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.
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
3.
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.
4.
Statistical mechanics
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Statistical mechanics is a branch of theoretical physics using probability theory to study the average behaviour of a mechanical system, where the state of the system is uncertain. A common use of mechanics is in explaining the thermodynamic behaviour of large systems. This branch of mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics. Statistical mechanics also finds use outside equilibrium, an important subbranch known as non-equilibrium statistical mechanics deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles, in physics there are two types of mechanics usually examined, classical mechanics and quantum mechanics. Statistical mechanics fills this disconnection between the laws of mechanics and the experience of incomplete knowledge, by adding some uncertainty about which state the system is in. The statistical ensemble is a probability distribution over all states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points, in quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a density matrix. These two meanings are equivalent for many purposes, and will be used interchangeably in this article, however the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself also evolves, as the systems in the ensemble continually leave one state. The ensemble evolution is given by the Liouville equation or the von Neumann equation, one special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium, Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics, non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems. The primary goal of thermodynamics is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles. Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium, Statistical equilibrium does not mean that the particles have stopped moving, rather, only that the ensemble is not evolving. A sufficient condition for statistical equilibrium with a system is that the probability distribution is a function only of conserved properties. There are many different equilibrium ensembles that can be considered, additional postulates are necessary to motivate why the ensemble for a given system should have one form or another. A common approach found in textbooks is to take the equal a priori probability postulate
Statistical mechanics
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Statistical mechanics
5.
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
6.
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
7.
Energy
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In physics, energy is the property that must be transferred to an object in order to perform work on – or to heat – the object, and can be converted in form, but not created or destroyed. The SI unit of energy is the joule, which is the transferred to an object by the mechanical work of moving it a distance of 1 metre against a force of 1 newton. Mass and energy are closely related, for example, with a sensitive enough scale, one could measure an increase in mass after heating an object. Living organisms require available energy to stay alive, such as the humans get from food. Civilisation gets the energy it needs from energy resources such as fuels, nuclear fuel. The processes of Earths climate and ecosystem are driven by the radiant energy Earth receives from the sun, the total energy of a system can be subdivided and classified in various ways. It may also be convenient to distinguish gravitational energy, thermal energy, several types of energy, electric energy. Many of these overlap, for instance, thermal energy usually consists partly of kinetic. Some types of energy are a mix of both potential and kinetic energy. An example is energy which is the sum of kinetic. Whenever physical scientists discover that a phenomenon appears to violate the law of energy conservation. Heat and work are special cases in that they are not properties of systems, in general we cannot measure how much heat or work are present in an object, but rather only how much energy is transferred among objects in certain ways during the occurrence of a given process. Heat and work are measured as positive or negative depending on which side of the transfer we view them from, the distinctions between different kinds of energy is not always clear-cut. In contrast to the definition, energeia was a qualitative philosophical concept, broad enough to include ideas such as happiness. The modern analog of this property, kinetic energy, differs from vis viva only by a factor of two, in 1807, Thomas Young was possibly the first to use the term energy instead of vis viva, in its modern sense. Gustave-Gaspard Coriolis described kinetic energy in 1829 in its modern sense, the law of conservation of energy was also first postulated in the early 19th century, and applies to any isolated system. It was argued for years whether heat was a physical substance, dubbed the caloric, or merely a physical quantity. In 1845 James Prescott Joule discovered the link between mechanical work and the generation of heat and these developments led to the theory of conservation of energy, formalized largely by William Thomson as the field of thermodynamics
Energy
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In a typical lightning strike, 500 megajoules of electric potential energy is converted into the same amount of energy in other forms, mostly light energy, sound energy and thermal energy.
Energy
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Thermal energy is energy of microscopic constituents of matter, which may include both kinetic and potential energy.
Energy
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Thomas Young – the first to use the term "energy" in the modern sense.
Energy
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A Turbo generator transforms the energy of pressurised steam into electrical energy
8.
Kinetic energy
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In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes, the same amount of work is done by the body in decelerating from its current speed to a state of rest. In classical mechanics, the energy of a non-rotating object of mass m traveling at a speed v is 12 m v 2. In relativistic mechanics, this is an approximation only when v is much less than the speed of light. The standard unit of energy is the joule. The adjective kinetic has its roots in the Greek word κίνησις kinesis, the dichotomy between kinetic energy and potential energy can be traced back to Aristotles concepts of actuality and potentiality. The principle in classical mechanics that E ∝ mv2 was first developed by Gottfried Leibniz and Johann Bernoulli, Willem s Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, Émilie du Châtelet recognized the implications of the experiment and published an explanation. The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century, early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled Du Calcul de lEffet des Machines outlining the mathematics of kinetic energy. William Thomson, later Lord Kelvin, is given the credit for coining the term kinetic energy c, energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. These can be categorized in two classes, potential energy and kinetic energy. Kinetic energy is the movement energy of an object, Kinetic energy can be transferred between objects and transformed into other kinds of energy. Kinetic energy may be best understood by examples that demonstrate how it is transformed to, for example, a cyclist uses chemical energy provided by food to accelerate a bicycle to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome air resistance, the chemical energy has been converted into kinetic energy, the energy of motion, but the process is not completely efficient and produces heat within the cyclist. The kinetic energy in the moving cyclist and the bicycle can be converted to other forms, for example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling, the energy is not destroyed, it has only been converted to another form by friction
Kinetic energy
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The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. When they start rising, the kinetic energy begins to be converted to gravitational potential energy. The sum of kinetic and potential energy in the system remains constant, ignoring losses to friction.
9.
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
10.
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
11.
Momentum
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In classical mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object, quantified in kilogram-meters per second. It is dimensionally equivalent to impulse, the product of force and time, Newtons second law of motion states that the change in linear momentum of a body is equal to the net impulse acting on it. If the truck were lighter, or moving slowly, then it would have less momentum. Linear momentum is also a quantity, meaning that if a closed system is not affected by external forces. In classical mechanics, conservation of momentum is implied by Newtons laws. It also holds in special relativity and, with definitions, a linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory. It is ultimately an expression of one of the symmetries of space and time. Linear momentum depends on frame of reference, observers in different frames would find different values of linear momentum of a system. But each would observe that the value of linear momentum does not change with time, momentum has a direction as well as magnitude. Quantities that have both a magnitude and a direction are known as vector quantities, because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds. Below, the properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations, the momentum of a particle is traditionally represented by the letter p. It is the product of two quantities, the mass and velocity, p = m v, the units of momentum are the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity in meters per second then the momentum is in kilogram meters/second, in cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters/second. Being a vector, momentum has magnitude and direction, for example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg m/s due north measured from the ground. The momentum of a system of particles is the sum of their momenta, if two particles have masses m1 and m2, and velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2. If all the particles are moving, the center of mass will generally be moving as well, if the center of mass is moving at velocity vcm, the momentum is, p = m v cm. This is known as Eulers first law, if a force F is applied to a particle for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t
Momentum
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In a game of pool, momentum is conserved; that is, if one ball stops dead after the collision, the other ball will continue away with all the momentum. If the moving ball continues or is deflected then both balls will carry a portion of the momentum from the collision.
12.
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
13.
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
14.
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.
15.
Virtual work
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Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements, among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the according to the principle of least action. The work of a force on a particle along a displacement is known as the virtual work. The principle of work had always been used in some form since antiquity in the study of statics. It was used by the Greeks, medieval Arabs and Latins, working with Leibnizian concepts, Johann Bernoulli systematized the virtual work principle and made explicit the concept of infinitesimal displacement. He was able to solve problems for both bodies as well as fluids. Bernoullis version of virtual work law appeared in his letter to Pierre Varignon in 1715 and this formulation of the principle is today known as the principle of virtual velocities and is commonly considered as the prototype of the contemporary virtual work principles. In 1743 DAlembert published his Traite de Dynamique where he applied the principle of work, based on the Bernoullis work. His idea was to convert a dynamical problem into static problem by introducing inertial force, consider a point particle that moves along a path which is described by a function r from point A, where r, to point B, where r. It is possible that the moves from A to B along a nearby path described by r + δr. The variation δr satisfies the requirement δr = δr =0, the components of the variation, δr1, δr2 and δr3, are called virtual displacements. This can be generalized to a mechanical system defined by the generalized coordinates qi. In which case, the variation of the qi is defined by the virtual displacements δqi. Virtual work is the work done by the applied forces. When considering forces applied to a body in equilibrium, the principle of least action requires the virtual work of these forces to be zero. Consider a particle P that moves from a point A to a point B along a trajectory r and it is important to notice that the value of the work W depends on the trajectory r. Suppose the force F is the same as F, the variation of the work δW associated with this nearby path, known as the virtual work, can be computed to be δ W = W ¯ − W = ∫ t 0 t 1 d t
Virtual work
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This is an engraving from Mechanics Magazine published in London in 1824.
Virtual work
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Illustration from Army Service Corps Training on Mechanical Transport, (1911), Fig. 112 Transmission of motion and force by gear wheels, compound train
16.
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).
17.
Lagrangian mechanics
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Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. No new physics is introduced in Lagrangian mechanics compared to Newtonian mechanics, Newtons laws can include non-conservative forces like friction, however, they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system, dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler-Lagrange equations. Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, as a special case of Noethers theorem. Lagrangian mechanics is important not just for its applications. It can also be applied to systems by analogy, for instance to coupled electric circuits with inductances and capacitances. Lagrangian mechanics is used to solve mechanical problems in physics. Lagrangian mechanics applies to the dynamics of particles, fields are described using a Lagrangian density, Lagranges equations are also used in optimisation problems of dynamic systems. In mechanics, Lagranges equations of the second kind are used more than those of the first kind. Suppose we have a bead sliding around on a wire, or a simple pendulum. This choice eliminates the need for the constraint force to enter into the resultant system of equations, there are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment. For a wide variety of systems, if the size and shape of a massive object are negligible. For a system of N point particles with masses m1, m2, MN, each particle has a position vector, denoted r1, r2. Cartesian coordinates are often sufficient, so r1 =, r2 =, in three dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system. These are all points in space to locate the particles. The velocity of particle is how fast the particle moves along its path of motion. In Newtonian mechanics, the equations of motion are given by Newtons laws, the second law net force equals mass times acceleration, Σ F = m d2r/dt2, applies to each particle. For an N particle system in 3d, there are 3N second order differential equations in the positions of the particles to solve for
Lagrangian mechanics
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Joseph-Louis Lagrange (1736—1813)
Lagrangian mechanics
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Isaac Newton (1642—1726)
Lagrangian mechanics
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Jean d'Alembert (1717—1783)
18.
Equations of motion
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In mathematical physics, equations of motion are equations that describe the behaviour of a physical system in terms of its motion as a function of time. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system, the functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions to the equations describing the motion of the dynamics. There are two descriptions of motion, dynamics and kinematics. Dynamics is general, since momenta, forces and energy of the particles are taken into account, in this instance, sometimes the term refers to the differential equations that the system satisfies, and sometimes to the solutions to those equations. However, kinematics is simpler as it concerns only variables derived from the positions of objects, equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the types of motion are translations, rotations, oscillations. A differential equation of motion, usually identified as some physical law, solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the initial values, to state this formally, in general an equation of motion M is a function of the position r of the object, its velocity, and its acceleration, and time t. Euclidean vectors in 3D are denoted throughout in bold and this is equivalent to saying an equation of motion in r is a second order ordinary differential equation in r, M =0, where t is time, and each overdot denotes one time derivative. The initial conditions are given by the constant values at t =0, r, r ˙, the solution r to the equation of motion, with specified initial values, describes the system for all times t after t =0. Sometimes, the equation will be linear and is likely to be exactly solvable. In general, the equation will be non-linear, and cannot be solved exactly so a variety of approximations must be used, the solutions to nonlinear equations may show chaotic behavior depending on how sensitive the system is to the initial conditions. Despite the great strides made in the development of geometry made by Ancient Greeks and surveys in Rome, the exposure of Europe to Arabic numerals and their ease in computations encouraged first the scholars to learn them and then the merchants and invigorated the spread of knowledge throughout Europe. These studies led to a new body of knowledge that is now known as physics, thomas Bradwardine, one of those scholars, extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested a law involving force, resistance, distance, velocity. Nicholas Oresme further extended Bradwardines arguments, for writers on kinematics before Galileo, since small time intervals could not be measured, the affinity between time and motion was obscure. They used time as a function of distance, and in free fall, de Sotos comments are shockingly correct regarding the definitions of acceleration and the observation that during the violent motion of ascent acceleration would be negative
Equations of motion
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Kinematic quantities of a classical particle of mass m: position r, velocity v, acceleration a.
19.
Fictitious force
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The force F does not arise from any physical interaction between two objects, but rather from the acceleration a of the non-inertial reference frame itself. As stated by Iro, Such an additional force due to relative motion of two reference frames is called a pseudo-force. Assuming Newtons second law in the form F = ma, fictitious forces are proportional to the mass m. A fictitious force on an object arises when the frame of reference used to describe the motion is accelerating compared to a non-accelerating frame. As a frame can accelerate in any way, so can fictitious forces be as arbitrary. Gravitational force would also be a force based upon a field model in which particles distort spacetime due to their mass. The role of forces in Newtonian mechanics is described by Tonnelat. To solve classical mechanics problems exactly in an Earth-bound reference frame, the Euler force is typically ignored because the variations in the angular velocity of the rotating Earth surface are usually insignificant. Both of the fictitious forces are weak compared to most typical forces in everyday life. For example, Léon Foucault was able to show that the Coriolis force results from the Earths rotation using the Foucault pendulum. If the Earth were to rotate a thousand times faster, people could easily get the impression that such forces are pulling on them. Other accelerations also give rise to forces, as described mathematically below. An example of the detection of a non-inertial, rotating reference frame is the precession of a Foucault pendulum, in the non-inertial frame of the Earth, the fictitious Coriolis force is necessary to explain observations. In an inertial frame outside the Earth, no such force is necessary. Figure 1 shows an accelerating car, when a car accelerates, a passenger feels like theyre being pushed back into the seat. In an inertial frame of reference attached to the road, there is no physical force moving the rider backward, however, in the riders non-inertial reference frame attached to the accelerating car, there is a backward fictitious force. We mention two possible reasons for the force to clarify its existence, Figure 1, to an observer at rest on an inertial reference frame, the car will seem to accelerate. In order for the passenger to stay inside the car, a force must be exerted on the passenger
Fictitious force
20.
Harmonic oscillator
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If a frictional force proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the coefficient, the system can, Oscillate with a frequency lower than in the non-damped case. Decay to the position, without oscillations. The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a value of the friction coefficient and is called critically damped. If an external time dependent force is present, the oscillator is described as a driven oscillator. Mechanical examples include pendulums, masses connected to springs, and acoustical systems, other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many devices, such as clocks. They are the source of virtually all sinusoidal vibrations and waves, a simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a m, which experiences a single force, F, which pulls the mass in the direction of the point x=0 and depends only on the masss position x. Balance of forces for the system is F = m a = m d 2 x d t 2 = m x ¨ = − k x. Solving this differential equation, we find that the motion is described by the function x = A cos , the motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude, A. The position at a time t also depends on the phase, φ. The period and frequency are determined by the size of the mass m, the velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the direction as the displacement. The potential energy stored in a harmonic oscillator at position x is U =12 k x 2. In real oscillators, friction, or damping, slows the motion of the system, due to frictional force, the velocity decreases in proportion to the acting frictional force. While simple harmonic motion oscillates with only the force acting on the system
Harmonic oscillator
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Another damped harmonic oscillator
Harmonic oscillator
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Dependence of the system behavior on the value of the damping ratio ζ
21.
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.
22.
Motion (physics)
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In physics, motion is a change in position of an object over time. Motion is described in terms of displacement, distance, velocity, acceleration, time, motion of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame. If the position of a body is not changing with respect to a frame of reference. An objects motion cannot change unless it is acted upon by a force, momentum is a quantity which is used for measuring motion of an object. As there is no frame of reference, absolute motion cannot be determined. Thus, everything in the universe can be considered to be moving, more generally, motion is a concept that applies to objects, bodies, and matter particles, to radiation, radiation fields and radiation particles, and to space, its curvature and space-time. One can also speak of motion of shapes and boundaries, so, the term motion in general signifies a continuous change in the configuration of a physical system. For example, one can talk about motion of a wave or about motion of a quantum particle, in physics, motion is described through two sets of apparently contradictory laws of mechanics. Motions of all large scale and familiar objects in the universe are described by classical mechanics, whereas the motion of very small atomic and sub-atomic objects is described by quantum mechanics. It produces very accurate results within these domains, and is one of the oldest and largest in science, engineering, classical mechanics is fundamentally based on Newtons laws of motion. These laws describe the relationship between the acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica and his three laws are, A body either is at rest or moves with constant velocity, until and unless an outer force is applied to it. An object will travel in one direction only until an outer force changes its direction, whenever one body exerts a force F onto a second body, the second body exerts the force −F on the first body. F and −F are equal in magnitude and opposite in sense, so, the body which exerts F will go backwards. Newtons three laws of motion, along with his Newtons law of motion, which were the first to provide a mathematical model for understanding orbiting bodies in outer space. This explanation unified the motion of bodies and motion of objects on earth. Classical mechanics was later enhanced by Albert Einsteins special relativity. Motion of objects with a velocity, approaching the speed of light
Motion (physics)
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Motion involves a change in position, such as in this perspective of rapidly leaving Yongsan Station.
23.
Newton's law of universal gravitation
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This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning. It is a part of classical mechanics and was formulated in Newtons work Philosophiæ Naturalis Principia Mathematica, in modern language, the law states, Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them, the first test of Newtons theory of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798. It took place 111 years after the publication of Newtons Principia, Newtons law of gravitation resembles Coulombs law of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Both are inverse-square laws, where force is proportional to the square of the distance between the bodies. Coulombs law has the product of two charges in place of the product of the masses, and the constant in place of the gravitational constant. Newtons law has since been superseded by Albert Einsteins theory of general relativity, at the same time Hooke agreed that the Demonstration of the Curves generated thereby was wholly Newtons. In this way, the question arose as to what, if anything and this is a subject extensively discussed since that time and on which some points, outlined below, continue to excite controversy. And that these powers are so much the more powerful in operating. Thus Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, Hookes statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hookes gravitation was also not yet universal, though it approached universality more closely than previous hypotheses and he also did not provide accompanying evidence or mathematical demonstration. It was later on, in writing on 6 January 1679|80 to Newton, Newton, faced in May 1686 with Hookes claim on the inverse square law, denied that Hooke was to be credited as author of the idea. Among the reasons, Newton recalled that the idea had been discussed with Sir Christopher Wren previous to Hookes 1679 letter, Newton also pointed out and acknowledged prior work of others, including Bullialdus, and Borelli. D T Whiteside has described the contribution to Newtons thinking that came from Borellis book, a copy of which was in Newtons library at his death. Newton further defended his work by saying that had he first heard of the inverse square proportion from Hooke, Hooke, without evidence in favor of the supposition, could only guess that the inverse square law was approximately valid at great distances from the center. Thus Newton gave a justification, otherwise lacking, for applying the inverse square law to large spherical planetary masses as if they were tiny particles, after his 1679-1680 correspondence with Hooke, Newton adopted the language of inward or centripetal force. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s, the lesson offered by Hooke to Newton here, although significant, was one of perspective and did not change the analysis. This background shows there was basis for Newton to deny deriving the inverse square law from Hooke, on the other hand, Newton did accept and acknowledge, in all editions of the Principia, that Hooke had separately appreciated the inverse square law in the solar system
Newton's law of universal gravitation
24.
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
25.
Circular motion
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In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, the rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body, since the objects velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, in physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times, though the bodys speed is constant, its velocity is not constant, velocity, a vector quantity, depends on both the bodys speed and its direction of travel. This changing velocity indicates the presence of an acceleration, this acceleration is of constant magnitude. This acceleration is, in turn, produced by a force which is also constant in magnitude. For motion in a circle of radius r, the circumference of the circle is C = 2π r, the axis of rotation is shown as a vector ω perpendicular to the plane of the orbit and with a magnitude ω = dθ / dt. The direction of ω is chosen using the right-hand rule, likewise, the acceleration is given by a = ω × v = ω ×, which is a vector perpendicular to both ω and v of magnitude ω |v| = ω2 r and directed exactly opposite to r. In the simplest case the speed, mass and radius are constant, consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second. The speed is one metre per second, the inward acceleration is one metre per square second, v2/r. It is subject to a force of one kilogram metre per square second. The momentum of the body is one kg·m·s−1, the moment of inertia is one kg·m2. The angular momentum is one kg·m2·s−1, the kinetic energy is 1/2 joule. The circumference of the orbit is 2π metres, the period of the motion is 2π seconds per turn. It is convenient to introduce the unit vector orthogonal to u ^ R as well and it is customary to orient u ^ θ to point in the direction of travel along the orbit. The velocity is the derivative of the displacement, v → = d d t r → = d R d t u ^ R + R d u ^ R d t. Because the radius of the circle is constant, the component of the velocity is zero
Circular motion
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Figure 1: Velocity v and acceleration a in uniform circular motion at angular rate ω; the speed is constant, but the velocity is always tangent to the orbit; the acceleration has constant magnitude, but always points toward the center of rotation
26.
Centrifugal force
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In Newtonian mechanics, the centrifugal force is an inertial force directed away from the axis of rotation that appears to act on all objects when viewed in a rotating reference frame. When they are analyzed in a coordinate system. The term has also been used for the force that is a reaction to a centripetal force. The centrifugal force is an outward force apparent in a reference frame. All measurements of position and velocity must be relative to some frame of reference. An inertial frame of reference is one that is not accelerating, the use of an inertial frame of reference, which will be the case for all elementary calculations, is often not explicitly stated but may generally be assumed unless stated otherwise. In terms of a frame of reference, the centrifugal force does not exist. All calculations can be performed using only Newtons laws of motion, in its current usage the term centrifugal force has no meaning in an inertial frame. In an inertial frame, an object that has no acting on it travels in a straight line. When measurements are made with respect to a reference frame, however. If it is desired to apply Newtons laws in the frame, it is necessary to introduce new, fictitious. Consider a stone being whirled round on a string, in a horizontal plane, the only real force acting on the stone in the horizontal plane is the tension in the string. There are no forces acting on the stone so there is a net force on the stone in the horizontal plane. In an inertial frame of reference, were it not for this net force acting on the stone, in order to keep the stone moving in a circular path, this force, known as the centripetal force, must be continuously applied to the stone. As soon as it is removed the stone moves in a straight line, in this inertial frame, the concept of centrifugal force is not required as all motion can be properly described using only real forces and Newtons laws of motion. In a frame of reference rotating with the stone around the axis as the stone. However, the tension in the string is still acting on the stone, if Newtons laws were applied in their usual form, the stone would accelerate in the direction of the net applied force, towards the axis of rotation, which it does not do. With this new the net force on the stone is zero, with the addition of this extra inertial or fictitious force Newtons laws can be applied in the rotating frame as if it were an inertial frame
Centrifugal force
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The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.
27.
Angular displacement
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Angular displacement of a body is the angle in radians through which a point or line has been rotated in a specified sense about a specified axis. When an object rotates about its axis, the motion cannot simply be analyzed as a particle, since in circular motion it undergoes a changing velocity, when dealing with the rotation of an object, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal, Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion. In the example illustrated to the right, a particle on object P is at a distance r from the origin, O. It becomes important to represent the position of particle P in terms of its polar coordinates. In this particular example, the value of θ is changing, if using radians, it provides a very simple relationship between distance traveled around the circle and the distance r from the centre. Therefore,1 revolution is 2 π radians, when object travels from point P to point Q, as it does in the illustration to the left, over δ t the radius of the circle goes around a change in angle. Δ θ = θ2 − θ1 which equals the Angular Displacement, in three dimensions, angular displacement is an entity with a direction and a magnitude. The direction specifies the axis of rotation, which exists by virtue of the Eulers rotation theorem. This entity is called an axis-angle, despite having direction and magnitude, angular displacement is not a vector because it does not obey the commutative law for addition. Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded, several ways to describe angular displacement exist, like rotation matrices or Euler angles. See charts on SO for others, given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being A0 and A f two matrices, the angular displacement matrix between them can be obtained as Δ A = A f, when this product is performed having a very small difference between both frames we will obtain a matrix close to the identity. In the limit, we will have a rotation matrix. An infinitesimal angular displacement is a rotation matrix, As any rotation matrix has a single real eigenvalue, which is +1. Its module can be deduced from the value of the infinitesimal rotation, when it is divided by the time, this will yield the angular velocity vector. Suppose we specify an axis of rotation by a unit vector, expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix ΔR is represented as, Δ R = + Δ θ = I + A Δ θ
Angular displacement
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Rotation of a rigid object P about a fixed object about a fixed axis O.
28.
Alexis Clairaut
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Alexis Claude Clairaut was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles, Clairaut was one of the key figures in the expedition to Lapland that helped to confirm Newtons theory for the figure of the Earth. In that context, Clairaut worked out a mathematical result now known as Clairauts theorem and he also tackled the gravitational three-body problem, being the first to obtain a satisfactory result for the apsidal precession of the Moons orbit. In mathematics he is credited with Clairauts equation and Clairauts relation. Clairaut was born in Paris, France, to Jean-Babtiste and Catherine Petit Clairaut, the couple had 20 children, however only a few of them survived childbirth. Alexis was a prodigy — at the age of ten he began studying calculus, Clairaut was unmarried, and known for leading an active social life. Though he led a social life, he was very prominent in the advancement of learning in young mathematicians. He was elected a Fellow of the Royal Society of London in November,1737, Clairaut died in Paris in 1765. In 1736, together with Pierre Louis Maupertuis, he took part in the expedition to Lapland, the goal of the excursion was to geometrically calculate the shape of the Earth, which Sir Issac Newton theorized in his book Principia was an ellipsoid shape. They sought to prove if Newtons theory and calculations were correct or not, before the expedition team returned to Paris, Clairaut sent his calculations to the Royal Society of London. The writing was published by the society in the 1736-37 volume of Philosophical Transactions. Initially, Clairaut disagrees with Newtons theory on the shape of the Earth, in the article, he outlines several key problems that effectively disprove Newtons calculations, and provides some solutions to the complications. The issues addressed include calculating gravitational attraction, the rotation of an ellipsoid on its axis, and this conclusion suggests not only that the Earth is of an oblate ellipsoid shape, but it is flattened more at the poles and is wider at the center. His article in Philosophical Transactions created much controversy, as he addressed the problems of Newtons theory, after his return, he published his treatise Théorie de la figure de la terre. This proved Sir Issac Newtons theory that the shape of the Earth was an oblate ellipsoid, in 1849 Stokes showed that Clairauts result was true whatever the interior constitution or density of the Earth, provided the surface was a spheroid of equilibrium of small ellipticity. In 1741, Alexis Clairaut wrote a book called Èléments de Géométrie, the book outlines the basic concepts of geometry. Geometry in the 1700s was complex to the average learner and it was considered to be a dry subject. Clairaut saw this trend, and wrote the book in an attempt to make the more interesting for the average learner
Alexis Clairaut
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Alexis Claude Clairaut
29.
Joseph-Louis Lagrange
–
Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia or Giuseppe Ludovico De la Grange Tournier, was an Italian and French Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, in 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life, Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He also extended the method to take into account possible constraints and he proved that every natural number is a sum of four squares. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, in calculus, Lagrange developed a novel approach to interpolation and Taylor series. Born as Giuseppe Lodovico Lagrangia, Lagrange was of Italian and French descent and his mother was from the countryside of Turin. He was raised as a Roman Catholic, a career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the University of Turin and his subject was classical Latin. At first he had no enthusiasm for mathematics, finding Greek geometry rather dull. It was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident. Alone and unaided he threw himself into mathematical studies, at the end of a years incessant toil he was already an accomplished mathematician, in that capacity, Lagrange was the first to teach calculus in an engineering school. In this Academy one of his students was François Daviet de Foncenex, Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of tautochrone, Lagrange wrote several letters to Leonhard Euler between 1754 and 1756 describing his results. He outlined his δ-algorithm, leading to the Euler–Lagrange equations of variational calculus, Lagrange also applied his ideas to problems of classical mechanics, generalizing the results of Euler and Maupertuis. Euler was very impressed with Lagranges results, Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773. Many of these are elaborate papers, the article concludes with a masterly discussion of echoes, beats, and compound sounds. Other articles in volume are on recurring series, probabilities. The next work he produced was in 1764 on the libration of the Moon, and an explanation as to why the face was always turned to the earth
Joseph-Louis Lagrange
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Joseph-Louis (Giuseppe Luigi), comte de Lagrange
Joseph-Louis Lagrange
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Lagrange's tomb in the crypt of the Panthéon
30.
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
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William Rowan Hamilton (1805–1865)
William Rowan Hamilton
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Irish commemorative coin celebrating the 200th Anniversary of his birth.
31.
Daniel Bernoulli
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Daniel Bernoulli FRS was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, Daniel Bernoulli was born in Groningen, in the Netherlands, into a family of distinguished mathematicians. The Bernoulli family came originally from Antwerp, at time in the Spanish Netherlands. After a brief period in Frankfurt the family moved to Basel, Daniel was the son of Johann Bernoulli, nephew of Jacob Bernoulli. He had two brothers, Niklaus and Johann II, Daniel Bernoulli was described by W. W. Rouse Ball as by far the ablest of the younger Bernoullis. He is said to have had a bad relationship with his father, Johann Bernoulli also plagiarized some key ideas from Daniels book Hydrodynamica in his own book Hydraulica which he backdated to before Hydrodynamica. Despite Daniels attempts at reconciliation, his father carried the grudge until his death, around schooling age, his father, Johann, encouraged him to study business, there being poor rewards awaiting a mathematician. However, Daniel refused, because he wanted to study mathematics and he later gave in to his fathers wish and studied business. His father then asked him to study in medicine, and Daniel agreed under the condition that his father would teach him mathematics privately, Daniel studied medicine at Basel, Heidelberg, and Strasbourg, and earned a PhD in anatomy and botany in 1721. He was a contemporary and close friend of Leonhard Euler and he went to St. Petersburg in 1724 as professor of mathematics, but was very unhappy there, and a temporary illness in 1733 gave him an excuse for leaving St. Petersberg. He returned to the University of Basel, where he held the chairs of medicine, metaphysics. In May,1750 he was elected a Fellow of the Royal Society and his earliest mathematical work was the Exercitationes, published in 1724 with the help of Goldbach. Two years later he pointed out for the first time the frequent desirability of resolving a compound motion into motions of translation and motion of rotation, together Bernoulli and Euler tried to discover more about the flow of fluids. In particular, they wanted to know about the relationship between the speed at which blood flows and its pressure, soon physicians all over Europe were measuring patients blood pressure by sticking point-ended glass tubes directly into their arteries. It was not until about 170 years later, in 1896 that an Italian doctor discovered a less painful method which is still in use today. However, Bernoullis method of measuring pressure is used today in modern aircraft to measure the speed of the air passing the plane. Taking his discoveries further, Daniel Bernoulli now returned to his work on Conservation of Energy. It was known that a moving body exchanges its kinetic energy for energy when it gains height
Daniel Bernoulli
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Daniel Bernoulli
32.
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
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Relation between Cartesian coordinates (x, y) and polar coordinates (r, θ).
33.
Position (vector)
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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)
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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.
34.
Time in physics
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Time in physics is defined by its measurement, time is what a clock reads. In classical, non-relativistic physics it is a quantity and, like length, mass. Time can be combined mathematically with other physical quantities to other concepts such as motion, kinetic energy. Timekeeping is a complex of technological and scientific issues, and part of the foundation of recordkeeping, simultaneously, our conception of time has evolved, as shown below. In the International System of Units, the unit of time is the second and this definition is based on the operation of a caesium atomic clock. The UTC timestamp in use worldwide is a time standard. The relative accuracy of such a standard is currently on the order of 10−15. The smallest time step considered observable is called the Planck time, both Galileo and Newton and most people up until the 20th century thought that time was the same for everyone everywhere. This is the basis for timelines, where time is a parameter, thus time is part of a coordinate, in this view. Physicists believe the entire Universe and therefore time itself began about 13.8 billion years ago in the big bang, whether it will ever come to an end is an open question. In order to time, one can record the number of occurrences of some periodic phenomenon. The regular recurrences of the seasons, the motions of the sun, moon and stars were noted and tabulated for millennia, in particular, the astronomical observatories maintained for religious purposes became accurate enough to ascertain the regular motions of the stars, and even some of the planets. At first, timekeeping was done by hand by priests, and then for commerce, the tabulation of the equinoxes, the sandglass, and the water clock became more and more accurate, and finally reliable. For ships at sea, boys were used to turn the sandglasses, richard of Wallingford, abbot of St. Albans abbey, famously built a mechanical clock as an astronomical orrery about 1330. At first, only kings could afford them, pendulum clocks were widely used in the 18th and 19th century. They have largely replaced in general use by quartz and digital clocks. Atomic clocks can theoretically keep accurate time for millions of years and they are appropriate for standards and scientific use. The Galilean transformations assume that time is the same for all reference frames, in this section, the relationships listed below treat time as a parameter which serves as an index to the behavior of the physical system under consideration
Time in physics
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Foucault 's pendulum in the Panthéon of Paris can measure time as well as demonstrate the rotation of Earth.
Time in physics
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Andromeda galaxy (M31) is two million light-years away. Thus we are viewing M31's light from two million years ago, a time before humans existed on Earth.
Time in physics
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WMAP fluctuations of the cosmic microwave background radiation.
Time in physics
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Key concepts
35.
Dimensional analysis
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Converting from one dimensional unit to another is often somewhat complex. Dimensional analysis, or more specifically the method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra. The concept of physical dimension was introduced by Joseph Fourier in 1822, Physical quantities that are measurable have the same dimension and can be directly compared to each other, even if they are originally expressed in differing units of measure. If physical quantities have different dimensions, they cannot be compared by similar units, hence, it is meaningless to ask whether a kilogram is greater than, equal to, or less than an hour. Any physically meaningful equation will have the dimensions on their left and right sides. Checking for dimensional homogeneity is an application of dimensional analysis. Dimensional analysis is routinely used as a check of the plausibility of derived equations and computations. It is generally used to categorize types of quantities and units based on their relationship to or dependence on other units. Many parameters and measurements in the sciences and engineering are expressed as a concrete number – a numerical quantity. Often a quantity is expressed in terms of other quantities, for example, speed is a combination of length and time. Compound relations with per are expressed with division, e. g.60 mi/1 h, other relations can involve multiplication, powers, or combinations thereof. A base unit is a unit that cannot be expressed as a combination of other units, for example, units for length and time are normally chosen as base units. Units for volume, however, can be factored into the units of length. Sometimes the names of units obscure that they are derived units, for example, an ampere is a unit of electric current, which is equivalent to electric charge per unit time and is measured in coulombs per second, so 1 A =1 C/s. Similarly, one newton is 1 kg⋅m/s2, percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as 1/100, derivatives with respect to a quantity add the dimensions of the variable one is differentiating with respect to on the denominator. Thus, position has the dimension L, derivative of position with respect to time has dimension LT−1 – length from position, time from the derivative, the second derivative has dimension LT−2. In economics, one distinguishes between stocks and flows, a stock has units of units, while a flow is a derivative of a stock, in some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions
Dimensional analysis
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Base quantity
36.
International System of 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
International System of 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).
International System of Units
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The seven base units in the International System of Units
International System of Units
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Carl Friedrich Gauss
International System of Units
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Thomson
37.
Kilometre per hour
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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
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A car speedometer that indicates measured speed in kilometres per hour.
Kilometre per hour
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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.
38.
Knot (unit)
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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)
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Graphic scale from a Mercator projection world map, showing the change with latitude
39.
Special relativity
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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
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Albert Einstein around 1905, the year his " Annus Mirabilis papers " – which included Zur Elektrodynamik bewegter Körper, the paper founding special relativity – were published.
40.
Speed of light
–
The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299792458 metres per second, it is exact because the unit of length, the metre, is defined from this constant, according to special relativity, c is the maximum speed at which all matter and hence information in the universe can travel. It is the speed at which all particles and changes of the associated fields travel in vacuum. Such particles and waves travel at c regardless of the motion of the source or the reference frame of the observer. In the theory of relativity, c interrelates space and time, the speed at which light propagates through transparent materials, such as glass or air, is less than c, similarly, the speed of radio waves in wire cables is slower than c. The ratio between c and the speed v at which light travels in a material is called the index n of the material. In communicating with distant space probes, it can take minutes to hours for a message to get from Earth to the spacecraft, the light seen from stars left them many years ago, allowing the study of the history of the universe by looking at distant objects. The finite speed of light limits the theoretical maximum speed of computers. The speed of light can be used time of flight measurements to measure large distances to high precision. Ole Rømer first demonstrated in 1676 that light travels at a speed by studying the apparent motion of Jupiters moon Io. In 1865, James Clerk Maxwell proposed that light was an electromagnetic wave, in 1905, Albert Einstein postulated that the speed of light c with respect to any inertial frame is a constant and is independent of the motion of the light source. He explored the consequences of that postulate by deriving the theory of relativity and in doing so showed that the parameter c had relevance outside of the context of light and electromagnetism. After centuries of increasingly precise measurements, in 1975 the speed of light was known to be 299792458 m/s with a measurement uncertainty of 4 parts per billion. In 1983, the metre was redefined in the International System of Units as the distance travelled by light in vacuum in 1/299792458 of a second, as a result, the numerical value of c in metres per second is now fixed exactly by the definition of the metre. The speed of light in vacuum is usually denoted by a lowercase c, historically, the symbol V was used as an alternative symbol for the speed of light, introduced by James Clerk Maxwell in 1865. In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used c for a different constant later shown to equal √2 times the speed of light in vacuum, in 1894, Paul Drude redefined c with its modern meaning. Einstein used V in his original German-language papers on special relativity in 1905, but in 1907 he switched to c, sometimes c is used for the speed of waves in any material medium, and c0 for the speed of light in vacuum. This article uses c exclusively for the speed of light in vacuum, since 1983, the metre has been defined in the International System of Units as the distance light travels in vacuum in 1⁄299792458 of a second
Speed of light
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One of the last and most accurate time of flight measurements, Michelson, Pease and Pearson's 1930-35 experiment used a rotating mirror and a one-mile (1.6 km) long vacuum chamber which the light beam traversed 10 times. It achieved accuracy of ±11 km/s
Speed of light
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Sunlight takes about 8 minutes 17 seconds to travel the average distance from the surface of the Sun to the Earth.
Speed of light
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Diagram of the Fizeau apparatus
Speed of light
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Rømer's observations of the occultations of Io from Earth
41.
Speedometer
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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
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A speedometer showing mph and km/h along with an odometer and a separate "trip" odometer (both showing distance traveled in miles).
42.
Tangent line
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In geometry, the tangent line to a plane curve at a given point is the straight line that just touches the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve, a similar definition applies to space curves and curves in n-dimensional Euclidean space. Similarly, the tangent plane to a surface at a point is the plane that just touches the surface at that point. The concept of a tangent is one of the most fundamental notions in geometry and has been extensively generalized. The word tangent comes from the Latin tangere, to touch, euclid makes several references to the tangent to a circle in book III of the Elements. In Apollonius work Conics he defines a tangent as being a line such that no other straight line could fall between it and the curve, archimedes found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself and these methods led to the development of differential calculus in the 17th century. Many people contributed, Roberval discovered a method of drawing tangents. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents, further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was a line which touches a curve. This old definition prevents inflection points from having any tangent and it has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve. The tangent at A is the limit when point B approximates or tends to A, the existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as differentiability. At most points, the tangent touches the curve without crossing it, a point where the tangent crosses the curve is called an inflection point. Circles, parabolas, hyperbolas and ellipses do not have any point, but more complicated curves do have, like the graph of a cubic function. Conversely, it may happen that the curve lies entirely on one side of a line passing through a point on it. This is the case, for example, for a passing through the vertex of a triangle. In convex geometry, such lines are called supporting lines, the geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century, suppose that a curve is given as the graph of a function, y = f
Tangent line
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Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.
43.
Chord (geometry)
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A chord of a circle is a straight line segment whose endpoints both lie on the circle. A secant line, or just secant, is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an ellipse, a chord that passes through a circles center point is the circles diameter. Every diameter is a chord, but not every chord is a diameter, the word chord is from the Latin chorda meaning bowstring. Among properties of chords of a circle are the following, Chords are equidistant from the center if, a chord that passes through the center of a circle is called a diameter, and is the longest chord. If the line extensions of chords AB and CD intersect at a point P, the area that a circular chord cuts off is called a circular segment. The midpoints of a set of chords of an ellipse are collinear. Chords were used extensively in the development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the function for every 7.5 degrees. The circle was of diameter 120, and the lengths are accurate to two base-60 digits after the integer part. The chord function is defined geometrically as shown in the picture, the chord of an angle is the length of the chord between two points on a unit circle separated by that angle. The last step uses the half-angle formula, much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve volume work on chords, all now lost, so presumably a great deal was known about them
Chord (geometry)
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The red segment BX is a chord (as is the diameter segment AB).
44.
Tangent lines to circles
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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
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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).
45.
Circumference
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The circumference of a closed curve or circular object is the linear distance around its edge. The circumference of a circle is of importance in geometry and trigonometry. Informally circumference may also refer to the edge rather than to the length of the edge. The circumference of a circle is the distance around it, the term is used when measuring physical objects, as well as when considering abstract geometric forms. The circumference of a circle relates to one of the most important mathematical constants in all of mathematics and this constant, pi, is represented by the Greek letter π. The numerical value of π is 3.141592653589793, pi is defined as the ratio of a circles circumference C to its diameter d, π = C d Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference, the use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. The constant ratio of circumference to radius C / r =2 π also has uses in mathematics, engineering. These uses include but are not limited to radians, computer programming, the Greek letter τ is sometimes used to represent this constant, but is not generally accepted as proper notation. The circumference of an ellipse can be expressed in terms of the elliptic integral of the second kind. In graph theory the circumference of a graph refers to the longest cycle contained in that graph, arc length Area Caccioppoli set Isoperimetric inequality Pythagorean theorem Volume Numericana - Circumference of an ellipse Circumference of a circle With interactive applet and animation
Circumference
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Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta. Circumference = π × diameter = 2 × π × radius.
46.
Conversion of units
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Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors. The process of conversion depends on the situation and the intended purpose. This may be governed by regulation, contract, technical specifications or other published standards, engineering judgment may include such factors as, The precision and accuracy of measurement and the associated uncertainty of measurement. The statistical confidence interval or tolerance interval of the initial measurement, the number of significant figures of the measurement. The intended use of the measurement including the engineering tolerances, historical definitions of the units and their derivatives used in old measurements, e. g. international foot vs. Some conversions from one system of units to another need to be exact and this is sometimes called soft conversion. It does not involve changing the configuration of the item being measured. By contrast, a conversion or an adaptive conversion may not be exactly equivalent. It changes the measurement to convenient and workable numbers and units in the new system and it sometimes involves a slightly different configuration, or size substitution, of the item. Nominal values are allowed and used. A conversion factor is used to change the units of a quantity without changing its value. The unity bracket method of unit conversion consists of a fraction in which the denominator is equal to the numerator, because of the identity property of multiplication, the value of a number will not change as long as it is multiplied by one. Also, if the numerator and denominator of a fraction are equal to each other, so as long as the numerator and denominator of the fraction are equivalent, they will not affect the value of the measured quantity. There are many applications that offer the thousands of the various units with conversions. For example, the free software movement offers a command line utility GNU units for Linux and this article gives lists of conversion factors for each of a number of physical quantities, which are listed in the index. For each physical quantity, a number of different units are shown, Conversion between units in the metric system can be discerned by their prefixes and are thus not listed in this article. Exceptions are made if the unit is known by another name. Within each table, the units are listed alphabetically, and the SI units are highlighted, notes, See Weight for detail of mass/weight distinction and conversion
Conversion of units
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Base units
47.
SI derived unit
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The International System of Units specifies a set of seven base units from which all other SI units of measurement are derived. Each of these units is either dimensionless or can be expressed as a product of powers of one or more of the base units. For example, the SI derived unit of area is the metre. The degree Celsius has an unclear status, and is arguably an exception to this rule. The names of SI units are written in lowercase, the symbols for units named after persons, however, are always written with an uppercase initial letter. In addition to the two dimensionless derived units radian and steradian,20 other derived units have special names, some other units such as the hour, litre, tonne, bar and electronvolt are not SI units, but are widely used in conjunction with SI units. Until 1995, the SI classified the radian and the steradian as supplementary units, but this designation was abandoned, International System of Quantities International System of Units International Vocabulary of Metrology Metric prefix Metric system Non-SI units mentioned in the SI Planck units SI base unit I. Mills, Tomislav Cvitas, Klaus Homann, Nikola Kallay, IUPAC, Quantities, Units and Symbols in Physical Chemistry. CS1 maint, Multiple names, authors list
SI derived unit
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Base units
48.
Kilometres per hour
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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
Kilometres per hour
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A car speedometer that indicates measured speed in kilometres per hour.
Kilometres per hour
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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.
49.
Mach number
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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
50.
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
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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
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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)
51.
Natural units
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In physics, natural units are physical units of measurement based only on universal physical constants. For example, the charge e is a natural unit of electric charge. It precludes the interpretation of an expression in terms of physical constants, such e and c. In this case, the reinsertion of the powers of e, c. Natural units are natural because the origin of their definition comes only from properties of nature, Planck units are often, without qualification, called natural units, although they constitute only one of several systems of natural units, albeit the best known such system. As with other systems of units, the units of a set of natural units will include definitions and values for length, mass, time, temperature. It is possible to disregard temperature as a physical quantity, since it states the energy per degree of freedom of a particle. Virtually every system of natural units normalizes Boltzmanns constant kB to 1, there are two common ways to relate charge to mass, length, and time, In Lorentz–Heaviside units, Coulombs law is F = q1q2/4πr2, and in Gaussian units, Coulombs law is F = q1q2/r2. Both possibilities are incorporated into different natural unit systems, where, α is the fine-structure constant,2 ≈0.007297, αG is the gravitational coupling constant,2 ≈ 6955175200000000000♠1. 752×10−45. Natural units are most commonly used by setting the units to one, for example, many natural unit systems include the equation c =1 in the unit-system definition, where c is the speed of light. If a velocity v is half the speed of light, then as v = c/2 and c =1, the equation v = 1/2 means the velocity v has the value one-half when measured in Planck units, or the velocity v is one-half the Planck unit of velocity. The equation c =1 can be plugged in anywhere else, for example, Einsteins equation E = mc2 can be rewritten in Planck units as E = m. This equation means The energy of a particle, measured in Planck units of energy, equals the mass of the particle, measured in Planck units of mass. For example, the special relativity equation E2 = p2c2 + m2c4 appears somewhat complicated, Physical interpretation, Natural unit systems automatically subsume dimensional analysis. For example, in Planck units, the units are defined by properties of quantum mechanics, not coincidentally, the Planck unit of length is approximately the distance at which quantum gravity effects become important. Likewise, atomic units are based on the mass and charge of an electron, no prototypes, A prototype is a physical object that defines a unit, such as the International Prototype Kilogram, a physical cylinder of metal whose mass is by definition exactly one kilogram. A prototype definition always has imperfect reproducibility between different places and between different times, and it is an advantage of natural systems that they use no prototypes. Less precise measurements, SI units are designed to be used in precision measurements, for example, the second is defined by an atomic transition frequency in cesium atoms, because this transition frequency can be precisely reproduced with atomic clock technology
Natural units
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Base units
52.
Sprint runner
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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
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Usain Bolt, world record holder in 100 m and 200 m sprints
Sprint runner
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Start of the women's 60 m at the 2010 World Indoor Championships
Sprint runner
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Tyson Gay completes a 100m race
Sprint runner
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A 200 m bend
53.
Usain Bolt
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Usain St Leo Bolt, OJ, CD is a Jamaican sprinter. He is the first person to both the 100 metres and 200 metres world records since fully automatic time became mandatory. He also holds the record as a part of the 4 ×100 metres relay. He is the world and Olympic champion in these three events. Due to his dominance and achievements in sprint competition, he is widely considered to be the greatest sprinter of all time. He gained worldwide popularity for his double sprint victory at the 2008 Beijing Olympics in world record times, Bolt is the only sprinter to win Olympic 100 m and 200 m titles at three consecutive olympics, a feat referred to as the triple double. An eleven-time World Champion, he won consecutive World Championship 100 m,200 m and 4 ×100 metres relay medals from 2009 to 2015. He is the most successful athlete of the World Championships and was the first athlete to win three titles in both the 100 m and 200 m at the competition. Bolt improved upon his first 100 m world record of 9.69 with 9.58 seconds in 2009 – the biggest improvement since the start of electronic timing. He has twice broken the 200 metres world record, setting 19.30 in 2008 and 19.19 in 2009 and he has helped Jamaica to three 4 ×100 metres relay world records, with the current record being 36.84 seconds set in 2012. Bolts most successful event is the 200 m, with three Olympic and four World titles, the 2008 Olympics was his international debut over 100 m, he had earlier won numerous 200 m medals and holds the world under-20 and world under-18 records for the event. Bolt has stated that he intends to retire from athletics after the 2017 World Championships. Bolt was born on 21 August 1986 in Sherwood Content, a town in Trelawny, Jamaica, and grew up with parents Wellesley and Jennifer Bolt, his brother Sadiki. As a child, Bolt attended Waldensia Primary, where he first began to show his sprinting potential, by the age of twelve, Bolt had become the schools fastest runner over the 100 metres distance. Upon his entry to William Knibb Memorial High School, Bolt continued to focus on other sports, pablo McNeil, a former Olympic sprint athlete, and Dwayne Jarrett coached Bolt, encouraging him to focus his energy on improving his athletic abilities. The school had a history of success in athletics with past students, Bolt won his first annual high school championships medal in 2001, taking the silver medal in the 200 metres with a time of 22.04 seconds. Performing for Jamaica in his first Caribbean regional event, Bolt clocked a personal best of 48.28 s in the 400 metres in the 2001 CARIFTA Games, the 200 m also yielded a silver, as Bolt finished in 21.81 s. He made his first appearance on the stage at the 2001 IAAF World Youth Championships in Debrecen
Usain Bolt
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Bolt at the 2013 World Championships in Athletics.
Usain Bolt
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Bolt at the Crystal Palace Meeting in 2007
Usain Bolt
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Bolt trailing behind Gay in the closing stages of the 200 m race, 2007
Usain Bolt
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Bolt (left) on the podium with his silver medal from the 200 m race in Osaka (2007)
54.
100 metres
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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
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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
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Usain Bolt breaking the world and Olympic records at the 2008 Beijing Olympics
100 metres
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Christine Arron (left) wins the 100 m at the Weltklasse meeting.
55.
Taipei 101
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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
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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
56.
Saffir-Simpson Hurricane Scale
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To be classified as a hurricane, a tropical cyclone must have maximum sustained winds of at least 74 mph. The highest classification in the scale, Category 5, is reserved for storms with winds exceeding 156 mph, the classifications can provide some indication of the potential damage and flooding a hurricane will cause upon landfall. Officially, the Saffir–Simpson hurricane wind scale is used only to describe hurricanes forming in the Atlantic Ocean, other areas use different scales to label these storms, which are called cyclones or typhoons, depending on the area. In 1967 Robert Simpson became the director of the National Hurricane Center, during 1968 Robert spoke to Herbert Saffir about work that he had just completed for the United Nations, about damage to structures that was expected by winds of different strengths. The scale was developed in 1971 by civil engineer Herbert Saffir and meteorologist Robert Simpson, the scale was introduced to the general public in 1973, and saw widespread use after Neil Frank replaced Simpson at the helm of the NHC in 1974. The initial scale was developed by Saffir, a structural engineer, while performing the study, Saffir realized there was no simple scale for describing the likely effects of a hurricane. Mirroring the utility of the Richter magnitude scale in describing earthquakes, Saffir gave the scale to the NHC, and Simpson added the effects of storm surge and flooding. In 2009, the NHC made moves to eliminate pressure and storm surge ranges from the categories, transforming it into a wind scale. The new scale became operational on May 15,2010, in 2012, the NHC expanded the windspeed range for Category 4 by 1 mph in both directions, to 130–156 mph, with corresponding changes in the other units, instead of 131–155 mph. The NHC and the Central Pacific Hurricane Center assign tropical cyclone intensities in 5 knot increments, so an intensity of 115 knots is rated Category 4, but the conversion to miles per hour would round down to 130 mph, making it appear to be a Category 3 storm. Likewise, an intensity of 135 knots is 250.02 km/h, to resolve these issues, the NHC had been obliged to incorrectly report storms with wind speeds of 115 kn as 135 mph, and 135 kn as 245 km/h. The change in definition allows storms of 115 kn to be rounded down to 130 mph, and storms of 135 kn to be correctly reported as 250 km/h. Since the NHC had previously rounded incorrectly to keep storms in Category 4 in each unit of measure, the new scale became operational on May 15,2012. The scale separates hurricanes into five different categories based on wind, the U. S. National Hurricane Center classifies hurricanes of Category 3 and above as major hurricanes, and the Joint Typhoon Warning Center classifies typhoons of 150 mph or greater as super typhoons. Central pressure and storm surge values are approximate and often dependent on other factors, such as the size of the storm, intensity of example hurricanes is from both the time of landfall and the maximum intensity. As a result, it is not uncommon for a pressure to be higher or lower than expected for a specific category. Generally, large storms with large radii of maximum winds have the lowest pressures relative to its intensity. Poorly attached roof shingles or tiles can blow off, coastal flooding and pier damage are often associated with Category 1 storms
Saffir-Simpson Hurricane Scale
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Hurricane Barbara in 2013 making landfall.
Saffir-Simpson Hurricane Scale
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Juan in 2003 approaching Nova Scotia
Saffir-Simpson Hurricane Scale
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Hurricane Isidore near its landfall on the Yucatán peninsula
Saffir-Simpson Hurricane Scale
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Daniel in the eastern Pacific
57.
Autoroutes of France
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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.
Flight airspeed record
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An air speed record is the highest airspeed attained by an aircraft of a particular class. The rules for all official records are defined by Fédération Aéronautique Internationale. Speed records are divided into classes with sub-divisions. There are three classes of aircraft, landplanes, seaplanes, and amphibians, then within these classes, there are still further sub-divisions for piston-engined, turbojet, turboprop, and rocket-engined aircraft. Within each of groups, records are defined for speed over a straight course. Records in gray font color are unofficial, including unconfirmed or unpublicized secrets, the Lockheed SR-71 Blackbird holds the official Air Speed Record for a manned airbreathing jet aircraft with a speed of 3,530 km/h. It was capable of taking off and landing unassisted on conventional runways, the record was set on 28 July 1976 by Eldon W. Joersz and George T. Morgan Jr. near Beale Air Force Base, California, US. SR-71 pilot Brian Shul reported in The Untouchables that he flew in excess of Mach 3.5 on April 15,1986, over Libya in order to avoid a missile. Whereas these were both demilitarised, modified fighters, the fastest piston-engined aeroplane in stock condition was the German Dornier Do 335 Pfeil, with a maximum speed of 474 mph in level flight. The unofficial record for fastest piston-engined aeroplane is held by a Supermarine Spitfire Mk IXX, the last new speed record ratified before the outbreak of World War II was set on 26 April 1939 with a Me 209 V1, at 755 km/h. The chaos, and secrecy, of World War II meant that new speed breakthroughs were not publicized nor ratified, in October 1941, an unofficial speed record of 1004 km/h was secretly set by a Messerschmitt Me 163 AV4 rocket aircraft. Continued research during the war extended the secret, unofficial speed record to 1130 km/h by July 1944, the first new official record in the post-war period was achieved by a Gloster Meteor in November 1945, at 976 km/h. The first aircraft to exceed the unofficial October 1941 record of the Me 163 AV4 was the Douglas Skystreak, the July 1944 unofficial record of the Me 163B V18 was officially surpassed in November 1947, when Chuck Yeager flew the Bell X-1 to 891 mph. The official speed record for a seaplane moved by piston engine - still valid today - is 709.209 km/h, from the seaplane Macchi-Castoldi M. C.72, attained on October 23,1934, by Francesco Agello. It was equipped with the Fiat AS.6 engine developing a power of 3100 hp at 3300 rpm, the original Macchi-Castoldi MC72 MM.181 seaplane that holds the record is kept in the Air Force Museum at Vigna di Valle in Italy. The fastest manned atmospheric vehicle of all time was the Apollo Command Module, while different from most peoples idea of an aircraft, the capsule did have a lift to drag ratio of around 0.368, which was used to control the flight trajectory. Flying between any two airports allow a number of combinations, so setting a speed record is fairly easy with an ordinary aircraft. List of vehicle speed records Lockheed X-7 - Mach 4.31 in the 1950s World record Allward, modern Combat Aircraft 4, F-86 Sabre
Flight airspeed record
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The SR-71 Blackbird is the current record-holder for a manned airbreathing jet aircraft.
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
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Luna 1, launched in 1959, was the first man-made object to attain escape velocity from Earth (see below table).
Escape velocity
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General
62.
Voyager 1
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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.
List of vehicle speed records
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The following is a list of speed records for various categories of vehicles. Note significant figures, unless otherwise indicated, trailing zeros to the left of a point are not significant. 7. ^ Mach number depends on ambient temperature, and thus altitude, as well as speed,8. ^ Unofficial helicopter speed records by the Sikorsky X2 and the Eurocopter X3 would surpass this record if accepted. In order to express the speed of a spacecraft, a frame of reference must be specified. Typically, this frame will be fixed to the body with the largest gravitational influence on the spacecraft. Velocities in different frames of reference are not directly comparable, the matter of the fastest spacecraft depends on the reference frame. Because of the influence of gravity, maximum velocities are usually attained just after launch, at entry, or at time of closest approach to a massive body
List of vehicle speed records
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ThrustSSC, which has held the land speed record since 15 October 1997
List of vehicle speed records
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Lt. Col. John P. Stapp rides the rocket sled at Edwards Air Force Base
List of vehicle speed records
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Sunswift IV, the world's fastest solar-powered land vehicle, during its world record run
List of vehicle speed records
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VeloX3, the world's fastest human-powered vehicle, is prepared for a run during the 2013 World Human Powered Speed Challenge
68.
V speeds
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In aviation, V-speeds are standard terms used to define airspeeds important or useful to the operation of all aircraft. Using them is considered a best practice to maximize aviation safety, the actual speeds represented by these designators are specific to a particular model of aircraft. They are expressed by the indicated airspeed, so that pilots may use them directly, without having to apply correction factors. In general aviation aircraft, the most commonly used and most safety-critical airspeeds are displayed as color-coded arcs, the lower ends of the green arc and the white arc are the stalling speed with wing flaps retracted, and stalling speed with wing flaps fully extended, respectively. These are the speeds for the aircraft at its maximum weight. The yellow range is the range in which the aircraft may be operated in air, and then only with caution to avoid abrupt control movement, and the red line is the Vne. Proper display of V speeds is a requirement for type-certificated aircraft in most countries. The most common V-speeds are often defined by a particular governments aviation regulations, in the United States, these are defined in title 14 of the United States Code of Federal Regulations, known as the Federal Aviation Regulations or FARs. In Canada, the body, Transport Canada, defines 26 commonly used V-speeds in their Aeronautical Information Manual. V-speed definitions in FAR23,25 and equivalent are for designing and certification of airplanes, the descriptions below are for use by pilots. These V-speeds are defined by regulations and they are typically defined with constraints such as weight, configuration, or phases of flight, some of these constraints have been omitted to simplify the description. Some of these V-speeds are specific to types of aircraft and are not defined by regulations. Whenever a limiting speed is expressed by a Mach number, it is expressed relative to the speed of sound, e. g. VMO, Maximum operating speed, MMO, V1 is the critical engine failure recognition speed or takeoff decision speed. It is the speed above which the takeoff will continue if a engine fails or another problem occurs. Aborting a takeoff after V1 is strongly discouraged because the aircraft will by definition not be able to stop before the end of the runway, Transport Canada defines it as, Critical engine failure recognition speed and adds, This definition is not restrictive. An operator may adopt any other definition outlined in the flight manual of TC type-approved aircraft as long as such definition does not compromise operational safety of the aircraft. Getting to grips with aircraft performance, flight Operations Support & Line Assistance
V speeds
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A single-engine Cessna 150L's airspeed indicator indicating its V speeds.
V speeds
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A flight envelope diagram showing V S (stall speed at 1G), V C (corner/maneuvering speed) and V D (dive speed)
69.
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.
70.
Absement
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In kinematics, absement is a measure of sustained displacement of an object from its initial position, i. e. a measure of how far away and for how long. Absement changes as an object remains displaced and stays constant as the object resides at the initial position and it is the first time-integral of the displacement, so the displacement is the rate of change of the absement. The dimension of absement is length multiplied by time and its SI unit is meter second, which corresponds to an object having been displaced by 1 meter for 1 second. This is not to be confused with a meter per second, a unit of velocity, for example, opening the gate of a gate valve by 1 mm for 10 seconds yields the same absement of 10 mm·s as opening it by 5 mm for 2 seconds. The amount of water having flowed through it is proportional to the absement of the gate. The word absement is a portmanteau of the absence and displacement. Similarly, absition is a portmanteau of the absence and position. Whenever the rate of change f of a quantity f is proportional to the displacement of an object, for example, when the fuel flow rate is proportional to the position of the throttle lever, then the total amount of fuel consumed is proportional to the levers absement. In this context, it gives rise to a new quantity called actergy, actergy has the same units as action but is the time-integral of total energy. Fluid flow in a throttle, A vehicles distance travelled results from its throttles absement, the further the throttle has been opened, and the longer its been open, the more the vehicles travelled. See Table 4, Analytic Displacement and Absement versus Piecewise Continuous Displacement and Absement
Absement
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Ryan Janzen, playing with Hart House Symphonic Band
Absement
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Waterflute (reedless) hydraulophone with 45 finger-embouchure holes, allowing an intricate but polyphonic embouchure-like control by inserting one finger into each of several of the instrument's 45 mouths at once
Absement
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Aquatune Hydraulophone at the main entrance to the Legoland waterpark in Carlsbad California. This hydraulophone is in the shape of giant lego blocks.
Absement
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Hot tub hydraulophone
71.
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
72.
Area
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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
Area
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A square metre quadrat made of PVC pipe.
Area
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The combined area of these three shapes is approximately 15.57 squares.
73.
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.
74.
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.
75.
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
–
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.
76.
Hertz
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The hertz is the unit of frequency in the International System of Units and is defined as one cycle per second. It is named for Heinrich Rudolf Hertz, the first person to provide proof of the existence of electromagnetic waves. Hertz are commonly expressed in SI multiples kilohertz, megahertz, gigahertz, kilo means thousand, mega meaning million, giga meaning billion and tera for trillion. Some of the units most common uses are in the description of waves and musical tones, particularly those used in radio-. It is also used to describe the speeds at which computers, the hertz is equivalent to cycles per second, i. e. 1/second or s −1. In English, hertz is also used as the plural form, as an SI unit, Hz can be prefixed, commonly used multiples are kHz, MHz, GHz and THz. One hertz simply means one cycle per second,100 Hz means one hundred cycles per second, and so on. The unit may be applied to any periodic event—for example, a clock might be said to tick at 1 Hz, the rate of aperiodic or stochastic events occur is expressed in reciprocal second or inverse second in general or, the specific case of radioactive decay, becquerels. Whereas 1 Hz is 1 cycle per second,1 Bq is 1 aperiodic radionuclide event per second, the conversion between a frequency f measured in hertz and an angular velocity ω measured in radians per second is ω =2 π f and f = ω2 π. This SI unit is named after Heinrich Hertz, 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, the hertz is named after the German physicist Heinrich Hertz, who made important scientific contributions to the study of electromagnetism. The name was established by the International Electrotechnical Commission in 1930, the term cycles per second was largely replaced by hertz by the 1970s. One hobby magazine, Electronics Illustrated, declared their intention to stick with the traditional kc. Mc. etc. units, sound is a traveling longitudinal wave which is an oscillation of pressure. Humans perceive frequency of waves as pitch. Each musical note corresponds to a frequency which can be measured in hertz. An infants ear is able to perceive frequencies ranging from 20 Hz to 20,000 Hz, the range of ultrasound, infrasound and other physical vibrations such as molecular and atomic vibrations extends from a few femtoHz into the terahertz range and beyond. Electromagnetic radiation is described by its frequency—the number of oscillations of the perpendicular electric and magnetic fields per second—expressed in hertz. Radio frequency radiation is measured in kilohertz, megahertz, or gigahertz
Hertz
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Details of a heartbeat as an example of a non- sinusoidal periodic phenomenon that can be described in terms of hertz. Two complete cycles are illustrated.
Hertz
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A sine wave with varying frequency
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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.
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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.