembert, pastel by Maurice Quentin de La Tour
36.
Pierre-Simon Laplace
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Pierre-Simon, marquis de Laplace was an influential French scholar whose work was important to the development of mathematics, statistics, physics and astronomy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste and this work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace, Laplace formulated Laplaces equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is named after him. Laplace is remembered as one of the greatest scientists of all time, sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries. Laplace became a count of the Empire in 1806 and was named a marquis in 1817, Laplace was born in Beaumont-en-Auge, Normandy on 23 March 1749, a village four miles west of Pont lEveque in Normandy. According to W. W. Rouse Ball, His father, Pierre de Laplace and his great-uncle, Maitre Oliver de Laplace, had held the title of Chirurgien Royal. It would seem that from a pupil he became an usher in the school at Beaumont, however, Karl Pearson is scathing about the inaccuracies in Rouse Balls account and states, Indeed Caen was probably in Laplaces day the most intellectually active of all the towns of Normandy. It was here that Laplace was educated and was provisionally a professor and it was here he wrote his first paper published in the Mélanges of the Royal Society of Turin, Tome iv. 1766–1769, at least two years before he went at 22 or 23 to Paris in 1771, thus before he was 20 he was in touch with Lagrange in Turin. He did not go to Paris a raw self-taught country lad with only a peasant background, the École Militaire of Beaumont did not replace the old school until 1776. His parents were from comfortable families and his father was Pierre Laplace, and his mother was Marie-Anne Sochon. The Laplace family was involved in agriculture until at least 1750, Pierre Simon Laplace attended a school in the village run at a Benedictine priory, his father intending that he be ordained in the Roman Catholic Church. At sixteen, to further his fathers intention, he was sent to the University of Caen to read theology, at the university, he was mentored by two enthusiastic teachers of mathematics, Christophe Gadbled and Pierre Le Canu, who awoke his zeal for the subject. Here Laplaces brilliance as a mathematician was recognised and while still at Caen he wrote a memoir Sur le Calcul integral aux differences infiniment petites et aux differences finies. About this time, recognizing that he had no vocation for the priesthood, in this connection reference may perhaps be made to the statement, which has appeared in some notices of him, that he broke altogether with the church and became an atheist. Laplace did not graduate in theology but left for Paris with a letter of introduction from Le Canu to Jean le Rond dAlembert who at time was supreme in scientific circles. According to his great-great-grandson, dAlembert received him rather poorly, and to get rid of him gave him a mathematics book
Pierre-Simon Laplace
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Pierre-Simon Laplace (1749–1827). Posthumous portrait by Jean-Baptiste Paulin Guérin, 1838.
Pierre-Simon Laplace
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Laplace's house at Arcueil.
Pierre-Simon Laplace
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Laplace.
Pierre-Simon Laplace
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Tomb of Pierre-Simon Laplace
37.
William Rowan Hamilton
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Sir William Rowan Hamilton PRIA FRSE was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and his best known contribution to mathematical physics is the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the study of classical field theories such as electromagnetism. In pure mathematics, he is best known as the inventor of quaternions, Hamilton is said to have shown immense talent at a very early age. Astronomer Bishop Dr. John Brinkley remarked of the 18-year-old Hamilton, This young man, I do not say will be, but is, Hamilton also invented icosian calculus, which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once. Hamilton was the fourth of nine born to Sarah Hutton and Archibald Hamilton. Hamiltons father, who was from Dunboyne, worked as a solicitor, by the age of three, Hamilton had been sent to live with his uncle James Hamilton, a graduate of Trinity College who ran a school in Talbots Castle in Trim, Co. His uncle soon discovered that Hamilton had an ability to learn languages. At the age of seven he had made very considerable progress in Hebrew. These included the classical and modern European languages, and Persian, Arabic, Hindustani, Sanskrit, in September 1813 the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, an older than Hamilton. The two were pitted against each other in a mental arithmetic contest with Colburn emerging the clear victor, in reaction to his defeat, Hamilton dedicated less time to studying languages and more time to studying mathematics. Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College, Dublin, which he entered at age 18. He studied both classics and mathematics, and was appointed Professor of Astronomy in 1827, prior to his taking up residence at Dunsink Observatory where he spent the rest of his life. Hamilton made important contributions to optics and to classical mechanics and his first discovery was in an early paper that he communicated in 1823 to Dr. Brinkley, who presented it under the title of Caustics in 1824 to the Royal Irish Academy. It was referred as usual to a committee, while their report acknowledged its novelty and value, they recommended further development and simplification before publication. Between 1825 and 1828 the paper grew to an immense size, but it also became more intelligible, and the features of the new method were now easily to be seen. Until this period Hamilton himself seems not to have fully understood either the nature or importance of optics and he proposed for it when he first predicted its existence in the third supplement to his Systems of Rays, read in 1832
William Rowan Hamilton
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Quaternion Plaque on Broom Bridge
William Rowan Hamilton
–
William Rowan Hamilton (1805–1865)
William Rowan Hamilton
–
Irish commemorative coin celebrating the 200th Anniversary of his birth.
38.
Augustin-Louis Cauchy
–
Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician who made pioneering contributions to analysis. He was one of the first to state and prove theorems of calculus rigorously and he almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had an influence over his contemporaries. His writings range widely in mathematics and mathematical physics, more concepts and theorems have been named for Cauchy than for any other mathematician. Cauchy was a writer, he wrote approximately eight hundred research articles. Cauchy was the son of Louis François Cauchy and Marie-Madeleine Desestre, Cauchy married Aloise de Bure in 1818. She was a relative of the publisher who published most of Cauchys works. By her he had two daughters, Marie Françoise Alicia and Marie Mathilde, Cauchys father was a high official in the Parisian Police of the New Régime. He lost his position because of the French Revolution that broke out one month before Augustin-Louis was born, the Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris, there Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, the famous mathematician Lagrange was also a friend of the Cauchy family. On Lagranges advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, most of the curriculum consisted of classical languages, the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of successes, Augustin-Louis chose an engineering career. In 1805 he placed second out of 293 applicants on this exam, one of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young, nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées. He graduated in engineering, with the highest honors. After finishing school in 1810, Cauchy accepted a job as an engineer in Cherbourg. Cauchys first two manuscripts were accepted, the one was rejected
Augustin-Louis Cauchy
–
Cauchy around 1840. Lithography by Zéphirin Belliard after a painting by Jean Roller.
Augustin-Louis Cauchy
–
The title page of a textbook by Cauchy.
Augustin-Louis Cauchy
–
Leçons sur le calcul différentiel, 1829
39.
Time derivative
–
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is written as t. A variety of notations are used to denote the time derivative, in addition to the normal notation, d x d t A very common short-hand notation used, especially in physics, is the over-dot. X ˙ Higher time derivatives are used, the second derivative with respect to time is written as d 2 x d t 2 with the corresponding shorthand of x ¨. As a generalization, the derivative of a vector, say. Time derivatives are a key concept in physics, for example, for a changing position x, its time derivative x ˙ is its velocity, and its second derivative with respect to time, x ¨, is its acceleration. Even higher derivatives are also used, the third derivative of position with respect to time is known as the jerk. A large number of equations in physics involve first or second time derivatives of quantities. A common occurrence in physics is the derivative of a vector. In dealing with such a derivative, both magnitude and orientation may depend upon time, for example, consider a particle moving in a circular path. With this form for the displacement, the velocity now is found, the time derivative of the displacement vector is the velocity vector. In general, the derivative of a vector is a made up of components each of which is the derivative of the corresponding component of the original vector. Thus, in case, the velocity vector is, v = d r d t = r = r =. Thus the velocity of the particle is nonzero even though the magnitude of the position is constant, the velocity is directed perpendicular to the displacement, as can be established using the dot product, v ⋅ r = ⋅ = − y x + x y =0. Acceleration is then the time-derivative of velocity, a = d v d t = = − r, the acceleration is directed inward, toward the axis of rotation. It points opposite to the vector and perpendicular to the velocity vector. This inward-directed acceleration is called centripetal acceleration, in economics, many theoretical models of the evolution of various economic variables are constructed in continuous time and therefore employ time derivatives. See for example exogenous growth model and ch, one situation involves a stock variable and its time derivative, a flow variable
Time derivative
–
Relation between Cartesian coordinates (x, y) and polar coordinates (r, θ).
40.
Position (vector)
–
Usually denoted x, r, or s, it corresponds to the straight-line distances along each axis from O to P, r = O P →. The term position vector is used mostly in the fields of geometry, mechanics. Frequently this is used in two-dimensional or three-dimensional space, but can be generalized to Euclidean spaces in any number of dimensions. These different coordinates and corresponding basis vectors represent the position vector. More general curvilinear coordinates could be used instead, and are in contexts like continuum mechanics, linear algebra allows for the abstraction of an n-dimensional position vector. The notion of space is intuitive since each xi can be any value, the dimension of the position space is n. The coordinates of the vector r with respect to the vectors ei are xi. The vector of coordinates forms the coordinate vector or n-tuple, each coordinate xi may be parameterized a number of parameters t. One parameter xi would describe a curved 1D path, two parameters xi describes a curved 2D surface, three xi describes a curved 3D volume of space, and so on. The linear span of a basis set B = equals the position space R, position vector fields are used to describe continuous and differentiable space curves, in which case the independent parameter needs not be time, but can be arc length of the curve. In the case of one dimension, the position has only one component and it could be, say, a vector in the x-direction, or the radial r-direction. Equivalent notations include, x ≡ x ≡ x, r ≡ r, s ≡ s ⋯ For a position vector r that is a function of time t and these derivatives have common utility in the study of kinematics, control theory, engineering and other sciences. Velocity v = d r d t where dr is a small displacement. By extension, the higher order derivatives can be computed in a similar fashion, study of these higher order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to represent the displacement function as a sum of an infinite sequence, enabling several analytical techniques in engineering. A displacement vector can be defined as the action of uniformly translating spatial points in a given direction over a given distance, thus the addition of displacement vectors expresses the composition of these displacement actions and scalar multiplication as scaling of the distance. With this in mind we may define a position vector of a point in space as the displacement vector mapping a given origin to that point. Note thus position vectors depend on a choice of origin for the space, affine space Six degrees of freedom Line element Parametric surface Keller, F. J, Gettys, W. E. et al
Position (vector)
–
Space curve in 3D. The position vector r is parameterized by a scalar t. At r = a the red line is the tangent to the curve, and the blue plane is normal to the curve.
41.
Limit (mathematics)
–
In mathematics, a limit is the value that a function or sequence approaches as the input or index approaches some value. Limits are essential to calculus and are used to define continuity, derivatives, the concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit is usually written as lim n → c f = L and is read as the limit of f of n as n approaches c equals L. Here lim indicates limit, and the fact that function f approaches the limit L as n approaches c is represented by the right arrow, suppose f is a real-valued function and c is a real number. Intuitively speaking, the lim x → c f = L means that f can be made to be as close to L as desired by making x sufficiently close to c. The first inequality means that the distance x and c is greater than 0 and that x ≠ c, while the second indicates that x is within distance δ of c. Note that the definition of a limit is true even if f ≠ L. Indeed. Now since x +1 is continuous in x at 1, we can now plug in 1 for x, in addition to limits at finite values, functions can also have limits at infinity. In this case, the limit of f as x approaches infinity is 2, in mathematical notation, lim x → ∞2 x −1 x =2. Consider the following sequence,1.79,1.799,1.7999 and it can be observed that the numbers are approaching 1.8, the limit of the sequence. Formally, suppose a1, a2. is a sequence of real numbers, intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the absolute value | an − L | is the distance between an and L. Not every sequence has a limit, if it does, it is called convergent, one can show that a convergent sequence has only one limit. The limit of a sequence and the limit of a function are closely related, on one hand, the limit as n goes to infinity of a sequence a is simply the limit at infinity of a function defined on the natural numbers n. On the other hand, a limit L of a function f as x goes to infinity, if it exists, is the same as the limit of any sequence a that approaches L. Note that one such sequence would be L + 1/n, in non-standard analysis, the limit of a sequence can be expressed as the standard part of the value a H of the natural extension of the sequence at an infinite hypernatural index n=H. Thus, lim n → ∞ a n = st , here the standard part function st rounds off each finite hyperreal number to the nearest real number. This formalizes the intuition that for very large values of the index. Conversely, the part of a hyperreal a = represented in the ultrapower construction by a Cauchy sequence, is simply the limit of that sequence
Limit (mathematics)
–
Whenever a point x is within δ units of c, f (x) is within ε units of L.
42.
International System of Units
–
The International System of Units is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, the system also establishes a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system was published in 1960 as the result of an initiative began in 1948. It is based on the system of units rather than any variant of the centimetre-gram-second system. The motivation for the development of the SI was the diversity of units that had sprung up within the CGS systems, the International System of Units has been adopted by most developed countries, however, the adoption has not been universal in all English-speaking countries. The metric system was first implemented during the French Revolution with just the metre and kilogram as standards of length, in the 1830s Carl Friedrich Gauss laid the foundations for a coherent system based on length, mass, and time. In the 1860s a group working under the auspices of the British Association for the Advancement of Science formulated the requirement for a coherent system of units with base units and derived units. Meanwhile, in 1875, the Treaty of the Metre passed responsibility for verification of the kilogram, in 1921, the Treaty was extended to include all physical quantities including electrical units originally defined in 1893. The units associated with these quantities were the metre, kilogram, second, ampere, kelvin, in 1971, a seventh base quantity, amount of substance represented by the mole, was added to the definition of SI. On 11 July 1792, the proposed the names metre, are, litre and grave for the units of length, area, capacity. The committee also proposed that multiples and submultiples of these units were to be denoted by decimal-based prefixes such as centi for a hundredth, on 10 December 1799, the law by which the metric system was to be definitively adopted in France was passed. Prior to this, the strength of the magnetic field had only been described in relative terms. The technique used by Gauss was to equate the torque induced on a magnet of known mass by the earth’s magnetic field with the torque induced on an equivalent system under gravity. The resultant calculations enabled him to assign dimensions based on mass, length, a French-inspired initiative for international cooperation in metrology led to the signing in 1875 of the Metre Convention. Initially the convention only covered standards for the metre and the kilogram, one of each was selected at random to become the International prototype metre and International prototype kilogram that replaced the mètre des Archives and kilogramme des Archives respectively. Each member state was entitled to one of each of the prototypes to serve as the national prototype for that country. Initially its prime purpose was a periodic recalibration of national prototype metres. The official language of the Metre Convention is French and the version of all official documents published by or on behalf of the CGPM is the French-language version
International System of Units
–
Stone marking the Austro-Hungarian /Italian border at Pontebba displaying myriametres, a unit of 10 km used in Central Europe in the 19th century (but since deprecated).
International System of Units
–
The seven base units in the International System of Units
International System of Units
–
Carl Friedrich Gauss
International System of Units
–
Thomson
43.
Kilometre per hour
–
The kilometre per hour is a unit of speed, expressing the number of kilometres travelled in one hour. Worldwide, it is the most commonly used unit of speed on road signs, the Dutch on the other hand adopted the kilometre in 1817 but gave it the local name of the mijl. The SI representations, classified as symbols, are km/h, km h−1 and km·h−1, the use of abbreviations dates back to antiquity, but abbreviations for kilometres per hour did not appear in the English language until the late nineteenth century. The kilometre, a unit of length, first appeared in English in 1810, kilometres per hour did not begin to be abbreviated in print until many years later, with several different abbreviations existing near-contemporaneously. For example, news organisations such as Reuters and The Economist require kph, in Australian unofficial usage, km/h is sometimes pronounced and written as klicks or clicks. The use of symbols to replace words dates back to at least the medieval era when Johannes Widman, writing in German in 1486. In the early 1800s Berzelius introduced a symbolic notation for the chemical elements derived from the elements Latin names, typically, Na was used for the element sodium and H2O for water. In 1879, four years after the signing of the Treaty of the Metre, among these were the use of the symbol km for kilometre. In 1948, as part of its work for the SI. The SI explicitly states that unit symbols are not abbreviations and are to be using a very specific set of rules. Hence the name of the unit can be replaced by a kind of algebraic symbol and this symbol is not merely an abbreviation but a symbol which, like chemical symbols, must be used in a precise and prescribed manner. SI, and hence the use of km/h has now been adopted around the world in areas related to health and safety. It is also the system of measure in academia and in education. During the early years of the car, each country developed its own system of road signs. In 1968 the Vienna Convention on Road Signs and Signals was drawn up under the auspices of the United Nations Economic, many countries have since signed the convention and adopted its proposals. The use of SI implicitly required that states use km/h as the shorthand for kilometres per hour on official documents. Examples include, Dutch, kilometer per uur, Portuguese, quilómetro por hora Greek, in 1988 the United States National Highway Traffic Safety Administration promulgated a rule stating that MPH and/or km/h were to be used in speedometer displays. On May 15,2000 this was clarified to read MPH, or MPH, however, the Federal Motor Vehicle Safety Standard number 101 allows any combination of upper- and lowercase letters to represent the units
Kilometre per hour
–
A car speedometer that indicates measured speed in kilometres per hour.
Kilometre per hour
–
Automobile speedometer, measuring speed in miles per hour on the outer track, and kilometres per hour on the inner track. In Canada "km/h" is shown on the outer track and "MPH" on the inner track.
44.
Miles per hour
–
Miles per hour is an imperial and United States customary unit of speed expressing the number of statute miles covered in one hour. Miles per hour is the also used in the Canadian rail system. In some countries mph may be used to express the speed of delivery of a ball in sporting events such as cricket, tennis, road traffic speeds in other countries are indicated in kilometres per hour, while occasionally both systems are used. For example, in Ireland, a considered a speeding case by examining speeds in both kilometres per hour and miles per hour. The judge was quoted as saying the speed seemed very excessive at 180 km/h but did not look as bad at 112 mph, nautical and aeronautical applications, however, favour the knot as a common unit of speed. 1 Mph =0.000277778 Mps Example, Apollo 11 attained speeds of 25,000 Mph, if Apollo 11 were to travel at 25,000 Mph from New York to Los Angeles it would reach Los Angeles in under 6 minutes
Miles per hour
–
Automobile speedometer, indicating speed in miles per hour on the outer scale and kilometres per hour on the inner scale
Miles per hour
–
United States road sign with maximum speed noted in standard Mph
45.
Knot (unit)
–
The knot is a unit of speed equal to one nautical mile per hour, approximately 1.151 mph. The ISO Standard symbol for the knot is kn, the same symbol is preferred by the IEEE, kt is also common. The knot is a unit that is accepted for use with the SI. Etymologically, the term derives from counting the number of knots in the line that unspooled from the reel of a log in a specific time. 1 international knot =1 nautical mile per hour,1.852 kilometres per hour,0.514 metres per second,1.151 miles per hour,20.254 inches per second,1852 m is the length of the internationally agreed nautical mile. The US adopted the definition in 1954, having previously used the US nautical mile. The UK adopted the international nautical mile definition in 1970, having used the UK Admiralty nautical mile. The speeds of vessels relative to the fluids in which they travel are measured in knots, for consistency, the speeds of navigational fluids are also measured in knots. Thus, speed over the ground and rate of progress towards a distant point are given in knots. Until the mid-19th century, vessel speed at sea was measured using a chip log, the chip log was cast over the stern of the moving vessel and the line allowed to pay out. Knots placed at a distance of 8 fathoms -47 feet 3 inches from each other, passed through a sailors fingers, the knot count would be reported and used in the sailing masters dead reckoning and navigation. This method gives a value for the knot of 20.25 in/s, the difference from the modern definition is less than 0. 02%. On a chart of the North Atlantic, the scale varies by a factor of two from Florida to Greenland, a single graphic scale, of the sort on many maps, would therefore be useless on such a chart. Recent British Admiralty charts have a latitude scale down the middle to make this even easier, speed is sometimes incorrectly expressed as knots per hour, which is in fact a measure of acceleration. Prior to 1969, airworthiness standards for aircraft in the United States Federal Aviation Regulations specified that distances were to be in statute miles. In 1969, these standards were amended to specify that distances were to be in nautical miles. At 11000 m, an airspeed of 300 kn may correspond to a true airspeed of 500 kn in standard conditions. Beaufort scale Hull speed, which deals with theoretical estimates of maximum speed of displacement hulls Knot count Knotted cord Metre per second Orders of magnitude Rope Kemp
Knot (unit)
–
Graphic scale from a Mercator projection world map, showing the change with latitude
46.
Special relativity
–
In physics, special relativity is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time. In Albert Einsteins original pedagogical treatment, it is based on two postulates, The laws of physics are invariant in all inertial systems, the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source. It was originally proposed in 1905 by Albert Einstein in the paper On the Electrodynamics of Moving Bodies, as of today, special relativity is the most accurate model of motion at any speed. Even so, the Newtonian mechanics model is useful as an approximation at small velocities relative to the speed of light. Not until Einstein developed general relativity, to incorporate general frames of reference, a translation that has often been used is restricted relativity, special really means special case. It has replaced the notion of an absolute universal time with the notion of a time that is dependent on reference frame. Rather than an invariant time interval between two events, there is an invariant spacetime interval, a defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other, rather space and time are interwoven into a single continuum known as spacetime. Events that occur at the time for one observer can occur at different times for another. The theory is special in that it applies in the special case where the curvature of spacetime due to gravity is negligible. In order to include gravity, Einstein formulated general relativity in 1915, Special relativity, contrary to some outdated descriptions, is capable of handling accelerations as well as accelerated frames of reference. e. At a sufficiently small scale and in conditions of free fall, a locally Lorentz-invariant frame that abides by special relativity can be defined at sufficiently small scales, even in curved spacetime. Galileo Galilei had already postulated that there is no absolute and well-defined state of rest, Einstein extended this principle so that it accounted for the constant speed of light, a phenomenon that had been recently observed in the Michelson–Morley experiment. He also postulated that it holds for all the laws of physics, Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light and the independence of physical laws from the choice of inertial system, the Principle of Invariant Light Speed –. Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. That is, light in vacuum propagates with the c in at least one system of inertial coordinates. Following Einsteins original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations, however, the most common set of postulates remains those employed by Einstein in his original paper
Special relativity
–
Albert Einstein around 1905, the year his " Annus Mirabilis papers " – which included Zur Elektrodynamik bewegter Körper, the paper founding special relativity – were published.
47.
Speedometer
–
A speedometer or a speed meter is a gauge that measures and displays the instantaneous speed of a vehicle. Now universally fitted to vehicles, they started to be available as options in the 1900s. Speedometers for other vehicles have specific names and use other means of sensing speed, for a boat, this is a pit log. For an aircraft, this is an airspeed indicator, charles Babbage is credited with creating an early type of a speedometer, which were usually fitted to locomotives. The electric speedometer was invented by the Croatian Josip Belušić in 1888, originally patented by Otto Schultze on October 7,1902, it uses a rotating flexible cable usually driven by gearing linked to the output of the vehicles transmission. The early Volkswagen Beetle and many motorcycles, however, use a cable driven from a front wheel, when the car or motorcycle is in motion, a speedometer gear assembly turns a speedometer cable, which then turns the speedometer mechanism itself. A small permanent magnet affixed to the speedometer cable interacts with an aluminum cup attached to the shaft of the pointer on the analogue speedometer instrument. As the magnet rotates near the cup, the magnetic field produces eddy currents in the cup. The effect is that the magnet exerts a torque on the cup, dragging it, the pointer shaft is held toward zero by a fine torsion spring. The torque on the cup increases with the speed of rotation of the magnet, thus an increase in the speed of the car will twist the cup and speedometer pointer against the spring. The cup and pointer will turn until the torque of the currents on the cup is balanced by the opposing torque of the spring. At a given speed the pointer will remain motionless and pointing to the number on the speedometers dial. The return spring is calibrated such that a given speed of the cable corresponds to a specific speed indication on the speedometer. The sensor is typically a set of one or more magnets mounted on the shaft or differential crownwheel. As the part in question turns, the magnets or teeth pass beneath the sensor, alternatively, in more recent designs, some manufactures rely on pulses coming from the ABS wheel sensors. Most modern electronic speedometers have the additional ability over the current type to show the vehicle speed when moving in reverse gear. A computer converts the pulses to a speed and displays this speed on an electronically controlled, another early form of electronic speedometer relies upon the interaction between a precision watch mechanism and a mechanical pulsator driven by the cars wheel or transmission. The watch mechanism endeavors to push the speedometer pointer toward zero, the position of the speedometer pointer reflects the relative magnitudes of the outputs of the two mechanisms
Speedometer
–
A speedometer showing mph and km/h along with an odometer and a separate "trip" odometer (both showing distance traveled in miles).
48.
Derivative
–
The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation, the reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration, differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small
Derivative
–
The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.
49.
Turntable
–
The phonograph is a device invented in 1877 for the mechanical recording and reproduction of sound. In its later forms it is called a gramophone. To recreate the sound, the surface is rotated while a playback stylus traces the groove and is therefore vibrated by it. In later electric phonographs, the motions of the stylus are converted into an electrical signal by a transducer. The phonograph was invented in 1877 by Thomas Edison, while other inventors had produced devices that could record sounds, Edisons phonograph was the first to be able to reproduce the recorded sound. His phonograph originally recorded sound onto a sheet wrapped around a rotating cylinder. A stylus responding to sound vibrations produced an up and down or hill-and-dale groove in the foil, in the 1890s, Emile Berliner initiated the transition from phonograph cylinders to flat discs with a spiral groove running from the periphery to near the center. Later improvements through the years included modifications to the turntable and its system, the stylus or needle. The disc phonograph record was the dominant audio recording format throughout most of the 20th century, from the mid-1980s on, phonograph use on a standard record player declined sharply because of the rise of the cassette tape, compact disc and other digital recording formats. Records are still a favorite format for some audiophiles and DJs, vinyl records are still used by some DJs and musicians in their concert performances. Musicians continue to release their recordings on vinyl records, the original recordings of musicians are sometimes re-issued on vinyl. Usage of terminology is not uniform across the English-speaking world, in more modern usage, the playback device is often called a turntable, record player, or record changer. When used in conjunction with a mixer as part of a DJ setup, the term phonograph was derived from the Greek words φωνή and γραφή. The similar related terms gramophone and graphophone have similar root meanings, the roots were already familiar from existing 19th-century words such as photograph, telegraph, and telephone. In British English, gramophone may refer to any sound-reproducing machine using disc records, the term phonograph was usually restricted to machines that used cylinder records. Gramophone generally referred to a wind-up machine, after the introduction of the softer vinyl records, 33 1⁄3-rpm LPs and 45-rpm single or two-song records, and EPs, the common name became record player or turntable. Often the home record player was part of a system that included a radio and, later, from about 1960, such a system began to be described as a hi-fi or a stereo. In American English, phonograph, properly specific to machines made by Edison, was used in a generic sense as early as the 1890s to include cylinder-playing machines made by others
Turntable
–
Edison cylinder phonograph, circa 1899
Turntable
–
Thomas Edison with his second phonograph, photographed by Mathew Brady in Washington, April 1878
Turntable
–
Close up of the mechanism of an Edison Amberola, manufactured circa 1915
Turntable
–
A late 20th-century turntable and record
50.
Tangent lines to circles
–
In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circles interior. Roughly speaking, it is a line through a pair of infinitely close points on the circle, Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines, a tangent line t to a circle C intersects the circle at a single point T. For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all and this property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the structure of the tangent line and circle, even though the line. The radius of a circle is perpendicular to the tangent line through its endpoint on the circles circumference, conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of circle and tangent line has a reflection symmetry about the axis of the radius, no tangent line can be drawn through a point within a circle, since any such line must be a secant line. However, two tangent lines can be drawn to a circle from a point P outside of the circle, the geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining P to the center point O of the circle. Thus the lengths of the segments from P to the two tangent points are equal, by the secant-tangent theorem, the square of this tangent length equals the power of the point P in the circle C. This power equals the product of distances from P to any two points of the circle with a secant line passing through P. The tangent line t and the tangent point T have a relationship to one another. The same reciprocal relation exists between a point P outside the circle and the secant line joining its two points of tangency. If a point P is exterior to a circle with center O, if a chord TM is drawn from the tangency point T of exterior point P and ∠PTM ≤ 90° then ∠PTM = ∠TOM. The intersection points T1 and T2 are the tangent points for passing through P. The line segments OT1 and OT2 are radii of the circle C, since both are inscribed in a semicircle, they are perpendicular to the line segments PT1 and PT2, but only a tangent line is perpendicular to the radial line. Hence, the two lines from P and passing through T1 and T2 are tangent to the circle C. Another method to construct the tangent lines to a point P external to the using only a straightedge. Let A1, A2, B1, B2, C1, C2 be the six points, with the same letter corresponding to the same line
Tangent lines to circles
–
By the power-of-a-point theorem, the product of lengths PM·PN for any ray PMN equals to the square of PT, the length of the tangent line segment (red).
51.
Mach number
–
In fluid dynamics, the Mach number is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound. M = u c, where, M is the Mach number, u is the flow velocity with respect to the boundaries. By definition, Mach 1 is equal to the speed of sound, Mach 0.65 is 65% of the speed of sound, and Mach 1.35 is 35% faster than the speed of sound. The local speed of sound, and thereby the Mach number, depends on the condition of the surrounding medium, the Mach number is primarily used to determine the approximation with which a flow can be treated as an incompressible flow. The medium can be a gas or a liquid, the boundary can be the boundary of an object immersed in the medium, or of a channel such as a nozzle, diffusers or wind tunnels channeling the medium. As the Mach number is defined as the ratio of two speeds, it is a dimensionless number, if M <0. 2–0.3 and the flow is quasi-steady and isothermal, compressibility effects will be small and simplified incompressible flow equations can be used. The Mach number is named after Austrian physicist and philosopher Ernst Mach, as the Mach number is a dimensionless quantity rather than a unit of measure, with Mach, the number comes after the unit, the second Mach number is Mach 2 instead of 2 Mach. This is somewhat reminiscent of the modern ocean sounding unit mark, which was also unit-first. In the decade preceding faster-than-sound human flight, aeronautical engineers referred to the speed of sound as Machs number, never Mach 1, Mach number is useful because the fluid behaves in a similar manner at a given Mach number, regardless of other variables. As modeled in the International Standard Atmosphere, dry air at sea level, standard temperature of 15 °C. For example, the atmosphere model lapses temperature to −56.5 °C at 11,000 meters altitude. In the following table, the regimes or ranges of Mach values are referred to, generally, NASA defines high hypersonic as any Mach number from 10 to 25, and re-entry speeds as anything greater than Mach 25. Aircraft operating in this include the Space Shuttle and various space planes in development. Flight can be classified in six categories, For comparison. At transonic speeds, the field around the object includes both sub- and supersonic parts. The transonic period begins when first zones of M >1 flow appear around the object, in case of an airfoil, this typically happens above the wing. Supersonic flow can decelerate back to only in a normal shock. As the speed increases, the zone of M >1 flow increases towards both leading and trailing edges
Mach number
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An F/A-18 Hornet creating a vapor cone at transonic speed just before reaching the speed of sound
52.
Speed of sound
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The speed of sound is the distance travelled per unit time by a sound wave as it propagates through an elastic medium. In dry air at 20 °C, the speed of sound is 343 metres per second, the speed of sound in an ideal gas depends only on its temperature and composition. The speed has a dependence on frequency and pressure in ordinary air. In common everyday speech, speed of sound refers to the speed of waves in air. However, the speed of sound varies from substance to substance, sound travels most slowly in gases, it travels faster in liquids, and faster still in solids. For example, sound travels at 343 m/s in air, it travels at 1,484 m/s in water, in an exceptionally stiff material such as diamond, sound travels at 12,000 m/s, which is around the maximum speed that sound will travel under normal conditions. Sound waves in solids are composed of waves, and a different type of sound wave called a shear wave. Shear waves in solids usually travel at different speeds, as exhibited in seismology, the speed of compression waves in solids is determined by the mediums compressibility, shear modulus and density. The speed of waves is determined only by the solid materials shear modulus. In fluid dynamics, the speed of sound in a medium is used as a relative measure for the speed of an object moving through the medium. The ratio of the speed of an object to the speed of sound in the fluid is called the objects Mach number, objects moving at speeds greater than Mach1 are said to be traveling at supersonic speeds. During the 17th century, there were attempts to measure the speed of sound accurately, including attempts by Marin Mersenne in 1630, Pierre Gassendi in 1635. In 1709, the Reverend William Derham, Rector of Upminster, published an accurate measure of the speed of sound. Measurements were made of gunshots from a number of local landmarks, the distance was known by triangulation, and thus the speed that the sound had travelled was calculated. The transmission of sound can be illustrated by using a model consisting of an array of balls interconnected by springs, for real material the balls represent molecules and the springs represent the bonds between them. Sound passes through the model by compressing and expanding the springs, transmitting energy to neighbouring balls, which transmit energy to their springs, the speed of sound through the model depends on the stiffness of the springs, and the mass of the balls. As long as the spacing of the balls remains constant, stiffer springs transmit energy more quickly, effects like dispersion and reflection can also be understood using this model. In a real material, the stiffness of the springs is called the modulus
Speed of sound
–
U.S. Navy F/A-18 traveling near the speed of sound. The white halo consists of condensed water droplets formed by the sudden drop in air pressure behind the shock cone around the aircraft (see Prandtl-Glauert singularity).
Speed of sound
–
Pressure-pulse or compression-type wave (longitudinal wave) confined to a plane. This is the only type of sound wave that travels in fluids (gases and liquids)
53.
Walk
–
Walking is one of the main gaits of locomotion among legged animals, and is typically slower than running and other gaits. Walking is defined by an inverted pendulum gait in which the vaults over the stiff limb or limbs with each step. This applies regardless of the number of limbs - even arthropods, with six, eight or more limbs, the word walk is descended from the Old English wealcan to roll. In humans and other bipeds, walking is generally distinguished from running in only one foot at a time leaves contact with the ground. In contrast, running begins when both feet are off the ground with each step and this distinction has the status of a formal requirement in competitive walking events. The most effective method to walking from running is to measure the height of a persons centre of mass using motion capture or a force plate at midstance. During walking, the centre of mass reaches a height at midstance, while during running. This distinction, however, only true for locomotion over level or approximately level ground. For walking up grades above 9%, this no longer holds for some individuals. Running humans and animals may have contact periods greater than 50% of a cycle when rounding corners. Speed is another factor that distinguishes walking from running, champion racewalkers can average more than 14 kilometres per hour over a distance of 20 kilometres. An average human child achieves independent walking ability at around 11 months old, regular, brisk exercise of any kind can improve confidence, stamina, energy, weight control and life expectancy and reduce stress. It can also reduce the risk of heart disease, strokes, diabetes, high blood pressure, bowel cancer. Life expectancy is also increased even for individuals suffering from obesity or high blood pressure, Walking also improves bone health, especially strengthening the hip bone, and lowering the harmful low-density lipoprotein cholesterol, and raising the useful high-density lipoprotein cholesterol. Studies have found that walking may also help prevent dementia and Alzheimers, the Centers for Disease Control and Preventions fact sheet on the Relationship of Walking to Mortality Among U. S. Adults with Diabetes states that those with diabetes who walked for 2 or more hours a week lowered their mortality rate from all causes by 39 per cent. Walking lengthened the life of people with diabetes regardless of age, sex, race, body mass index, length of time since diagnosis, and presence of complications or functional limitations. It has been suggested there is a relationship between the speed of walking and health, and that the best results are obtained with a speed of more than 2.5 mph
Walk
–
Computer simulation of a human walk cycle. In this model the head keeps the same level at all times, whereas the hip follows a sine curve.
Walk
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Racewalkers at the World Cup Trials in 1987
Walk
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Nordic walkers
Walk
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Walking in Shilda, Georgia.
54.
Sprint runner
–
Sprinting is running over a short distance in a limited period of time. It is used in sports that incorporate running, typically as a way of quickly reaching a target or goal. In athletics and track and field, sprints are races over short distances and they are among the oldest running competitions. The first 13 editions of the Ancient Olympic Games featured only one event—the stadion race, there are three sprinting events which are currently held at the Summer Olympics and outdoor World Championships, the 100 metres,200 metres, and 400 metres. The set position differs depending on the start, body alignment is of key importance in producing the optimal amount of force. Ideally the athlete should begin in a 4-point stance and push off using both legs for maximum force production, athletes remain in the same lane on the running track throughout all sprinting events, with the sole exception of the 400 m indoors. Races up to 100 m are largely focused upon acceleration to a maximum speed. All sprints beyond this distance increasingly incorporate an element of endurance, the 60 metres is a khaled and Rion indoor event and it is an indoor world championship event. Less common events include the 50 metres,55 metres,300 metres, biological factors that determine a sprinters potential include, The 60 metres is normally run indoors, on a straight section of an indoor athletic track. Since races at this distance can last around six or seven seconds, having good reflexes and this is roughly the distance required for a human to reach maximum speed and can be run with one breath. It is popular for training and testing in other sports, the world record in this event is held by American sprinter Maurice Greene with a time of 6.39 seconds. 60-metres is used as a distance by younger athletes when starting sprint racing. Note, Indoor distances are less standardized as many facilities run shorter or occasionally longer distances depending on available space, the 100 metres sprint takes place on one length of the home straight of a standard outdoor 400 m track. Often, the holder in this race is considered the worlds fastest man/woman. The current world record of 9.58 seconds is held by Usain Bolt of Jamaica and was set on 16 August 2009, the womens world record is 10.49 seconds and was set by Florence Griffith-Joyner. World class male sprinters need 41 to 50 strides to cover the whole 100 metres distances, the 200 metres begins on the curve of a standard track, and ends on the home straight. The ability to run a good bend is key at the distance, as a well conditioned runner will typically be able to run 200 m in an average speed higher than their 100 m speed. Usain Bolt, however, ran 200 m in the time of 19.19 sec, an average speed of 10.422 m/s, whereas he ran 100 m in the world-record time of 9.58 sec
Sprint runner
–
Usain Bolt, world record holder in 100 m and 200 m sprints
Sprint runner
–
Start of the women's 60 m at the 2010 World Indoor Championships
Sprint runner
–
Tyson Gay completes a 100m race
Sprint runner
–
A 200 m bend
55.
100 metres
–
The 100 metres, or 100-metre dash, is a sprint race in track and field competitions. The shortest common outdoor running distance, it is one of the most popular and it has been contested at the Summer Olympics since 1896 for men and since 1928 for women. The reigning 100 m Olympic champion is named the fastest runner in the world. The World Championships 100 metres has been contested since 1983, jamaicans Usain Bolt and Shelly-Ann Fraser-Pryce are the reigning world champions, Bolt and Elaine Thompson are the Olympic champions in the mens and womens 100 metres, respectively. On an outdoor 400 metres running track, the 100 m is run on the home straight, runners begin in the starting blocks and the race begins when an official fires the starters pistol. Sprinters typically reach top speed after somewhere between 50–60 m and their speed then slows towards the finish line. The 10-second barrier has historically been a barometer of fast mens performances, the current mens world record is 9.58 seconds, set by Jamaicas Usain Bolt in 2009, while the womens world record of 10.49 seconds set by American Florence Griffith-Joyner in 1988 remains unbroken. The 100 m emerged from the metrication of the 100 yards, the event is largely held outdoors as few indoor facilities have a 100 m straight. US athletes have won the mens Olympic 100 metres title more times than any country,16 out of the 28 times that it has been run. US women have dominated the event winning 9 out of 21 times. At the start, some athletes play psychological games such as trying to be last to the starting blocks, at high level meets, the time between the gun and first kick against the starting block is measured electronically, via sensors built in the gun and the blocks. A reaction time less than 0.1 s is considered a false start, the 0. 2-second interval accounts for the sum of the time it takes for the sound of the starters pistol to reach the runners ears, and the time they take to react to it. For many years a sprinter was disqualified if responsible for two false starts individually, however, this rule allowed some major races to be restarted so many times that the sprinters started to lose focus. The next iteration of the rule, introduced in February 2003, meant that one false start was allowed among the field, but anyone responsible for a subsequent false start was disqualified. To avoid such abuse and to improve spectator enjoyment, the IAAF implemented a change in the 2010 season – a false starting athlete now receives immediate disqualification. This proposal was met with objections when first raised in 2005, justin Gatlin commented, Just a flinch or a leg cramp could cost you a years worth of work. The rule had an impact at the 2011 World Championships. Runners normally reach their top speed just past the point of the race
100 metres
–
Start of the 100 metres final at the 2012 Olympic Games.
100 metres
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Male sprinters await the starter's instructions
100 metres
–
Usain Bolt breaking the world and Olympic records at the 2008 Beijing Olympics
100 metres
–
Christine Arron (left) wins the 100 m at the Weltklasse meeting.
56.
Taipei 101
–
Taipei 101 – stylized as TAIPEI101 and formerly known as the Taipei World Financial Center – is a landmark supertall skyscraper in Xinyi District, Taipei, Taiwan. The building was classified as the worlds tallest in 2004. It used to have the fastest elevator in the world, traveling at 60.6 km/h, in 2016, the title for the fastest elevator was given to one in Shanghai Tower. Construction on the 101-story tower started in 1999 and finished in 2004, the tower has served as an icon of modern Taiwan ever since its opening. The building was created as a symbol of the evolution of technology. Its postmodernist approach to style incorporates traditional design elements and gives them modern treatments, the tower is designed to withstand typhoons and earthquakes. A multi-level shopping mall adjoining the tower houses hundreds of stores, restaurants, fireworks launched from Taipei 101 feature prominently in international New Years Eve broadcasts and the structure appears frequently in travel literature and international media. Taipei 101 is primarily owned by pan-government shareholders, the name that was originally planned for the building, Taipei World Financial Center, until 2003, was derived from the name of the owner. The original name in Chinese was Taipei International Financial Center, Taipei 101 comprises 101 floors above ground, as well as 5 basement levels. It was not only the first building in the world to break the mark in height. As of 28 July 2011, it is still the worlds largest and highest-use green building. It also surpassed the 85-story,347.5 m Tuntex Sky Tower in Kaohsiung as the tallest building in Taiwan, Taipei 101 claimed the official records for the worlds tallest sundial and the worlds largest New Years Eve countdown clock. Various sources, including the owners, give the height of Taipei 101 as 508 m, roof height. This lower figure is derived by measuring from the top of a 1.2 m platform at the base. CTBUH standards, though, include the height of the platform in calculating the overall height, Taipei 101 displaced the Petronas Towers as the tallest building in the world by 57.3 m. The record it claimed for greatest height from ground to pinnacle was surpassed by the Burj Khalifa in Dubai, which is 829.8 m in height. Taipei 101s records for roof height and highest occupied floor briefly passed to the Shanghai World Financial Center in 2008, Taipei 101 is designed to withstand the typhoon winds and earthquake tremors that are common in the area east of Taiwan. Evergreen Consulting Engineering, the engineer, designed Taipei 101 to withstand gale winds of 60 metres per second, as well as the strongest earthquakes in a 2
Taipei 101
–
Taipei 101 Tower in August 2008
Taipei 101
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Base of the tower
Taipei 101
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Location of Taipei 101's largest tuned mass damper
Taipei 101
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Tuned mass damper
57.
Autoroutes of France
–
The Autoroute system in France consists largely of toll roads, except around large cities and in parts of the north. It is a network of 11,882 km worth of motorways in 2014, Autoroute destinations are shown in blue, while destinations reached through a combination of autoroutes are shown with an added autoroute logo. Toll autoroutes are signalled with the word péage, unlike other motorway systems, there is no systematic numbering system, but there is a clustering of Autoroute numbers based on region. A1, A3, A4, A5, A6, A10, A13, A14, A15, A16 radiate from Paris with A2, A11, and A12 branching from A1, A10, A7 begins in Lyon, where A6 ends. A8 and A9 begin respectively near Aix-en-Provence and Avignon, the 20s are found in northern France. The 30s are found in eastern France, the 40s are found near the Alps. The 50s are near the French Riviera, the 60s are found in southern France. The 70s are found in the centre of the country, the 80s are found west of Paris. Some of the autoroutes have their own name in addition to a number, A4 is the autoroute de lEst. A6 and A7 are autoroutes du Soleil, for lead from northern to southern France. A8 is named La provençale as it cross Provence, A9 is named La Languedocienne as it crosses the Languedoc A10 is named LAquitaine because it leads to Bordeaux, which is situated in the part of France named Aquitaine. The A13 is named the autoroute de Normandie as it traverses Normandy, a20 is named Loccitane as it leads to the south-west of France, this part of France was historically called Occitanie. The A26 is the autoroute des Anglais as it leads from Calais and it continues to Troyes, and just happens to pass straight through the Champagne region, whose wines are so loved by the British. It also passes sites of earlier UK interest such as Crecy, the A29 is part of the route des Estuaires, a chain of motorways crossing the estuaries of the English Channel. The A40 is named the autoroute blanche because it is the road goes to Chamonix. The A62 and A61 are named autoroute des deux mers because these roads connect the Atlantic Ocean, a68 is called autoroute du Pastel because it leads to Albi and to the Lauragais where woad was cultivated to produce pastel. The N104, one of Pariss beltways, is known as La Francilienne because it circles the region of Ile-de-France. The status of motorways in France has been the subject of debate through years, originally, the autoroutes were built by private companies mandated by the French government, and followed strict construction rules as described below
Autoroutes of France
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Toll barrier in Toulouse-Sud (south of Toulouse), on autoroute A61
Autoroutes of France
Autoroutes of France
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The French Autoroute A1
Autoroutes of France
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A French motorway.
58.
Recumbent bicycle
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A recumbent bicycle is a bicycle that places the rider in a laid-back reclining position. Most recumbent riders choose this type of design for ergonomic reasons, on a traditional upright bicycle, the body weight rests entirely on a small portion of the sitting bones, the feet, and the hands. Most recumbent models also have an advantage, the reclined. A variant with three wheels is a recumbent tricycle, Recumbents can be categorized by their wheelbase, wheel sizes, steering system, faired or unfaired, and front-wheel or rear-wheel drive. Within these categories are variations, intermediate types, and even convertible designs – there is no standard recumbent, the rear wheel of a recumbent is usually behind the rider and may be any size, from around 16 inches to the 700c of an upright racing cycle. The front wheel is smaller than the rear, although a number of recumbents feature dual 26-inch, ISO571, ISO622. Larger diameter wheels generally have lower rolling resistance but a higher profile leading to air resistance. Another advantage of both wheels being the size is that the bike requires only one size of inner tube. One common arrangement is an ISO559 rear wheel and an ISO406 or ISO451 front wheel. The small front wheel and large rear wheel combination is used to keep the pedals and front wheel clear of each other, a pivoting-boom front-wheel drive configuration also overcomes heel strike since the pedals and front wheel turn together. PBFWD bikes may have dual 26-inch wheels or larger, steering for recumbent bikes can be generally categorized as over-seat or above seat steering, under-seat, or center steering or pivot steering. Chopper-style bars are seen on LWB bikes. USS is usually indirect — the bars link to the headset through a system of rods or cables, center steered or pivot steered recumbents, such as Flevobikes and Pythons, may have no handlebars at all. In addition, some such as the Sidewinder have used rear-wheel steer. They can provide good maneuverability at low speeds, but have been reported to be unstable at speeds above 25 mph. Most recumbents have the attached to a boom fixed to the frame. However, due to the proximity of the crank to the front wheel, front wheel drive can be an option, one style requires the chain to twist slightly to allow for steering. Another style, Pivoting-boom FWD, has the crankset connected to, in addition to the much shorter chain, the advantages to PBFWD are use of a larger front wheel for lower rolling resistance without heel strike and use of the upper body when sprinting or climbing
Recumbent bicycle
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Bacchetta Corsa, a short-wheelbase high racer
Recumbent bicycle
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A RANS V2 Formula long-wheelbase recumbent bike fitted with a front fairing
Recumbent bicycle
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Woman riding a Cruzbike Sofrider (PBFWD recumbent) near the end of the 500-mile (800 km) "Ride Across North Carolina" 2007
Recumbent bicycle
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Long-wheel-base low-rider recumbent with steering u-joint (UA)
59.
Kelvins
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The kelvin is a unit of measure for temperature based upon an absolute scale. It is one of the seven units in the International System of Units and is assigned the unit symbol K. The kelvin is defined as the fraction 1⁄273.16 of the temperature of the triple point of water. In other words, it is defined such that the point of water is exactly 273.16 K. The Kelvin scale is named after the Belfast-born, Glasgow University engineer and physicist William Lord Kelvin, unlike the degree Fahrenheit and degree Celsius, the kelvin is not referred to or typeset as a degree. The kelvin is the unit of temperature measurement in the physical sciences, but is often used in conjunction with the Celsius degree. The definition implies that absolute zero is equivalent to −273.15 °C, Kelvin calculated that absolute zero was equivalent to −273 °C on the air thermometers of the time. This absolute scale is known today as the Kelvin thermodynamic temperature scale, when spelled out or spoken, the unit is pluralised using the same grammatical rules as for other SI units such as the volt or ohm. When reference is made to the Kelvin scale, the word kelvin—which is normally a noun—functions adjectivally to modify the noun scale and is capitalized, as with most other SI unit symbols there is a space between the numeric value and the kelvin symbol. Before the 13th CGPM in 1967–1968, the unit kelvin was called a degree and it was distinguished from the other scales with either the adjective suffix Kelvin or with absolute and its symbol was °K. The latter term, which was the official name from 1948 until 1954, was ambiguous since it could also be interpreted as referring to the Rankine scale. Before the 13th CGPM, the form was degrees absolute. The 13th CGPM changed the name to simply kelvin. Its measured value was 0.01028 °C with an uncertainty of 60 µK, the use of SI prefixed forms of the degree Celsius to express a temperature interval has not been widely adopted. In 2005 the CIPM embarked on a program to redefine the kelvin using a more experimentally rigorous methodology, the current definition as of 2016 is unsatisfactory for temperatures below 20 K and above 1300 K. In particular, the committee proposed redefining the kelvin such that Boltzmanns constant takes the exact value 1. 3806505×10−23 J/K, from a scientific point of view, this will link temperature to the rest of SI and result in a stable definition that is independent of any particular substance. From a practical point of view, the redefinition will pass unnoticed, the kelvin is often used in the measure of the colour temperature of light sources. Colour temperature is based upon the principle that a black body radiator emits light whose colour depends on the temperature of the radiator, black bodies with temperatures below about 4000 K appear reddish, whereas those above about 7500 K appear bluish
Kelvins
–
Lord Kelvin, the namesake of the unit
Kelvins
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A thermometer calibrated in degrees Celsius (left) and kelvins (right).
60.
Space shuttle
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The Space Shuttle was a partially reusable low Earth orbital spacecraft system operated by the U. S. National Aeronautics and Space Administration, as part of the Space Shuttle program. Its official program name was Space Transportation System, taken from a 1969 plan for a system of reusable spacecraft of which it was the only item funded for development, the first of four orbital test flights occurred in 1981, leading to operational flights beginning in 1982. Five complete Shuttle systems were built and used on a total of 135 missions from 1981 to 2011, the Shuttle fleets total mission time was 1322 days,19 hours,21 minutes and 23 seconds. Shuttle components included the Orbiter Vehicle, a pair of solid rocket boosters. The Shuttle was launched vertically, like a rocket, with the two SRBs operating in parallel with the OVs three main engines, which were fueled from the ET. The SRBs were jettisoned before the vehicle reached orbit, and the ET was jettisoned just before orbit insertion, at the conclusion of the mission, the orbiter fired its OMS to de-orbit and re-enter the atmosphere. The orbiter then glided as a spaceplane to a landing, usually at the Shuttle Landing Facility of KSC or Rogers Dry Lake in Edwards Air Force Base. After landing at Edwards, the orbiter was back to the KSC on the Shuttle Carrier Aircraft. The first orbiter, Enterprise, was built in 1976, used in Approach, four fully operational orbiters were initially built, Columbia, Challenger, Discovery, and Atlantis. Of these, two were lost in accidents, Challenger in 1986 and Columbia in 2003, with a total of fourteen astronauts killed. A fifth operational orbiter, Endeavour, was built in 1991 to replace Challenger, the Space Shuttle was retired from service upon the conclusion of Atlantiss final flight on July 21,2011. Nixons post-Apollo NASA budgeting withdrew support of all components except the Shuttle. The vehicle consisted of a spaceplane for orbit and re-entry, fueled by liquid hydrogen and liquid oxygen tanks. The first of four orbital test flights occurred in 1981, leading to operational flights beginning in 1982, all launched from the Kennedy Space Center, Florida. The system was retired from service in 2011 after 135 missions, the program ended after Atlantis landed at the Kennedy Space Center on July 21,2011. Major missions included launching numerous satellites and interplanetary probes, conducting space science experiments, the first orbiter vehicle, named Enterprise, was built for the initial Approach and Landing Tests phase and lacked engines, heat shielding, and other equipment necessary for orbital flight. A total of five operational orbiters were built, and of these and it was used for orbital space missions by NASA, the US Department of Defense, the European Space Agency, Japan, and Germany. The United States funded Shuttle development and operations except for the Spacelab modules used on D1, sL-J was partially funded by Japan
Space shuttle
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Discovery lifts off at the start of STS-120.
Space shuttle
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STS-129 ready for launch
Space shuttle
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President Nixon (right) with NASA Administrator Fletcher in January 1972, three months before Congress approved funding for the Shuttle program
Space shuttle
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STS-1 on the launch pad, December 1980
61.
Escape velocity
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The escape velocity from Earth is about 11.186 km/s at the surface. More generally, escape velocity is the speed at which the sum of a kinetic energy. With escape velocity in a direction pointing away from the ground of a massive body, once escape velocity is achieved, no further impulse need be applied for it to continue in its escape. When given a speed V greater than the speed v e. In these equations atmospheric friction is not taken into account, escape velocity is only required to send a ballistic object on a trajectory that will allow the object to escape the gravity well of the mass M. The existence of escape velocity is a consequence of conservation of energy, by adding speed to the object it expands the possible places that can be reached until with enough energy they become infinite. For a given gravitational potential energy at a position, the escape velocity is the minimum speed an object without propulsion needs to be able to escape from the gravity. Escape velocity is actually a speed because it does not specify a direction, no matter what the direction of travel is, the simplest way of deriving the formula for escape velocity is to use conservation of energy. Imagine that a spaceship of mass m is at a distance r from the center of mass of the planet and its initial speed is equal to its escape velocity, v e. At its final state, it will be a distance away from the planet. The same result is obtained by a calculation, in which case the variable r represents the radial coordinate or reduced circumference of the Schwarzschild metric. All speeds and velocities measured with respect to the field, additionally, the escape velocity at a point in space is equal to the speed that an object would have if it started at rest from an infinite distance and was pulled by gravity to that point. In common usage, the point is on the surface of a planet or moon. On the surface of the Earth, the velocity is about 11.2 km/s. However, at 9,000 km altitude in space, it is less than 7.1 km/s. The escape velocity is independent of the mass of the escaping object and it does not matter if the mass is 1 kg or 1,000 kg, what differs is the amount of energy required. For an object of mass m the energy required to escape the Earths gravitational field is GMm / r, a related quantity is the specific orbital energy which is essentially the sum of the kinetic and potential energy divided by the mass. An object has reached escape velocity when the orbital energy is greater or equal to zero
Escape velocity
–
Luna 1, launched in 1959, was the first man-made object to attain escape velocity from Earth (see below table).
Escape velocity
–
General
62.
Voyager 1
–
Voyager 1 is a space probe launched by NASA on September 5,1977. Part of the Voyager program to study the outer Solar System, Voyager 1 launched 16 days after its twin, having operated for 39 years,6 months and 30 days, the spacecraft still communicates with the Deep Space Network to receive routine commands and return data. At a distance of 138 AU from the Sun as of March 2017, the probes primary mission objectives included flybys of Jupiter, Saturn, and Saturns large moon, Titan. It studied the weather, magnetic fields, and rings of the two planets and was the first probe to provide detailed images of their moons. After completing its mission with the flyby of Saturn on November 20,1980, Voyager 1 began an extended mission to explore the regions. On August 25,2012, Voyager 1 crossed the heliopause to become the first spacecraft to enter interstellar space, in the 1960s, a Grand Tour to study the outer planets was proposed which prompted NASA to begin work on a mission in the early 1970s. Information gathered by the Pioneer 10 spacecraft helped Voyagers engineers design Voyager to cope effectively with the intense radiation environment around Jupiter. Initially, Voyager 1 was planned as Mariner 11 of the Mariner program, due to budget cuts, the mission was scaled back to be a flyby of Jupiter and Saturn and renamed the Mariner Jupiter-Saturn probes. As the program progressed, the name was changed to Voyager. Voyager 1 was constructed by the Jet Propulsion Laboratory and it has 16 hydrazine thrusters, three-axis stabilization gyroscopes, and referencing instruments to keep the probes radio antenna pointed toward Earth. Collectively, these instruments are part of the Attitude and Articulation Control Subsystem, the spacecraft also included 11 scientific instruments to study celestial objects such as planets as it travels through space. The radio communication system of Voyager 1 was designed to be used up to, the communication system includes a 3. 7-meter diameter parabolic dish high-gain antenna to send and receive radio waves via the three Deep Space Network stations on the Earth. The craft normally transmits data to Earth over Deep Space Network Channel 18, using a frequency of either 2.3 GHz or 8.4 GHz, while signals from Earth to Voyager are broadcast at 2.1 GHz. When Voyager 1 is unable to communicate directly with the Earth, signals from Voyager 1 take over 19 hours to reach Earth. Voyager 1 has three radioisotope thermoelectric generators mounted on a boom, each MHW-RTG contains 24 pressed plutonium-238 oxide spheres. The RTGs generated about 470 W of electric power at the time of launch, the power output of the RTGs declines over time, but the crafts RTGs will continue to support some of its operations until 2025. As of 2017-04-04, Voyager 1 has 73. 14% of the plutonium-238 that it had at launch, by 2050, it will have 56. 5% left. Since the 1990s, space probes usually have completely autonomous cameras, the computer command subsystem controls the cameras
Voyager 1
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Voyager 1, artist's impression
Voyager 1
Voyager 1
Voyager 1
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Voyager 1 lifted off with a Titan IIIE
63.
Earth
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Earth, otherwise known as the World, or the Globe, is the third planet from the Sun and the only object in the Universe known to harbor life. It is the densest planet in the Solar System and the largest of the four terrestrial planets, according to radiometric dating and other sources of evidence, Earth formed about 4.54 billion years ago. Earths gravity interacts with objects in space, especially the Sun. During one orbit around the Sun, Earth rotates about its axis over 365 times, thus, Earths axis of rotation is tilted, producing seasonal variations on the planets surface. The gravitational interaction between the Earth and Moon causes ocean tides, stabilizes the Earths orientation on its axis, Earths lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earths surface is covered with water, mostly by its oceans, the remaining 29% is land consisting of continents and islands that together have many lakes, rivers and other sources of water that contribute to the hydrosphere. The majority of Earths polar regions are covered in ice, including the Antarctic ice sheet, Earths interior remains active with a solid iron inner core, a liquid outer core that generates the Earths magnetic field, and a convecting mantle that drives plate tectonics. Within the first billion years of Earths history, life appeared in the oceans and began to affect the Earths atmosphere and surface, some geological evidence indicates that life may have arisen as much as 4.1 billion years ago. Since then, the combination of Earths distance from the Sun, physical properties, in the history of the Earth, biodiversity has gone through long periods of expansion, occasionally punctuated by mass extinction events. Over 99% of all species that lived on Earth are extinct. Estimates of the number of species on Earth today vary widely, over 7.4 billion humans live on Earth and depend on its biosphere and minerals for their survival. Humans have developed diverse societies and cultures, politically, the world has about 200 sovereign states, the modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most often spelled eorðe. It has cognates in every Germanic language, and their proto-Germanic root has been reconstructed as *erþō, originally, earth was written in lowercase, and from early Middle English, its definite sense as the globe was expressed as the earth. By early Modern English, many nouns were capitalized, and the became the Earth. More recently, the name is simply given as Earth. House styles now vary, Oxford spelling recognizes the lowercase form as the most common, another convention capitalizes Earth when appearing as a name but writes it in lowercase when preceded by the. It almost always appears in lowercase in colloquial expressions such as what on earth are you doing, the oldest material found in the Solar System is dated to 4. 5672±0.0006 billion years ago. By 4. 54±0.04 Gya the primordial Earth had formed, the formation and evolution of Solar System bodies occurred along with the Sun
Earth
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" The Blue Marble " photograph of Earth, taken during the Apollo 17 lunar mission in 1972
Earth
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Artist's impression of the early Solar System's planetary disk
Earth
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World map color-coded by relative height
Earth
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The summit of Chimborazo, in Ecuador, is the point on Earth's surface farthest from its center.
64.
Vacuum
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Vacuum is space void of matter. The word stems from the Latin adjective vacuus for vacant or void, an approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. In engineering and applied physics on the hand, vacuum refers to any space in which the pressure is lower than atmospheric pressure. The Latin term in vacuo is used to describe an object that is surrounded by a vacuum, the quality of a partial vacuum refers to how closely it approaches a perfect vacuum. Other things equal, lower gas pressure means higher-quality vacuum, for example, a typical vacuum cleaner produces enough suction to reduce air pressure by around 20%. Ultra-high vacuum chambers, common in chemistry, physics, and engineering, operate below one trillionth of atmospheric pressure, outer space is an even higher-quality vacuum, with the equivalent of just a few hydrogen atoms per cubic meter on average. In the electromagnetism in the 19th century, vacuum was thought to be filled with a medium called aether, in modern particle physics, the vacuum state is considered the ground state of matter. Vacuum has been a frequent topic of debate since ancient Greek times. Evangelista Torricelli produced the first laboratory vacuum in 1643, and other techniques were developed as a result of his theories of atmospheric pressure. A torricellian vacuum is created by filling a glass container closed at one end with mercury. Vacuum became an industrial tool in the 20th century with the introduction of incandescent light bulbs and vacuum tubes. The recent development of human spaceflight has raised interest in the impact of vacuum on human health, the word vacuum comes from Latin an empty space, void, noun use of neuter of vacuus, meaning empty, related to vacare, meaning be empty. Vacuum is one of the few words in the English language that contains two consecutive letters u. Historically, there has been dispute over whether such a thing as a vacuum can exist. Ancient Greek philosophers debated the existence of a vacuum, or void, in the context of atomism, Aristotle believed that no void could occur naturally, because the denser surrounding material continuum would immediately fill any incipient rarity that might give rise to a void. Almost two thousand years after Plato, René Descartes also proposed a geometrically based alternative theory of atomism, without the problematic nothing–everything dichotomy of void, by the ancient definition however, directional information and magnitude were conceptually distinct. The explanation of a clepsydra or water clock was a topic in the Middle Ages. Although a simple wine skin sufficed to demonstrate a partial vacuum, in principle and he concluded that airs volume can expand to fill available space, and he suggested that the concept of perfect vacuum was incoherent. However, according to Nader El-Bizri, the physicist Ibn al-Haytham and the Mutazili theologians disagreed with Aristotle and Al-Farabi, using geometry, Ibn al-Haytham mathematically demonstrated that place is the imagined three-dimensional void between the inner surfaces of a containing body
Vacuum
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Pump to demonstrate vacuum
Vacuum
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A large vacuum chamber
Vacuum
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The Crookes tube, used to discover and study cathode rays, was an evolution of the Geissler tube.
Vacuum
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A glass McLeod gauge, drained of mercury
65.
Metre
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The metre or meter, is the base unit of length in the International System of Units. The metre is defined as the length of the path travelled by light in a vacuum in 1/299792458 seconds, the metre was originally defined in 1793 as one ten-millionth of the distance from the equator to the North Pole. In 1799, it was redefined in terms of a metre bar. In 1960, the metre was redefined in terms of a number of wavelengths of a certain emission line of krypton-86. In 1983, the current definition was adopted, the imperial inch is defined as 0.0254 metres. One metre is about 3 3⁄8 inches longer than a yard, Metre is the standard spelling of the metric unit for length in nearly all English-speaking nations except the United States and the Philippines, which use meter. Measuring devices are spelled -meter in all variants of English, the suffix -meter has the same Greek origin as the unit of length. This range of uses is found in Latin, French, English. Thus calls for measurement and moderation. In 1668 the English cleric and philosopher John Wilkins proposed in an essay a decimal-based unit of length, as a result of the French Revolution, the French Academy of Sciences charged a commission with determining a single scale for all measures. In 1668, Wilkins proposed using Christopher Wrens suggestion of defining the metre using a pendulum with a length which produced a half-period of one second, christiaan Huygens had observed that length to be 38 Rijnland inches or 39.26 English inches. This is the equivalent of what is now known to be 997 mm, no official action was taken regarding this suggestion. In the 18th century, there were two approaches to the definition of the unit of length. One favoured Wilkins approach, to define the metre in terms of the length of a pendulum which produced a half-period of one second. The other approach was to define the metre as one ten-millionth of the length of a quadrant along the Earths meridian, that is, the distance from the Equator to the North Pole. This means that the quadrant would have defined as exactly 10000000 metres at that time. To establish a universally accepted foundation for the definition of the metre, more measurements of this meridian were needed. This portion of the meridian, assumed to be the length as the Paris meridian, was to serve as the basis for the length of the half meridian connecting the North Pole with the Equator
Metre
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Belfry, Dunkirk —the northern end of the meridian arc
Metre
Metre
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Fortress of Montjuïc —the southerly end of the meridian arc
Metre
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Creating the metre-alloy in 1874 at the Conservatoire des Arts et Métiers. Present Henri Tresca, George Matthey, Saint-Claire Deville and Debray
66.
Jean Piaget
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Jean Piaget was a Swiss clinical psychologist known for his pioneering work in child development. Piagets theory of development and epistemological view are together called genetic epistemology. Piaget placed great importance on the education of children, as the Director of the International Bureau of Education, he declared in 1934 that only education is capable of saving our societies from possible collapse, whether violent, or gradual. Piagets theory and research influenced several people and his theory of child development is studied in pre-service education programs. Educators continue to incorporate constructionist-based strategies, Piaget created the International Center for Genetic Epistemology in Geneva in 1955 while on the faculty of the University of Geneva and directed the Center until his death in 1980. The number of collaborations that its founding made possible, and their impact, according to Ernst von Glasersfeld, Jean Piaget was the great pioneer of the constructivist theory of knowing. However, his ideas did not become widely popularized until the 1960s and this then led to the emergence of the study of development as a major sub-discipline in psychology. By the end of the 20th century, Piaget was second only to B. F. Skinner as the most cited psychologist of that era, Piaget was born in Neuchâtel, in the Francophone region of Switzerland. He was the oldest son of Arthur Piaget, a professor of literature at the University of Neuchâtel. Piaget was a child who developed an interest in biology. His early interest in zoology earned him a reputation among those in the field after he had published articles on mollusks by the age of 15. He was educated at the University of Neuchâtel, and studied briefly at the University of Zürich, during this time, he published two philosophical papers that showed the direction of his thinking at the time, but which he later dismissed as adolescent thought. His interest in psychoanalysis, at the time a burgeoning strain of psychology, Piaget moved from Switzerland to Paris, France after his graduation and he taught at the Grange-Aux-Belles Street School for Boys. The school was run by Alfred Binet, the developer of the Binet intelligence Test and it was while he was helping to mark some of these tests that Piaget noticed that young children consistently gave wrong answers to certain questions. Piaget did not focus so much on the fact of the answers being wrong. This led him to the theory that young childrens cognitive processes are different from those of adults. Ultimately, he was to propose a theory of cognitive developmental stages in which individuals exhibit certain common patterns of cognition in each period of development. In 1921, Piaget returned to Switzerland as director of the Rousseau Institute in Geneva, at this time, the institute was directed by Édouard Claparède
Jean Piaget
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Bust of Jean Piaget in the Parc des Bastions, Geneva
Jean Piaget
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Photo of the Jean Piaget Foundation with Pierre Bovet (1878–1965) first row (with large beard) and Jean Piaget (1896–1980) first row (on the right, with glasses) in front of the Rousseau Institute (Geneva), 1925
67.
Projectile
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A projectile is any object thrown into space by the exertion of a force. Although any object in motion through space may be called a projectile, mathematical equations of motion are used to analyze projectile trajectory. Blowguns and pneumatic rifles use compressed gases, while most other guns and cannons utilize expanding gases liberated by sudden chemical reactions, light-gas guns use a combination of these mechanisms. Railguns utilize electromagnetic fields to provide a constant acceleration along the length of the device. Some projectiles provide propulsion during flight by means of an engine or jet engine. In military terminology, a rocket is unguided, while a missile is guided, note the two meanings of rocket, an ICBM is a guided missile with a rocket engine. An explosion, whether or not by a weapon, causes the debris to act as high velocity projectiles. An explosive weapon, or device may also be designed to produce high velocity projectiles by the break-up of its casing. Many projectiles, e. g. shells, may carry a charge or another chemical or biological substance. Aside from explosive payload, a projectile can be designed to cause damage, e. g. fire. Typical kinetic energy weapons are blunt projectiles such as rocks and round shots, pointed ones such as arrows, among projectiles that do not contain explosives are those launched from railguns, coilguns, and mass drivers, as well as kinetic energy penetrators. Other types of weapons are accelerated over time by a rocket engine. In either case, it is the energy of the projectile that destroys its target. Some kinetic weapons for targeting objects in spaceflight are anti-satellite weapons, since in order to reach an object in orbit it is necessary to attain an extremely high velocity, their released kinetic energy alone is enough to destroy their target, explosives are not necessary. For example, the energy of TNT is 4.6 MJ/kg, and this saves costly weight and there is no detonation to be precisely timed. This method, however, requires direct contact with the target, some hit-to-kill warheads are additionally equipped with an explosive directional warhead to enhance the kill probability. With regard to weapons, the Arrow missile and MIM-104 Patriot PAC-2 have explosives, while the Kinetic Energy Interceptor, Lightweight Exo-Atmospheric Projectile. A kinetic projectile can also be dropped from aircraft and this is applied by replacing the explosives of a regular bomb with a non-explosive material, for a precision hit with less collateral damage
Projectile
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Projectile and cartridge case for the massive World War II German 80cm Schwerer Gustav railway gun
Projectile
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Ball speeds of 105 miles per hour (169 km/h) have been recorded in baseball.
68.
Richard Feynman
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For his contributions to the development of quantum electrodynamics, Feynman, jointly with Julian Schwinger and Sinichirō Tomonaga, received the Nobel Prize in Physics in 1965. Feynman developed a widely used pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, during his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World he was ranked as one of the ten greatest physicists of all time. Along with his work in physics, Feynman has been credited with pioneering the field of quantum computing. Tolman professorship in physics at the California Institute of Technology. They were not religious, and by his youth, Feynman described himself as an avowed atheist, like Albert Einstein and Edward Teller, Feynman was a late talker, and by his third birthday had yet to utter a single word. He retained a Brooklyn accent as an adult and that accent was thick enough to be perceived as an affectation or exaggeration – so much so that his good friends Wolfgang Pauli and Hans Bethe once commented that Feynman spoke like a bum. The young Feynman was heavily influenced by his father, who encouraged him to ask questions to challenge orthodox thinking, from his mother, he gained the sense of humor that he had throughout his life. As a child, he had a talent for engineering, maintained a laboratory in his home. When he was in school, he created a home burglar alarm system while his parents were out for the day running errands. When Richard was five years old, his mother gave birth to a brother, Henry Philips. Four years later, Richards sister Joan was born, and the moved to Far Rockaway. Though separated by nine years, Joan and Richard were close and their mother thought that women did not have the cranial capacity to comprehend such things. Despite their mothers disapproval of Joans desire to study astronomy, Richard encouraged his sister to explore the universe, Joan eventually became an astrophysicist specializing in interactions between the Earth and the solar wind. Feynman attended Far Rockaway High School, a school in Far Rockaway, Queens, upon starting high school, Feynman was quickly promoted into a higher math class. A high-school-administered IQ test estimated his IQ at 125—high, but merely respectable according to biographer James Gleick and his sister Joan did better, allowing her to claim that she was smarter. Years later he declined to join Mensa International, saying that his IQ was too low, physicist Steve Hsu stated of the test, I suspect that this test emphasized verbal, as opposed to mathematical, ability. Feynman received the highest score in the United States by a margin on the notoriously difficult Putnam mathematics competition exam
Richard Feynman
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Richard Feynman
Richard Feynman
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Feynman (center) with Robert Oppenheimer (right) relaxing at a Los Alamos social function during the Manhattan Project
Richard Feynman
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The Feynman section at the Caltech bookstore
Richard Feynman
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Mention of Feynman's prize on the monument at the American Museum of Natural History in New York City. Because the monument is dedicated to American Laureates, Tomonaga is not mentioned.
69.
Addison-Wesley
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Addison-Wesley is a publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing, in addition to publishing books, Addison-Wesley also distributes its technical titles through the Safari Books Online e-reference service. Addison-Wesleys majority of sales derive from the United States and Europe, the Addison-Wesley Professional Imprint produces content including books, eBooks, and video for the professional IT worker including developers, programmers, managers, system administrators. Classic titles include The Art of Computer Programming, The C++ Programming Language, The Mythical Man-Month, Addison-Wesley Professional is also a partner with Safari Books Online. Its first computer book was Programs for an Electronic Digital Computer, by Wilkes, Wheeler, in 1977, Addison-Wesley acquired W. A. Benjamin Company, and merged it with the Cummings division of the company to form Benjamin Cummings. It was purchased by the publishing and education company, Pearson PLC in 1988. The trade publishing division of Addison-Wesley was sold to Perseus Books Group in 1997, Pearson acquired the educational division of Simon & Schuster in 1998, and merged it with Addison Wesley Longman to form Pearson Education and subsequently rebranded to Pearson in 2011. Pearson moved the former Addison Wesley Longman offices from Reading, Massachusetts to Boston in 2004 and its current executives hail from the original Addison-Wesley with a storied history of their own. Addison-Wesley Secondary Math, An Integrated Approach, Focus on Algebra The Art of Computer Programming by Donald Knuth The Feynman Lectures on Physics by Richard Feynman, Robert B. Leighton, and Matthew Sands Concrete Mathematics, A Foundation For Computer Science by Ronald Graham, Donald Knuth, exploratory data analysis by John W. Tukey, based on a course taught at Princeton. The Mythical Man-Month by Fred P. Brooks
Addison-Wesley
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Addison-Wesley
70.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
71.
Integral
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
Integral
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A definite integral of a function can be represented as the signed area of the region bounded by its graph.
72.
Jerk (physics)
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Jerk is a vector, and there is no generally used term to describe its scalar magnitude. According to the result of analysis of jerk, the SI units are m/s3. Where a → is acceleration, v → is velocity, r → is position, there is no universal agreement on the symbol for jerk, but j is commonly used. Newtons notation for the derivative is also applied. The fourth derivative of position, equivalent to the first derivative of jerk, is jounce, because of involving third derivatives, in mathematics differential equations of the form J =0 are called jerk equations. This motivates mathematical interest in jerk systems, systems involving a fourth or higher derivative are accordingly called hyperjerk systems. In balancing some given force the postcentral gyrus establishes a control loop to achieve equilibrium by adjusting the muscular tension according to the sensed position of the actuator. As an everyday example, driving in a car can show effects of acceleration, the more experienced drivers accelerate smoothly, but beginners provide a jerky ride. High-powered sports cars offer the feeling of being pressed into the cushioning, note that there would be no jerk if the car started to move backwards with the same acceleration. Every experienced driver knows how to start and how to stop braking with low jerk, see also below in the motion profile, segment 7, Deceleration ramp-down. X itself, zeroth derivative The most prominent force F associated with the position of a particle relates through Hookes law to the stiffness k r of a spring. This is a force opposing the increase in displacement, the drag coefficient depends on the scalable shape of the object and on the Reynolds number, which itself depends on the speed. The acceleration a is according to Newtons second law F = m ⋅ a bound to a force F by the proportionality given by the mass m. It is often reported that NASA in designing the Hubble Telescope not only limited the jerk in their requirement specification, but also the next higher derivative, the jounce. For a recoil force on accelerating charged particles emitting radiation, which is proportional to their jerk, a more advanced theory, applicable in a relativistic and quantum environment, accounting for self-energy is provided in Wheeler–Feynman absorber theory. In real world environments, because of deformation, granularity at least at the Planck scale, i. e. quanta-effects, extrapolating from the idealized settings, the effect of jerk in real situations can be qualitatively described, explained and predicted. The jump-discontinuity in acceleration may be modeled by a Dirac delta in the jerk, assume a path along a circular arc with radius r, which tangentially connects to a straight line. The whole path is continuous and its pieces are smooth, see below for a more concrete application
Jerk (physics)
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Timing diagram over one rev. for angle, angular velocity, angular acceleration, and angular jerk
73.
SI units
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The International System of Units is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, the system also establishes a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system was published in 1960 as the result of an initiative began in 1948. It is based on the system of units rather than any variant of the centimetre-gram-second system. The motivation for the development of the SI was the diversity of units that had sprung up within the CGS systems, the International System of Units has been adopted by most developed countries, however, the adoption has not been universal in all English-speaking countries. The metric system was first implemented during the French Revolution with just the metre and kilogram as standards of length, in the 1830s Carl Friedrich Gauss laid the foundations for a coherent system based on length, mass, and time. In the 1860s a group working under the auspices of the British Association for the Advancement of Science formulated the requirement for a coherent system of units with base units and derived units. Meanwhile, in 1875, the Treaty of the Metre passed responsibility for verification of the kilogram, in 1921, the Treaty was extended to include all physical quantities including electrical units originally defined in 1893. The units associated with these quantities were the metre, kilogram, second, ampere, kelvin, in 1971, a seventh base quantity, amount of substance represented by the mole, was added to the definition of SI. On 11 July 1792, the proposed the names metre, are, litre and grave for the units of length, area, capacity. The committee also proposed that multiples and submultiples of these units were to be denoted by decimal-based prefixes such as centi for a hundredth, on 10 December 1799, the law by which the metric system was to be definitively adopted in France was passed. Prior to this, the strength of the magnetic field had only been described in relative terms. The technique used by Gauss was to equate the torque induced on a magnet of known mass by the earth’s magnetic field with the torque induced on an equivalent system under gravity. The resultant calculations enabled him to assign dimensions based on mass, length, a French-inspired initiative for international cooperation in metrology led to the signing in 1875 of the Metre Convention. Initially the convention only covered standards for the metre and the kilogram, one of each was selected at random to become the International prototype metre and International prototype kilogram that replaced the mètre des Archives and kilogramme des Archives respectively. Each member state was entitled to one of each of the prototypes to serve as the national prototype for that country. Initially its prime purpose was a periodic recalibration of national prototype metres. The official language of the Metre Convention is French and the version of all official documents published by or on behalf of the CGPM is the French-language version
SI units
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Stone marking the Austro-Hungarian /Italian border at Pontebba displaying myriametres, a unit of 10 km used in Central Europe in the 19th century (but since deprecated).
SI units
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The seven base units in the International System of Units
SI units
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Carl Friedrich Gauss
SI units
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Thomson
74.
Angle
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
Angle
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An angle enclosed by rays emanating from a vertex.
75.
Radian
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The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
Radian
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A chart to convert between degrees and radians
Radian
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An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to an angle of 2 π radians.
76.
Solid angle
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In geometry, a solid angle is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large the object appears to an observer looking from that point, in the International System of Units, a solid angle is expressed in a dimensionless unit called a steradian. A small object nearby may subtend the same angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, indeed, as viewed from any point on Earth, both objects have approximately the same solid angle as well as apparent size. This is evident during a solar eclipse, an objects solid angle in steradians is equal to the area of the segment of a unit sphere, centered at the angles vertex, that the object covers. A solid angle in steradians equals the area of a segment of a sphere in the same way a planar angle in radians equals the length of an arc of a unit circle. Solid angles are used in physics, in particular astrophysics. The solid angle of an object that is far away is roughly proportional to the ratio of area to squared distance. Here area means the area of the object when projected along the viewing direction. The solid angle of a sphere measured from any point in its interior is 4π sr, Solid angles can also be measured in square degrees, in square minutes and square seconds, or in fractions of the sphere, also known as spat. In spherical coordinates there is a formula for the differential, d Ω = sin θ d θ d φ where θ is the colatitude, at the equator you see all of the celestial sphere, at either pole only one half. Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where a →, b →, c → are the positions of the vertices A, B and C. Define the vertex angle θa to be the angle BOC and define θb, let φab be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define φac, φbc correspondingly. When implementing the above equation care must be taken with the function to avoid negative or incorrect solid angles. One source of errors is that the scalar triple product can be negative if a, b, c have the wrong winding. Computing abs is a sufficient solution since no other portion of the equation depends on the winding, the other pitfall arises when the scalar triple product is positive but the divisor is negative. Indices are cycled, s0 = sn and s1 = sn +1, the solid angle of a latitude-longitude rectangle on a globe is s r, where φN and φS are north and south lines of latitude, and θE and θW are east and west lines of longitude. Mathematically, this represents an arc of angle φN − φS swept around a sphere by θE − θW radians, when longitude spans 2π radians and latitude spans π radians, the solid angle is that of a sphere
Solid angle
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Any area on a sphere which is equal in area to the square of its radius, when observed from its center, subtends precisely one steradian.
77.
Steradian
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The steradian or square radian is the SI unit of solid angle. It is used in geometry, and is analogous to the radian which quantifies planar angles. The name is derived from the Greek stereos for solid and the Latin radius for ray and it is useful, however, to distinguish between dimensionless quantities of a different nature, so the symbol sr is used to indicate a solid angle. For example, radiant intensity can be measured in watts per steradian, the steradian was formerly an SI supplementary unit, but this category was abolished in 1995 and the steradian is now considered an SI derived unit. A steradian can be defined as the angle subtended at the center of a unit sphere by a unit area on its surface. For a general sphere of radius r, any portion of its surface with area A = r2 subtends one steradian, because the surface area A of a sphere is 4πr2, the definition implies that a sphere measures 4π steradians. By the same argument, the solid angle that can be subtended at any point is 4π sr. Since A = r2, it corresponds to the area of a cap. Therefore one steradian corresponds to the angle of the cross-section of a simple cone subtending the plane angle 2θ, with θ given by, θ = arccos = arccos = arccos ≈0.572 rad. This angle corresponds to the plane angle of 2θ ≈1.144 rad or 65. 54°. A steradian is also equal to the area of a polygon having an angle excess of 1 radian, to 1/4π of a complete sphere. The solid angle of a cone whose cross-section subtends the angle 2θ is, Ω =2 π s r. In two dimensions, an angle is related to the length of the arc that it spans, θ = l r r a d where l is arc length, r is the radius of the circle. For example, a measurement of the width of an object would be given in radians. At the same time its visible area over ones visible field would be given in steradians. Just as the area of a circle is related to its diameter or radius. One-dimensional circular measure has units of radians or degrees, while two-dimensional spherical measure is expressed in steradians, in higher dimensional mathematical spaces, units for analogous solid angles have not been explicitly named. When they are used, they are dealt with by analogy with the circular or spherical cases and that is, as a proportion of the relevant unit hypersphere taken up by the generalized angle, or point set expressed in spherical coordinates
Steradian
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A graphical representation of 1 steradian. The sphere has radius r, and in this case the area A of the highlighted surface patch is r 2. The solid angle Ω equals A sr/ r 2 which is 1 sr in this example. The entire sphere has a solid angle of 4π sr.
78.
Frequency
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Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as frequency, which emphasizes the contrast to spatial frequency. The period is the duration of time of one cycle in a repeating event, for example, if a newborn babys heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as vibrations, audio signals, radio waves. For cyclical processes, such as rotation, oscillations, or waves, in physics and engineering disciplines, such as optics, acoustics, and radio, frequency is usually denoted by a Latin letter f or by the Greek letter ν or ν. For a simple motion, the relation between the frequency and the period T is given by f =1 T. The SI unit of frequency is the hertz, named after the German physicist Heinrich Hertz, a previous name for this unit was cycles per second. The SI unit for period is the second, a traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. As a matter of convenience, longer and slower waves, such as ocean surface waves, short and fast waves, like audio and radio, are usually described by their frequency instead of period. Spatial frequency is analogous to temporal frequency, but the axis is replaced by one or more spatial displacement axes. Y = sin = sin d θ d x = k Wavenumber, in the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has a relationship to the wavelength. Even in dispersive media, the frequency f of a wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave. In the special case of electromagnetic waves moving through a vacuum, then v = c, where c is the speed of light in a vacuum, and this expression becomes, f = c λ. When waves from a monochrome source travel from one medium to another, their remains the same—only their wavelength. For example, if 71 events occur within 15 seconds the frequency is, the latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called gating error and causes an error in the calculated frequency of Δf = 1/, or a fractional error of Δf / f = 1/ where Tm is the timing interval. This error decreases with frequency, so it is a problem at low frequencies where the number of counts N is small, an older method of measuring the frequency of rotating or vibrating objects is to use a stroboscope
Frequency
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A resonant-reed frequency meter, an obsolete device used from about 1900 to the 1940s for measuring the frequency of alternating current. It consists of a strip of metal with reeds of graduated lengths, vibrated by an electromagnet. When the unknown frequency is applied to the electromagnet, the reed which is resonant at that frequency will vibrate with large amplitude, visible next to the scale.
Frequency
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As time elapses – represented here as a movement from left to right, i.e. horizontally – the five sinusoidal waves shown vary regularly (i.e. cycle), but at different rates. The red wave (top) has the lowest frequency (i.e. varies at the slowest rate) while the purple wave (bottom) has the highest frequency (varies at the fastest rate).
Frequency
Frequency
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Modern frequency counter
79.
Kilogram square metre
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It depends on the bodys mass distribution and the axis chosen, with larger moments requiring more torque to change the bodys rotation. It is a property, the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems. One of its definitions is the moment of mass with respect to distance from an axis r, I = ∫ Q r 2 d m. For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about a perpendicular to the plane. When a body is rotating, or free to rotate, around an axis, the amount of torque needed to cause any given angular acceleration is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram metre squared in SI units, moment of inertia plays the role in rotational kinetics that mass plays in linear kinetics - both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, for a point-like mass, the moment of inertia about some axis is given by mr2, where r is the distance to the axis, and m is the mass. For an extended body, the moment of inertia is just the sum of all the pieces of mass multiplied by the square of their distances from the axis in question. For an extended body of a shape and uniform density. In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, the term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Eulers second law. Comparison of this frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. Moment of inertia appears in momentum, kinetic energy, and in Newtons laws of motion for a rigid body as a physical parameter that combines its shape. There is a difference in the way moment of inertia appears in planar. The moment of inertia of a flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis, if the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their arms or divers curl their bodies into a tuck position during a dive. For a simple pendulum, this yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as. Thus, moment of inertia depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation
Kilogram square metre
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Tightrope walker Samuel Dixon using the long rod's moment of inertia for balance while crossing the Niagara River in 1890.
Kilogram square metre
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Flywheels have large moments of inertia to smooth out mechanical motion. This example is in a Russian museum.
Kilogram square metre
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Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to conservation of angular momentum.
Kilogram square metre
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Pendulums used in Mendenhall gravimeter apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth.
80.
List of equations in classical mechanics
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Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. It is the most familiar of the theories of physics, the concepts it covers, such as mass, acceleration, and force, are commonly used and known. The subject is based upon a three-dimensional Euclidean space with fixed axes, the point of concurrency of the three axes is known as the origin of the particular space. Classical mechanics utilises many equations—as well as other mathematical concepts—which relate various physical quantities to one another and these include differential equations, manifolds, Lie groups, and ergodic theory. This page gives a summary of the most important of these and this article lists equations from Newtonian mechanics, see analytical mechanics for the more general formulation of classical mechanics. Every conservative force has a potential energy, by following two principles one can consistently assign a non-relative value to U, Wherever the force is zero, its potential energy is defined to be zero as well. Whenever the force does work, potential energy is lost, in the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use θ, but this does not have to be the angle used in polar coordinate systems. The unit axial vector n ^ = e ^ r × e ^ θ defines the axis of rotation, the precession angular speed of a spinning top is given by, Ω = w r I ω where w is the weight of the spinning flywheel. Euler also worked out analogous laws of motion to those of Newton and these extend the scope of Newtons laws to rigid bodies, but are essentially the same as above. A new equation Euler formulated is, I ⋅ α + ω × = τ where I is the moment of inertia tensor, the previous equations for planar motion can be used here, corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane, r = r = r e ^ r the following results apply to the particle. If acceleration is not constant then the general calculus equations above must be used, found by integrating the definitions of position, velocity, for classical mechanics, the transformation law from one inertial or accelerating frame to another is the Galilean transform. Conversely F moves at velocity relative to F, the situation is similar for relative accelerations. SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator, mathematical Methods of Classical Mechanics, Springer, ISBN 978-0-387-96890-2 Berkshire, Frank H. Kibble, T. W. B
List of equations in classical mechanics
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Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.
81.
Newton (unit)
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The newton is the International System of Units derived unit of force. It is named after Isaac Newton in recognition of his work on classical mechanics, see below for the conversion factors. One newton is the force needed to one kilogram of mass at the rate of one metre per second squared in direction of the applied force. In 1948, the 9th CGPM resolution 7 adopted the name newton for this force, the MKS system then became the blueprint for todays SI system of units. The newton thus became the unit of force in le Système International dUnités. This SI unit is named after Isaac Newton, as with every International System of Units unit named for a person, the first letter of its symbol is upper case. Note that degree Celsius conforms to this rule because the d is lowercase. — Based on The International System of Units, section 5.2. Newtons second law of motion states that F = ma, where F is the applied, m is the mass of the object receiving the force. The newton is therefore, where the symbols are used for the units, N for newton, kg for kilogram, m for metre. In dimensional analysis, F = M L T2 where F is force, M is mass, L is length, at average gravity on earth, a kilogram mass exerts a force of about 9.8 newtons. An average-sized apple exerts about one newton of force, which we measure as the apples weight, for example, the tractive effort of a Class Y steam train and the thrust of an F100 fighter jet engine are both around 130 kN. One kilonewton,1 kN, is 102.0 kgf,1 kN =102 kg ×9.81 m/s2 So for example, a platform rated at 321 kilonewtons will safely support a 32,100 kilograms load. Specifications in kilonewtons are common in safety specifications for, the values of fasteners, Earth anchors. Working loads in tension and in shear, thrust of rocket engines and launch vehicles clamping forces of the various moulds in injection moulding machines used to manufacture plastic parts
Newton (unit)
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Base units