1.
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
2.
Classical mechanics
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In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. Classical mechanics describes the motion of objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases, Classical mechanics also provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When both quantum and classical mechanics cannot apply, such as at the level with high speeds. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, however, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and accurate form. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and these advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newtons work, particularly through their use of analytical mechanics. The following introduces the concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, the motion of a point particle is characterized by a small number of parameters, its position, mass, and the forces applied to it. Each of these parameters is discussed in turn, in reality, the kind of objects that classical mechanics can describe always have a non-zero size. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the degrees of freedom. However, the results for point particles can be used to such objects by treating them as composite objects. The center of mass of a composite object behaves like a point particle, Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space, non-relativistic mechanics also assumes that forces act instantaneously. The position of a point particle is defined with respect to a fixed reference point in space called the origin O, in space. A simple coordinate system might describe the position of a point P by means of a designated as r. In general, the point particle need not be stationary relative to O, such that r is a function of t, the time
3.
Second law of motion
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Newtons laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a measure of the force. These three laws have been expressed in different ways, over nearly three centuries, and can be summarised as follows. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation. Newtons laws are applied to objects which are idealised as single point masses, in the sense that the size and this can be done when the object is small compared to the distances involved in its analysis, or the deformation and rotation of the body are of no importance. In this way, even a planet can be idealised as a particle for analysis of its orbital motion around a star, in their original form, Newtons laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newtons laws of motion for rigid bodies called Eulers laws of motion, if a body is represented as an assemblage of discrete particles, each governed by Newtons laws of motion, then Eulers laws can be derived from Newtons laws. Eulers laws can, however, be taken as axioms describing the laws of motion for extended bodies, Newtons laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Other authors do treat the first law as a corollary of the second, the explicit concept of an inertial frame of reference was not developed until long after Newtons death. In the given mass, acceleration, momentum, and force are assumed to be externally defined quantities. This is the most common, but not the interpretation of the way one can consider the laws to be a definition of these quantities. Newtonian mechanics has been superseded by special relativity, but it is useful as an approximation when the speeds involved are much slower than the speed of light. The first law states that if the net force is zero, the first law can be stated mathematically when the mass is a non-zero constant, as, ∑ F =0 ⇔ d v d t =0. Consequently, An object that is at rest will stay at rest unless a force acts upon it, an object that is in motion will not change its velocity unless a force acts upon it. This is known as uniform motion, an object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest, if an object is moving, it continues to move without turning or changing its speed
4.
Continuum mechanics
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Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century, research in the area continues till today. Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies, Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical properties are represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience, Materials, such as solids, liquids and gases, are composed of molecules separated by space. On a microscopic scale, materials have cracks and discontinuities, a continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material. More specifically, the continuum hypothesis/assumption hinges on the concepts of an elementary volume. This condition provides a link between an experimentalists and a viewpoint on constitutive equations as well as a way of spatial and statistical averaging of the microstructure. The latter then provide a basis for stochastic finite elements. The levels of SVE and RVE link continuum mechanics to statistical mechanics, the RVE may be assessed only in a limited way via experimental testing, when the constitutive response becomes spatially homogeneous. Specifically for fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made, consider car traffic on a highway---with just one lane for simplicity. Somewhat surprisingly, and in a tribute to its effectiveness, continuum mechanics effectively models the movement of cars via a differential equation for the density of cars. The familiarity of this situation empowers us to understand a little of the continuum-discrete dichotomy underlying continuum modelling in general. To start modelling define that, x measure distance along the highway, t is time, ρ is the density of cars on the highway, cars do not appear and disappear. Consider any group of cars, from the car at the back of the group located at x = a to the particular car at the front located at x = b. The total number of cars in this group N = ∫ a b ρ d x, since cars are conserved d N / d t =0. The only way an integral can be zero for all intervals is if the integrand is zero for all x, consequently, conservation derives the first order nonlinear conservation PDE ∂ ρ ∂ t + ∂ ∂ x =0 for all positions on the highway. This conservation PDE applies not only to car traffic but also to fluids, solids, crowds, animals, plants, bushfires, financial traders and this PDE is one equation with two unknowns, so another equation is needed to form a well posed problem
5.
Kinematics
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Kinematics as a field of study is often referred to as the geometry of motion and as such may be seen as a branch of mathematics. The study of the influence of forces acting on masses falls within the purview of kinetics, for further details, see analytical dynamics. Kinematics is used in astrophysics to describe the motion of celestial bodies, in mechanical engineering, robotics, and biomechanics kinematics is used to describe the motion of systems composed of joined parts such as an engine, a robotic arm or the human skeleton. Kinematic analysis is the process of measuring the quantities used to describe motion. In addition, kinematics applies geometry to the study of the mechanical advantage of a mechanical system or mechanism. The term kinematic is the English version of A. M, ampères cinématique, which he constructed from the Greek κίνημα kinema, itself derived from κινεῖν kinein. Kinematic and cinématique are related to the French word cinéma, particle kinematics is the study of the trajectory of a particle. The position of a particle is defined to be the vector from the origin of a coordinate frame to the particle. If the tower is 50 m high, then the vector to the top of the tower is r=. In the most general case, a coordinate system is used to define the position of a particle. However, if the particle is constrained to move in a surface, all observations in physics are incomplete without those observations being described with respect to a reference frame. The position vector of a particle is a vector drawn from the origin of the frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin, the magnitude of the position vector |P| gives the distance between the point P and the origin. | P | = x P2 + y P2 + z P2, the direction cosines of the position vector provide a quantitative measure of direction. It is important to note that the vector of a particle isnt unique. The position vector of a particle is different relative to different frames of reference. The velocity of a particle is a quantity that describes the direction of motion. More mathematically, the rate of change of the vector of a point
6.
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
7.
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
8.
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
9.
Angular momentum
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In physics, angular momentum is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque. The definition of momentum for a point particle is a pseudovector r×p. This definition can be applied to each point in continua like solids or fluids, unlike momentum, angular momentum does depend on where the origin is chosen, since the particles position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object via the moment of inertia I. However, while ω always points in the direction of the rotation axis, Angular momentum is additive, the total angular momentum of a system is the vector sum of the angular momenta. For continua or fields one uses integration, torque can be defined as the rate of change of angular momentum, analogous to force. Applications include the gyrocompass, control moment gyroscope, inertial systems, reaction wheels, flying discs or Frisbees. In general, conservation does limit the motion of a system. In quantum mechanics, angular momentum is an operator with quantized eigenvalues, Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the spin of elementary particles does not correspond to literal spinning motion, Angular momentum is a vector quantity that represents the product of a bodys rotational inertia and rotational velocity about a particular axis. Angular momentum can be considered an analog of linear momentum. Thus, where momentum is proportional to mass m and linear speed v, p = m v, angular momentum is proportional to moment of inertia I. Unlike mass, which only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation. Unlike linear speed, which occurs in a line, angular speed occurs about a center of rotation. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center and this simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case, L = r m v ⊥, where v ⊥ = v sin θ is the component of the motion. It is this definition, × to which the moment of momentum refers
10.
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
11.
D'Alembert's principle
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DAlemberts principle, also known as the Lagrange–dAlembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond dAlembert and it is the dynamic analogue to the principle of virtual work for applied forces in a static system and in fact is more general than Hamiltons principle, avoiding restriction to holonomic systems. A holonomic constraint depends only on the coordinates and time and it does not depend on the velocities. The principle does not apply for irreversible displacements, such as sliding friction, DAlemberts contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish. That is to say that the generalized forces Q j need not include constraint forces and it is equivalent to the somewhat more cumbersome Gausss principle of least constraint. The general statement of dAlemberts principle mentions the time derivatives of the momenta of the system. The momentum of the mass is the product of its mass and velocity, p i = m i v i. In many applications, the masses are constant and this reduces to p i ˙ = m i v ˙ i = m i a i. However, some applications involve changing masses and in those cases both terms m ˙ i v i and m i v ˙ i have to remain present, to date, nobody has shown that DAlemberts principle is equivalent to Newtons Second Law. This is true only for very special cases e. g. rigid body constraints. However, a solution to this problem does exist. Consider Newtons law for a system of particles, i, if arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces, the constraint forces do no work. Such displacements are said to be consistent with the constraints and this leads to the formulation of dAlemberts principle, which states that the difference of applied forces and inertial forces for a dynamic system does no virtual work. There is also a principle for static systems called the principle of virtual work for applied forces. DAlembert showed that one can transform an accelerating rigid body into an equivalent static system by adding the so-called inertial force, the inertial force must act through the center of mass and the inertial torque can act anywhere. The system can then be analyzed exactly as a static system subjected to this force and moment. The advantage is that, in the equivalent static system one can take moments about any point and this often leads to simpler calculations because any force can be eliminated from the moment equations by choosing the appropriate point about which to apply the moment equation. Even in the course of Fundamentals of Dynamics and Kinematics of machines, in textbooks of engineering dynamics this is sometimes referred to as dAlemberts principle
12.
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
13.
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
14.
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
15.
Force
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In physics, a force is any interaction that, when unopposed, will change the motion of an object. In other words, a force can cause an object with mass to change its velocity, force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity and it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newtons second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. In an extended body, each part usually applies forces on the adjacent parts, such internal mechanical stresses cause no accelation of that body as the forces balance one another. Pressure, the distribution of small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate. Stress usually causes deformation of materials, or flow in fluids. In part this was due to an understanding of the sometimes non-obvious force of friction. A fundamental error was the belief that a force is required to maintain motion, most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved-on for nearly three hundred years, the Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known, in order of decreasing strength, they are, strong, electromagnetic, weak, high-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was famous for formulating a treatment of buoyant forces inherent in fluids. Aristotle provided a discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotles view, the sphere contained four elements that come to rest at different natural places therein. Aristotle believed that objects on Earth, those composed mostly of the elements earth and water, to be in their natural place on the ground. He distinguished between the tendency of objects to find their natural place, which led to natural motion, and unnatural or forced motion
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Impulse (physics)
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In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Since force is a quantity, impulse is also a vector in the same direction. Impulse applied to an object produces an equivalent vector change in its linear momentum, the SI unit of impulse is the newton second, and the dimensionally equivalent unit of momentum is the kilogram meter per second. The corresponding English engineering units are the pound-second and the slug-foot per second, a resultant force causes acceleration and a change in the velocity of the body for as long as it acts. Conversely, a force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly. This is often called the impulse-momentum theorem, as a result, an impulse may also be regarded as the change in momentum of an object to which a resultant force is applied. Impulse has the units and dimensions as momentum. In the International System of Units, these are kg·m/s = N·s, in English engineering units, they are slug·ft/s = lbf·s. The term impulse is also used to refer to a force or impact. This type of impulse is often idealized so that the change in momentum produced by the force happens with no change in time and this sort of change is a step change, and is not physically possible. However, this is a model for computing the effects of ideal collisions. The application of Newtons second law for variable mass allows impulse, in the case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse. This fact can be used to derive the Tsiolkovsky rocket equation, which relates the vehicles propulsive change in velocity to the specific impulse. Wave–particle duality defines the impulse of a wave collision, the preservation of momentum in the collision is then called phase matching. Applications include, Compton effect nonlinear optics Acousto-optic modulator Electron phonon scattering Serway, Raymond A. Jewett, John W. Physics for Scientists, Physics for Scientists and Engineers, Mechanics, Oscillations and Waves, Thermodynamics
17.
Inertia
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Inertia is the resistance of any physical object to any change in its state of motion, this includes changes to its speed, direction, or state of rest. It is the tendency of objects to keep moving in a line at constant velocity. The principle of inertia is one of the principles of classical physics that are used to describe the motion of objects. Inertia comes from the Latin word, iners, meaning idle, Inertia is one of the primary manifestations of mass, which is a quantitative property of physical systems. In common usage, the inertia may refer to an objects amount of resistance to change in velocity, or sometimes to its momentum. Thus, an object will continue moving at its current velocity until some force causes its speed or direction to change. On the surface of the Earth, inertia is often masked by the effects of friction and air resistance, both of which tend to decrease the speed of moving objects, and gravity. Aristotle explained the continued motion of projectiles, which are separated from their projector, by the action of the surrounding medium, Aristotle concluded that such violent motion in a void was impossible. Despite its general acceptance, Aristotles concept of motion was disputed on several occasions by notable philosophers over nearly two millennia, for example, Lucretius stated that the default state of matter was motion, not stasis. Philoponus proposed that motion was not maintained by the action of a surrounding medium, although this was not the modern concept of inertia, for there was still the need for a power to keep a body in motion, it proved a fundamental step in that direction. This view was opposed by Averroes and by many scholastic philosophers who supported Aristotle. However, this view did not go unchallenged in the Islamic world, in the 14th century, Jean Buridan rejected the notion that a motion-generating property, which he named impetus, dissipated spontaneously. Buridans position was that an object would be arrested by the resistance of the air. Buridan also maintained that impetus increased with speed, thus, his idea of impetus was similar in many ways to the modern concept of momentum. Buridan also believed that impetus could be not only linear, but also circular in nature, buridans thought was followed up by his pupil Albert of Saxony and the Oxford Calculators, who performed various experiments that further undermined the classical, Aristotelian view. Their work in turn was elaborated by Nicole Oresme who pioneered the practice of demonstrating laws of motion in the form of graphs, benedetti cites the motion of a rock in a sling as an example of the inherent linear motion of objects, forced into circular motion. The law of inertia states that it is the tendency of an object to resist a change in motion. According to Newton, an object will stay at rest or stay in motion unless acted on by a net force, whether it results from gravity, friction, contact
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Moment of inertia
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It depends on the bodys mass distribution and the axis chosen, with larger moments requiring more torque to change the bodys rotation. It is a property, the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems. One of its definitions is the moment of mass with respect to distance from an axis r, I = ∫ Q r 2 d m. For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about a perpendicular to the plane. When a body is rotating, or free to rotate, around an axis, the amount of torque needed to cause any given angular acceleration is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram metre squared in SI units, moment of inertia plays the role in rotational kinetics that mass plays in linear kinetics - both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, for a point-like mass, the moment of inertia about some axis is given by mr2, where r is the distance to the axis, and m is the mass. For an extended body, the moment of inertia is just the sum of all the pieces of mass multiplied by the square of their distances from the axis in question. For an extended body of a shape and uniform density. In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, the term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Eulers second law. Comparison of this frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. Moment of inertia appears in momentum, kinetic energy, and in Newtons laws of motion for a rigid body as a physical parameter that combines its shape. There is a difference in the way moment of inertia appears in planar. The moment of inertia of a flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis, if the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their arms or divers curl their bodies into a tuck position during a dive. For a simple pendulum, this yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as. Thus, moment of inertia depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation
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Mass
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In physics, mass is a property of a physical body. It is the measure of a resistance to acceleration when a net force is applied. It also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram, Mass is not the same as weight, even though mass is often determined by measuring the objects weight using a spring scale, rather than comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity and this is because weight is a force, while mass is the property that determines the strength of this force. In Newtonian physics, mass can be generalized as the amount of matter in an object, however, at very high speeds, special relativity postulates that energy is an additional source of mass. Thus, any body having mass has an equivalent amount of energy. In addition, matter is a defined term in science. There are several distinct phenomena which can be used to measure mass, active gravitational mass measures the gravitational force exerted by an object. Passive gravitational mass measures the force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force, according to Newtons second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A bodys mass also determines the degree to which it generates or is affected by a gravitational field and this is sometimes referred to as gravitational mass. The standard International System of Units unit of mass is the kilogram, the kilogram is 1000 grams, first defined in 1795 as one cubic decimeter of water at the melting point of ice. Then in 1889, the kilogram was redefined as the mass of the prototype kilogram. As of January 2013, there are proposals for redefining the kilogram yet again. In this context, the mass has units of eV/c2, the electronvolt and its multiples, such as the MeV, are commonly used in particle physics. The atomic mass unit is 1/12 of the mass of a carbon-12 atom, the atomic mass unit is convenient for expressing the masses of atoms and molecules. Outside the SI system, other units of mass include, the slug is an Imperial unit of mass, the pound is a unit of both mass and force, used mainly in the United States
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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
21.
Work (physics)
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In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and then dropped, the SI unit of work is the joule. The SI unit of work is the joule, which is defined as the work expended by a force of one newton through a distance of one metre. The dimensionally equivalent newton-metre is sometimes used as the unit for work, but this can be confused with the unit newton-metre. Usage of N⋅m is discouraged by the SI authority, since it can lead to confusion as to whether the quantity expressed in newton metres is a torque measurement, or a measurement of energy. Non-SI units of work include the erg, the foot-pound, the foot-poundal, the hour, the litre-atmosphere. Due to work having the physical dimension as heat, occasionally measurement units typically reserved for heat or energy content, such as therm, BTU. The work done by a constant force of magnitude F on a point that moves a distance s in a line in the direction of the force is the product W = F s. For example, if a force of 10 newtons acts along a point that travels 2 meters and this is approximately the work done lifting a 1 kg weight from ground level to over a persons head against the force of gravity. Notice that the work is doubled either by lifting twice the weight the distance or by lifting the same weight twice the distance. Work is closely related to energy, the work-energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in energy is caused by an equal amount of negative work done by the resultant force. From Newtons second law, it can be shown that work on a free, rigid body, is equal to the change in energy of the velocity and rotation of that body. The work of forces generated by a function is known as potential energy. These formulas demonstrate that work is the associated with the action of a force, so work subsequently possesses the physical dimensions. The work/energy principles discussed here are identical to Electric work/energy principles, constraint forces determine the movement of components in a system, constraining the object within a boundary. Constraint forces ensure the velocity in the direction of the constraint is zero and this only applies for a single particle system. For example, in an Atwood machine, the rope does work on each body, there are, however, cases where this is not true
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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
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Space
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Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, in Isaac Newtons view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the visibility of spatial depth in his Essay Towards a New Theory of Vision. Kant referred to the experience of space in his Critique of Pure Reason as being a pure a priori form of intuition. In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in space is conceived as curved. According to Albert Einsteins theory of relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a model for the shape of space. In the seventeenth century, the philosophy of space and time emerged as an issue in epistemology. At its heart, Gottfried Leibniz, the German philosopher-mathematician, and Isaac Newton, unoccupied regions are those that could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete. Space could be thought of in a way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people, but since there would be no observational way of telling these universes apart then, according to the identity of indiscernibles, there would be no real difference between them. According to the principle of sufficient reason, any theory of space that implied that there could be two possible universes must therefore be wrong. Newton took space to be more than relations between objects and based his position on observation and experimentation
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Speed
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In everyday use and in kinematics, the speed of an object is the magnitude of its velocity, it is thus a scalar quantity. Speed has the dimensions of distance divided by time, the SI unit of speed is the metre per second, but the most common unit of speed in everyday usage is the kilometre per hour or, in the US and the UK, miles per hour. For air and marine travel the knot is commonly used, the fastest possible speed at which energy or information can travel, according to special relativity, is the speed of light in a vacuum c =299792458 metres per second. Matter cannot quite reach the speed of light, as this would require an amount of energy. In relativity physics, the concept of rapidity replaces the classical idea of speed, italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time, in equation form, this is v = d t, where v is speed, d is distance, and t is time. A cyclist who covers 30 metres in a time of 2 seconds, objects in motion often have variations in speed. If s is the length of the path travelled until time t, in the special case where the velocity is constant, this can be simplified to v = s / t. The average speed over a time interval is the total distance travelled divided by the time duration. Speed at some instant, or assumed constant during a short period of time, is called instantaneous speed. By looking at a speedometer, one can read the speed of a car at any instant. A car travelling at 50 km/h generally goes for less than one hour at a constant speed, if the vehicle continued at that speed for half an hour, it would cover half that distance. If it continued for one minute, it would cover about 833 m. Different from instantaneous speed, average speed is defined as the distance covered divided by the time interval. For example, if a distance of 80 kilometres is driven in 1 hour, likewise, if 320 kilometres are travelled in 4 hours, the average speed is also 80 kilometres per hour. When a distance in kilometres is divided by a time in hours, average speed does not describe the speed variations that may have taken place during shorter time intervals, and so average speed is often quite different from a value of instantaneous speed. If the average speed and the time of travel are known, using this equation for an average speed of 80 kilometres per hour on a 4-hour trip, the distance covered is found to be 320 kilometres. Linear speed is the distance travelled per unit of time, while speed is the linear speed of something moving along a circular path
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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
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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
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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
28.
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
29.
Newton's laws of motion
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Newtons laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a measure of the force. These three laws have been expressed in different ways, over nearly three centuries, and can be summarised as follows. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation. Newtons laws are applied to objects which are idealised as single point masses, in the sense that the size and this can be done when the object is small compared to the distances involved in its analysis, or the deformation and rotation of the body are of no importance. In this way, even a planet can be idealised as a particle for analysis of its orbital motion around a star, in their original form, Newtons laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newtons laws of motion for rigid bodies called Eulers laws of motion, if a body is represented as an assemblage of discrete particles, each governed by Newtons laws of motion, then Eulers laws can be derived from Newtons laws. Eulers laws can, however, be taken as axioms describing the laws of motion for extended bodies, Newtons laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Other authors do treat the first law as a corollary of the second, the explicit concept of an inertial frame of reference was not developed until long after Newtons death. In the given mass, acceleration, momentum, and force are assumed to be externally defined quantities. This is the most common, but not the interpretation of the way one can consider the laws to be a definition of these quantities. Newtonian mechanics has been superseded by special relativity, but it is useful as an approximation when the speeds involved are much slower than the speed of light. The first law states that if the net force is zero, the first law can be stated mathematically when the mass is a non-zero constant, as, ∑ F =0 ⇔ d v d t =0. Consequently, An object that is at rest will stay at rest unless a force acts upon it, an object that is in motion will not change its velocity unless a force acts upon it. This is known as uniform motion, an object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest, if an object is moving, it continues to move without turning or changing its speed
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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
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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
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Routhian mechanics
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In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions, the Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, and has a similar mathematical form to the Hamiltonian, but is not exactly the same. The difference between the Lagrangian, Hamiltonian, and Routhian functions are their variables, the Routhian differs from these functions in that some coordinates are chosen to have corresponding generalized velocities, the rest to have corresponding generalized momenta. This choice is arbitrary, and can be done to simplify the problem, in each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian. The full set thus has the advantages of both sets of equations, with the convenience of splitting one set of coordinates to the Hamilton equations, and the rest to the Lagrangian equations. The Lagrangian equations are powerful results, used frequently in theory, however, if cyclic coordinates occur there will still be equations to solve for all the coordinates, including the cyclic coordinates despite their absence in the Lagrangian. Overall fewer equations need to be solved compared to the Lagrangian approach, as with the rest of analytical mechanics, Routhian mechanics is completely equivalent to Newtonian mechanics, all other formulations of classical mechanics, and introduces no new physics. It offers a way to solve mechanical problems. The velocities dqi/dt are expressed as functions of their corresponding momenta by inverting their defining relation, in this context, pi is said to be the momentum canonically conjugate to qi. The Routhian is intermediate between L and H, some coordinates q1, q2, qn are chosen to have corresponding generalized momenta p1, p2. Pn, the rest of the coordinates ζ1, ζ2, ζs to have generalized velocities dζ1/dt, dζ2/dt. Dζs/dt, and time may appear explicitly, where again the generalized velocity dqi/dt is to be expressed as a function of generalized momentum pi via its defining relation. The choice of n coordinates are to have corresponding momenta. The above is used by Landau and Lifshitz, and Goldstein, some authors may define the Routhian to be the negative of the above definition. Below, the Routhian equations of motion are obtained in two ways, in the other useful derivatives are found that can be used elsewhere. Consider the case of a system with two degrees of freedom, q and ζ, with generalized velocities dq/dt and dζ/dt, now change variables, from the set to, simply switching the velocity dq/dt to the momentum p. This change of variables in the differentials is the Legendre transformation, the differential of the new function to replace L will be a sum of differentials in dq, dζ, dp, d, and dt. Notice the Routhian replaces the Hamiltonian and Lagrangian functions in all the equations of motion, the remaining equation states the partial time derivatives of L and R are negatives ∂ L ∂ t = − ∂ R ∂ t. n, and j =1,2
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Damping
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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
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Damping ratio
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In engineering, the damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium, a mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system is trying to return to its equilibrium position, sometimes losses damp the system and can cause the oscillations to gradually decay in amplitude towards zero or attenuate. The damping ratio is a measure of describing how rapidly the oscillations decay from one bounce to the next, where the spring–mass system is completely lossless, the mass would oscillate indefinitely, with each bounce of equal height to the last. This hypothetical case is called undamped, If the system contained high losses, for example if the spring–mass experiment were conducted in a viscous fluid, the mass could slowly return to its rest position without ever overshooting. Commonly, the mass tends to overshoot its starting position, and then return, with each overshoot, some energy in the system is dissipated, and the oscillations die towards zero. Between the overdamped and underdamped cases, there exists a level of damping at which the system will just fail to overshoot. This case is called critical damping, the key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time. The damping ratio is a parameter, usually denoted by ζ and it is particularly important in the study of control theory. It is also important in the harmonic oscillator, the damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. This equation can be solved with the approach, X = C e s t, where C and s are both complex constants. That approach assumes a solution that is oscillatory and/or decaying exponentially, using it in the ODE gives a condition on the frequency of the damped oscillations, s = − ω n. Undamped, Is the case where ζ →0 corresponds to the simple harmonic oscillator. Underdamped, If s is a number, then the solution is a decaying exponential combined with an oscillatory portion that looks like exp . This case occurs for ζ <1, and is referred to as underdamped, overdamped, If s is a real number, then the solution is simply a decaying exponential with no oscillation. This case occurs for ζ >1, and is referred to as overdamped, critically damped, The case where ζ =1 is the border between the overdamped and underdamped cases, and is referred to as critically damped. This turns out to be an outcome in many cases where engineering design of a damped oscillator is required. The factors Q, damping ratio ζ, and exponential decay rate α are related such that ζ =12 Q = α ω0, a lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times
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Displacement (vector)
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A displacement is a vector that is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a line from the initial position to the final position of the point. The velocity then is distinct from the speed which is the time rate of change of the distance traveled along a specific path. The velocity may be defined as the time rate of change of the position vector. For motion over an interval of time, the displacement divided by the length of the time interval defines the average velocity. In dealing with the motion of a body, the term displacement may also include the rotations of the body. In this case, the displacement of a particle of the body is called linear displacement, for a position vector s that is a function of time t, the derivatives can be computed with respect to t. These derivatives have common utility in the study of kinematics, control theory, vibration sensing and other sciences, by extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the displacement function. Such higher-order terms are required in order to represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering. The fourth order derivative is called jounce
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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
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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
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Friction
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Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction, Dry friction resists relative lateral motion of two surfaces in contact. Dry friction is subdivided into static friction between non-moving surfaces, and kinetic friction between moving surfaces, fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other. Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces, skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body. Internal friction is the force resisting motion between the making up a solid material while it undergoes deformation. When surfaces in contact move relative to other, the friction between the two surfaces converts kinetic energy into thermal energy. This property can have consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to thermal energy whenever motion with friction occurs, another important consequence of many types of friction can be wear, which may lead to performance degradation and/or damage to components. Friction is a component of the science of tribology, Friction is not itself a fundamental force. Dry friction arises from a combination of adhesion, surface roughness, surface deformation. The complexity of interactions makes the calculation of friction from first principles impractical and necessitates the use of empirical methods for analysis. Friction is a non-conservative force - work done against friction is path dependent, in the presence of friction, some energy is always lost in the form of heat. Thus mechanical energy is not conserved, the Greeks, including Aristotle, Vitruvius, and Pliny the Elder, were interested in the cause and mitigation of friction. They were aware of differences between static and kinetic friction with Themistius stating in 350 A. D. that it is easier to further the motion of a moving body than to move a body at rest. The classic laws of sliding friction were discovered by Leonardo da Vinci in 1493, a pioneer in tribology and these laws were rediscovered by Guillaume Amontons in 1699. Amontons presented the nature of friction in terms of surface irregularities, the understanding of friction was further developed by Charles-Augustin de Coulomb. Coulomb further considered the influence of sliding velocity, temperature and humidity, the distinction between static and dynamic friction is made in Coulombs friction law, although this distinction was already drawn by Johann Andreas von Segner in 1758. Leslie was equally skeptical about the role of adhesion proposed by Desaguliers, in Leslies view, friction should be seen as a time-dependent process of flattening, pressing down asperities, which creates new obstacles in what were cavities before
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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
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Mechanics of planar particle motion
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This article describes a particle in planar motion when observed from non-inertial reference frames. The most famous examples of motion are related to the motion of two spheres that are gravitationally attracted to one another, and the generalization of this problem to planetary motion. See centrifugal force, two-body problem, orbit and Keplers laws of planetary motion and those problems fall in the general field of analytical dynamics, the determination of orbits from given laws of force. This article is focused more on the issues surrounding planar motion. The Lagrangian approach to fictitious forces is introduced, unlike real forces such as electromagnetic forces, fictitious forces do not originate from physical interactions between objects. The appearance of fictitious forces normally is associated with use of a frame of reference. For solving problems of mechanics in non-inertial reference frames, the advice given in textbooks is to treat the fictitious forces like real forces, elaboration of this point and some citations on the subject follow. Examples are Cartesian coordinates, polar coordinates and curvilinear coordinates, or as seen from a rotating frame. A time-dependent description of observations does not change the frame of reference in which the observations are made, in discussion of a particle moving in a circular orbit, in an inertial frame of reference one can identify the centripetal and tangential forces. It then seems to be no problem to switch hats, change perspective and that switch is unconscious, but real. Suppose we sit on a particle in planar motion. What analysis underlies a switch of hats to introduce fictitious centrifugal, to explore that question, begin in an inertial frame of reference. In Figure 1, the arc length s is the distance the particle has traveled along its path in time t, the path r with components x, y in Cartesian coordinates is described using arc length s as, r =. One way to look at the use of s is to think of the path of the particle as sitting in space, like the left by a skywriter. Any position on this path is described by stating its distance s from some starting point on the path, then an incremental displacement along the path ds is described by, d r = = d s, where primes are introduced to denote derivatives with respect to s. The magnitude of this displacement is ds, showing that, =1, the unit magnitude of these vectors is a consequence of Eq.1. As an aside, notice that the use of vectors that are not aligned along the Cartesian xy-axes does not mean we are no longer in an inertial frame. All it means is that we are using unit vectors that vary with s to describe the path, the radius of curvature is introduced completely formally as,1 ρ = d θ d s
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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
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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
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Relative velocity
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The relative velocity v → B | A is the velocity of an object or observer B in the rest frame of another object or observer A. We begin with relative motion in the classical, that all speeds are less than the speed of light. This limit is associated with the Galilean transformation, the figure shows a man on top of a train, at the back edge. At 1,00 pm he begins to forward at a walking speed of 10 km/hr. The train is moving at 40 km/hr, the figure depicts the man and train at two different times, first, when the journey began, and also one hour later at 2,00 pm. The figure suggests that the man is 50 km from the point after having traveled for one hour. This, by definition, is 50 km/hour, which suggests that the prescription for calculating relative velocity in this fashion is to add the two velocities. The figure displays clocks and rulers to remind the reader that while the logic behind this calculation seem flawless, it makes false assumptions about how clocks, V → M | T is the velocity of the Man relative to the Train. V → T | E is the velocity of the Train relative to Earth, fully legitimate expressions for the velocity of A relative to B include the velocity of A with respect to B and the velocity of A in the coordinate system where B is always at rest. The violation of special relativity occurs because this equation for relative velocity falsely predicts that different observers will measure different speeds when observing the motion of light, the figure shows two objects moving at constant velocity. The equations of motion are, r → A = r → A i + v → A t r → B = r → B i + v → B t, where the subscript i refers to the initial displacement. The difference between the two displacement vectors, r → B − r → A, represents the location of B as seen from A, to construct a theory of relative motion consistent with the theory of special relativity, we must adopt a different convention. Recall that v is the motion of an object in the primed frame. As in classical mechanics, in Special Relativity the relative velocity v → B | A is the velocity of an object or observer B in the rest frame of object or observer A. This rotation has no effect on the magnitude of a vector, beer and Johnston, Statics and Dynamics. McGraw Hill Dictionary of Physics and Mathematics, Mechanics, Engineering Mechanics, Statics, Dynamics Relative Motion at HyperPhysics A Java applet illustrating Relative Velocity, by Andrew Duffy Relatív mozgás. Sebességek összegzése Relative tranquility of trout in creek
44.
Rigid body
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In physics, a rigid body is an idealization of a solid body in which deformation is neglected. In other words, the distance between any two points of a rigid body remains constant in time regardless of external forces exerted on it. Even though such an object cannot physically exist due to relativity, in classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechanics a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics molecules are often seen as rigid bodies, the position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, if the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, however, typically a different, mathematically more convenient, but equivalent approach is used. Thus, the position of a body has two components, linear and angular, respectively. The same is true for other kinematic and kinetic quantities describing the motion of a body, such as linear and angular velocity, acceleration, momentum, impulse. This reference point may define the origin of a coordinate system fixed to the body, there are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix. In general, when a body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation of the body starting from a hypothetic reference position. Velocity and angular velocity are measured with respect to a frame of reference, the linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a point fixed to the body. During purely translational motion, all points on a body move with the same velocity. However, when motion involves rotation, the velocity of any two points on the body will generally not be the same. Two points of a body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation. Angular velocity is a quantity that describes the angular speed at which the orientation of the rigid body is changing. All points on a rigid body experience the same velocity at all times
45.
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
46.
Vibration
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Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem, the oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road. Vibration can be desirable, for example, the motion of a fork, the reed in a woodwind instrument or harmonica. In many cases, however, vibration is undesirable, wasting energy, for example, the vibrational motions of engines, electric motors, or any mechanical device in operation are typically unwanted. Such vibrations could be caused by imbalances in the parts, uneven friction. Careful designs usually minimize unwanted vibrations, the studies of sound and vibration are closely related. Sound, or pressure waves, are generated by vibrating structures, hence, attempts to reduce noise are often related to issues of vibration. Free vibration occurs when a system is set in motion with an initial input. Examples of this type of vibration are pulling a child back on a swing and letting go, or hitting a tuning fork, the mechanical system vibrates at one or more of its natural frequencies and damps down to motionlessness. Forced vibration is when a disturbance is applied to a mechanical system. The disturbance can be a periodic and steady-state input, a transient input, the periodic input can be a harmonic or a non-harmonic disturbance. Damped vibration, When the energy of a system is gradually dissipated by friction and other resistances. The vibrations gradually reduce or change in frequency or intensity or cease, Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. Alternately, a DUT is attached to the table of a shaker, Vibration testing is performed to examine the response of a device under test to a defined vibration environment. The measured response may be life, resonant frequencies or squeak. Squeak and rattle testing is performed with a type of quiet shaker that produces very low sound levels while under operation. For relatively low frequency forcing, servohydraulic shakers are used, for higher frequencies, electrodynamic shakers are used. Generally, one or more input or control points located on the DUT-side of a fixture is kept at a specified acceleration, other response points experience maximum vibration level or minimum vibration level
47.
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
48.
Centripetal force
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A centripetal force is a force that makes a body follow a curved path. Its direction is orthogonal to the motion of the body. Isaac Newton described it as a force by which bodies are drawn or impelled, or in any way tend, in Newtonian mechanics, gravity provides the centripetal force responsible for astronomical orbits. One common example involving centripetal force is the case in which a body moves with uniform speed along a circular path, the centripetal force is directed at right angles to the motion and also along the radius towards the centre of the circular path. The mathematical description was derived in 1659 by the Dutch physicist Christiaan Huygens, the direction of the force is toward the center of the circle in which the object is moving, or the osculating circle. The speed in the formula is squared, so twice the speed needs four times the force, the inverse relationship with the radius of curvature shows that half the radial distance requires twice the force. Expressed using the orbital period T for one revolution of the circle, the rope example is an example involving a pull force. The centripetal force can also be supplied as a push force, newtons idea of a centripetal force corresponds to what is nowadays referred to as a central force. Another example of centripetal force arises in the helix that is traced out when a particle moves in a uniform magnetic field in the absence of other external forces. In this case, the force is the centripetal force that acts towards the helix axis. Below are three examples of increasing complexity, with derivations of the formulas governing velocity and acceleration, uniform circular motion refers to the case of constant rate of rotation. Here are two approaches to describing this case, assume uniform circular motion, which requires three things. The object moves only on a circle, the radius of the circle r does not change in time. The object moves with constant angular velocity ω around the circle, therefore, θ = ω t where t is time. Now find the velocity v and acceleration a of the motion by taking derivatives of position with respect to time, consequently, a = − ω2 r. negative shows that the acceleration is pointed towards the center of the circle, hence it is called centripetal. While objects naturally follow a path, this centripetal acceleration describes the circular motion path caused by a centripetal force. The image at right shows the relationships for uniform circular motion. In this subsection, dθ/dt is assumed constant, independent of time, consequently, d r d t = lim Δ t →0 r − r Δ t = d ℓ d t
49.
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
50.
Reactive centrifugal force
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In classical mechanics, a reactive centrifugal force forms part of an action–reaction pair with a centripetal force. In accordance with Newtons first law of motion, an object moves in a line in the absence of any external forces acting on the object. A curved path may however ensue when a physical acts on it, the two forces will only have the same magnitude in the special cases where circular motion arises and where the axis of rotation is the origin of the rotating frame of reference. It is the force that is the subject of this article. Any force directed away from a center can be called centrifugal, centrifugal simply means directed outward from the center. Similarly, centripetal means directed toward the center, the reactive centrifugal force discussed in this article is not the same thing as the centrifugal pseudoforce, which is usually whats meant by the term centrifugal force. The figure at right shows a ball in circular motion held to its path by a massless string tied to an immovable post. The figure is an example of a real force. In this system a centripetal force upon the ball provided by the string maintains the motion. In this model, the string is assumed massless and the rotational motion frictionless, the string transmits the reactive centrifugal force from the ball to the fixed post, pulling upon the post. Again according to Newtons third law, the post exerts a reaction upon the string, labeled the post reaction, the two forces upon the string are equal and opposite, exerting no net force upon the string, but placing the string under tension. It should be noted, however, that the reason the post appears to be immovable is because it is fixed to the earth. If the rotating ball was tethered to the mast of a boat, for example, even though the reactive centrifugal is rarely used in analyses in the physics literature, the concept is applied within some mechanical engineering concepts. An example of this kind of engineering concept is an analysis of the stresses within a rapidly rotating turbine blade, the blade can be treated as a stack of layers going from the axis out to the edge of the blade. Each layer exerts a force on the immediately adjacent, radially inward layer. At the same time the inner layer exerts a centripetal force on the middle layer, while and the outer layer exerts an elastic centrifugal force. It is the stresses in the blade and their causes that mainly interest mechanical engineers in this situation, another example of a rotating device in which a reactive centrifugal force can be identified used to describe the system behavior is the centrifugal clutch. A centrifugal clutch is used in small engine-powered devices such as saws, go-karts
51.
Coriolis force
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In physics, the Coriolis force is an inertial force that acts on objects that are in motion relative to a rotating reference frame. In a reference frame with clockwise rotation, the acts to the left of the motion of the object. In one with anticlockwise rotation, the acts to the right. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology, deflection of an object due to the Coriolis force is called the Coriolis effect. Newtons laws of motion describe the motion of an object in a frame of reference. When Newtons laws are transformed to a frame of reference. Both forces are proportional to the mass of the object, the Coriolis force is proportional to the rotation rate and the centrifugal force is proportional to its square. The Coriolis force acts in a perpendicular to the rotation axis. The centrifugal force acts outwards in the direction and is proportional to the distance of the body from the axis of the rotating frame. These additional forces are termed inertial forces, fictitious forces or pseudo forces and they allow the application of Newtons laws to a rotating system. They are correction factors that do not exist in a non-accelerating or inertial reference frame, a commonly encountered rotating reference frame is the Earth. The Coriolis effect is caused by the rotation of the Earth, such motions are constrained by the surface of the Earth, so only the horizontal component of the Coriolis force is generally important. This force causes moving objects on the surface of the Earth to be deflected to the right in the Northern Hemisphere, the horizontal deflection effect is greater near the poles, since the effective rotation rate about a local vertical axis is largest there, and smallest at the equator. This effect is responsible for the rotation of large cyclones, riccioli, Grimaldi, and Dechales all described the effect as part of an argument against the heliocentric system of Copernicus. In other words, they argued that the Earths rotation should create the effect, the effect was described in the tidal equations of Pierre-Simon Laplace in 1778. Gaspard-Gustave Coriolis published a paper in 1835 on the yield of machines with rotating parts. That paper considered the forces that are detected in a rotating frame of reference. Coriolis divided these forces into two categories
52.
Pendulum (mathematics)
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The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allows the equations of motion to be solved analytically for small-angle oscillations. e, the bob does not trace an ellipse but an arc. The motion does not lose energy to friction or air resistance, the differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However adding a restriction to the size of the oscillations amplitude gives a form whose solution can be easily obtained, the error due to the approximation is of order θ3. Given the initial conditions θ = θ0 and dθ/dt =0, the period of the motion, the time for a complete oscillation is which is known as Christiaan Huygenss law for the period. Note that under the small-angle approximation, the period is independent of the amplitude θ0, T0 =2 π ℓ g can be expressed as ℓ = g π2 T024. If SI units are used, and assuming the measurement is taking place on the Earths surface, then g ≈9.81 m/s2, and g/π2 ≈1. Therefore, a reasonable approximation for the length and period are. Note that this integral diverges as θ0 approaches the vertical lim θ0 → π T = ∞, so that a pendulum with just the right energy to go vertical will never actually get there. For comparison of the approximation to the solution, consider the period of a pendulum of length 1 m on Earth at initial angle 10 degrees is 41 m g K ≈2.0102 s. The linear approximation gives 2 π1 m g ≈2.0064 s, the difference between the two values, less than 0. 2%, is much less than that caused by the variation of g with geographical location. From here there are ways to proceed to calculate the elliptic integral. Figure 4 shows the relative errors using the power series, T0 is the linear approximation, and T2 to T10 include respectively the terms up to the 2nd to the 10th powers. The resulting power series is, T =2 π ℓ g, given Eq.3 and the arithmetic–geometric mean solution of the elliptic integral, K = π2 M, where M is the arithmetic-geometric mean of x and y. This yields an alternative and faster-converging formula for the period, T =2 π M ℓ g, the animations below depict the motion of a simple pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity. The small graph above each pendulum is the phase plane diagram, the horizontal axis is displacement. With a large enough initial velocity the pendulum does not oscillate back and forth, a compound pendulum is one where the rod is not massless, and may have extended size, that is, an arbitrarily shaped rigid body swinging by a pivot. In this case the period depends on its moment of inertia I around the pivot point
53.
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
54.
Angular velocity
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This speed can be measured in the SI unit of angular velocity, radians per second, or in terms of degrees per second, degrees per hour, etc. Angular velocity is usually represented by the symbol omega, the direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction that is usually specified by the right-hand rule. The angular velocity of a particle is measured around or relative to a point, called the origin. As shown in the diagram, if a line is drawn from the origin to the particle, then the velocity of the particle has a component along the radius, if there is no radial component, then the particle moves in a circle. On the other hand, if there is no cross-radial component, a radial motion produces no change in the direction of the particle relative to the origin, so, for the purpose of finding the angular velocity, the radial component can be ignored. Therefore, the rotation is completely produced by the perpendicular motion around the origin, the angular velocity in two dimensions is a pseudoscalar, a quantity that changes its sign under a parity inversion. The positive direction of rotation is taken, by convention, to be in the direction towards the y axis from the x axis, if the parity is inverted, but the orientation of a rotation is not, then the sign of the angular velocity changes. There are three types of angular velocity involved in the movement on an ellipse corresponding to the three anomalies, in three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in case is generally thought of as a vector, or more precisely. It now has not only a magnitude, but a direction as well, the magnitude is the angular speed, and the direction describes the axis of rotation that Eulers rotation theorem guarantees must exist. The right-hand rule indicates the direction of the angular velocity pseudovector. Let u be a vector along the instantaneous rotation axis. This is the definition of a vector space, the only property that presents difficulties to prove is the commutativity of the addition. This can be proven from the fact that the velocity tensor W is skew-symmetric, therefore, R = e W t is a rotation matrix and in a time dt is an infinitesimal rotation matrix. Therefore, it can be expanded as R = I + W ⋅ d t +122 +, in such a frame, each vector is a particular case of the previous case, in which the module of the vector is constant. Though it just a case of a moving particle, this is a very important one for its relationship with the rigid body study. There are two ways to describe the angular velocity of a rotating frame, the angular velocity vector. Both entities are related and they can be calculated from each other, in a consistent way with the general definition, the angular velocity of a frame is defined as the angular velocity of each of the three vectors of the frame
55.
Galileo Galilei
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Galileo Galilei was an Italian polymath, astronomer, physicist, engineer, philosopher, and mathematician. He played a role in the scientific revolution of the seventeenth century. Galileo also worked in applied science and technology, inventing an improved military compass, Galileos championing of heliocentrism and Copernicanism was controversial during his lifetime, when most subscribed to either geocentrism or the Tychonic system. He met with opposition from astronomers, who doubted heliocentrism because of the absence of a stellar parallax. He was tried by the Inquisition, found vehemently suspect of heresy and he spent the rest of his life under house arrest. He has been called the father of observational astronomy, the father of modern physics, the father of scientific method, and the father of science. Galileo was born in Pisa, Italy, on 15 February 1564, the first of six children of Vincenzo Galilei, a famous lutenist, composer, and music theorist, and Giulia, three of Galileos five siblings survived infancy. The youngest, Michelangelo, also became a noted lutenist and composer although he contributed to financial burdens during Galileos young adulthood, Michelangelo was unable to contribute his fair share of their fathers promised dowries to their brothers-in-law, who would later attempt to seek legal remedies for payments due. Michelangelo would also occasionally have to borrow funds from Galileo to support his musical endeavours and these financial burdens may have contributed to Galileos early fire to develop inventions that would bring him additional income. When Galileo Galilei was eight, his family moved to Florence and he then was educated in the Vallombrosa Abbey, about 30 km southeast of Florence. Galileo Bonaiuti was buried in the church, the Basilica of Santa Croce in Florence. It was common for mid-sixteenth century Tuscan families to name the eldest son after the parents surname, hence, Galileo Galilei was not necessarily named after his ancestor Galileo Bonaiuti. The Italian male given name Galileo derives from the Latin Galilaeus, meaning of Galilee, the biblical roots of Galileos name and surname were to become the subject of a famous pun. In 1614, during the Galileo affair, one of Galileos opponents, in it he made a point of quoting Acts 1,11, Ye men of Galilee, why stand ye gazing up into heaven. Despite being a genuinely pious Roman Catholic, Galileo fathered three children out of wedlock with Marina Gamba and they had two daughters, Virginia and Livia, and a son, Vincenzo. Their only worthy alternative was the religious life, both girls were accepted by the convent of San Matteo in Arcetri and remained there for the rest of their lives. Virginia took the name Maria Celeste upon entering the convent and she died on 2 April 1634, and is buried with Galileo at the Basilica of Santa Croce, Florence. Livia took the name Sister Arcangela and was ill for most of her life, Vincenzo was later legitimised as the legal heir of Galileo and married Sestilia Bocchineri
56.
Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. Newtons Principia formulated the laws of motion and universal gravitation that dominated scientists view of the universe for the next three centuries. Newtons work on light was collected in his influential book Opticks. He also formulated a law of cooling, made the first theoretical calculation of the speed of sound. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He was knighted by Queen Anne in 1705 and he spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint and his father, also named Isaac Newton, had died three months before. Born prematurely, he was a child, his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Newtons mother had three children from her second marriage. From the age of twelve until he was seventeen, Newton was educated at The Kings School, Grantham which taught Latin and Greek. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, Henry Stokes, master at the Kings School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a bully, he became the top-ranked student. In June 1661, he was admitted to Trinity College, Cambridge and he started as a subsizar—paying his way by performing valets duties—until he was awarded a scholarship in 1664, which guaranteed him four more years until he would get his M. A. He set down in his notebook a series of Quaestiones about mechanical philosophy as he found it, in 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his B. A. degree in August 1665, in April 1667, he returned to Cambridge and in October was elected as a fellow of Trinity. Fellows were required to become ordained priests, although this was not enforced in the restoration years, however, by 1675 the issue could not be avoided and by then his unconventional views stood in the way. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II. A and he was elected a Fellow of the Royal Society in 1672. Newtons work has been said to distinctly advance every branch of mathematics then studied and his work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers
57.
Johannes Kepler
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Johannes Kepler was a German mathematician, astronomer, and astrologer. A key figure in the 17th-century scientific revolution, he is best known for his laws of motion, based on his works Astronomia nova, Harmonices Mundi. These works also provided one of the foundations for Isaac Newtons theory of universal gravitation, Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague and he was also a mathematics teacher in Linz, and an adviser to General Wallenstein. Kepler lived in an era when there was no distinction between astronomy and astrology, but there was a strong division between astronomy and physics. Kepler was born on December 27, the feast day of St John the Evangelist,1571 and his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline and his father, Heinrich Kepler, earned a precarious living as a mercenary, and he left the family when Johannes was five years old. He was believed to have died in the Eighty Years War in the Netherlands and his mother Katharina Guldenmann, an innkeepers daughter, was a healer and herbalist. Born prematurely, Johannes claimed to have weak and sickly as a child. Nevertheless, he often impressed travelers at his grandfathers inn with his phenomenal mathematical faculty and he was introduced to astronomy at an early age, and developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, in 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being called outdoors to see it and that the moon appeared quite red. However, childhood smallpox left him with vision and crippled hands. In 1589, after moving through grammar school, Latin school, there, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who also taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590. He proved himself to be a mathematician and earned a reputation as a skilful astrologer. Under the instruction of Michael Maestlin, Tübingens professor of mathematics from 1583 to 1631 and he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and he accepted the position in April 1594, at the age of 23. Keplers first major work, Mysterium Cosmographicum, was the first published defense of the Copernican system
58.
Jeremiah Horrocks
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Jeremiah Horrocks, sometimes given as Jeremiah Horrox, was an English astronomer. Jeremiah Horrocks was born at Lower Lodge Farm in Toxteth Park and his father James had moved to Toxteth Park to be apprenticed to Thomas Aspinwall, a watchmaker, and subsequently married his masters daughter Mary. Both families were well educated Puritans, the Horrocks sent their sons to the University of Cambridge. For their unorthodox beliefs the Puritans were excluded from public office, in 1632 Horrocks matriculated at Emmanuel College at the University of Cambridge as a sizar. At Cambridge he associated with the mathematician John Wallis and the platonist John Worthington, at that time he was one of only a few at Cambridge to accept Copernicuss revolutionary heliocentric theory, and he studied the works of Johannes Kepler, Tycho Brahe and others. In 1635 for reasons not clear Horrocks left Cambridge without graduating, now committed to the study of astronomy, Horrocks began to collect astronomical books and equipment, by 1638 he owned the best telescope he could find. Liverpool was a town so navigational instruments such as the astrolabe. But there was no market for the very specialised astronomical instruments he needed and he was well placed to do this, his father and uncles were watchmakers with expertise in creating precise instruments. While a youth he read most of the treatises of his day and marked their weaknesses. Tradition has it that after he left home he supported himself by holding a curacy in Much Hoole, near Preston in Lancashire, according to local tradition in Much Hoole, he lived at Carr House, within the Bank Hall Estate, Bretherton. Carr House was a property owned by the Stones family who were prosperous farmers and merchants. Horrocks was the first to demonstrate that the Moon moved in a path around the Earth. He anticipated Isaac Newton in suggesting the influence of the Sun as well as the Earth on the moons orbit, in the Principia Newton acknowledged Horrockss work in relation to his theory of lunar motion. In the final months of his life Horrocks made detailed studies of tides in attempting to explain the nature of causation of tidal movements. Keplers tables had predicted a near-miss of a transit of Venus in 1639 but, having made his own observations of Venus for years, Horrocks predicted a transit would indeed occur. Horrocks made a simple helioscope by focusing the image of the Sun through a telescope onto a plane surface, from his location in Much Hoole he calculated the transit would begin at approximately 3,00 pm on 24 November 1639, Julian calendar. The weather was cloudy but he first observed the tiny black shadow of Venus crossing the Sun at about 3,15 pm, the 1639 transit was also observed by William Crabtree from his home in Broughton near Manchester. His figure of 95 million kilometres was far from the 150 million kilometres known today and it presented Horrocks enthusiastic and romantic nature, including humorous comments and passages of original poetry
59.
Edmond Halley
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Edmond Halley, FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of Halleys Comet. He was the second Astronomer Royal in Britain, succeeding John Flamsteed, Halley was born in Haggerston, in east London. His father, Edmond Halley Sr. came from a Derbyshire family and was a wealthy soap-maker in London, as a child, Halley was very interested in mathematics. He studied at St Pauls School, and from 1673 at The Queens College, while still an undergraduate, Halley published papers on the Solar System and sunspots. While at Oxford University, Halley was introduced to John Flamsteed, influenced by Flamsteeds project to compile a catalog of northern stars, Halley proposed to do the same for the Southern Hemisphere. In 1676, Halley visited the south Atlantic island of Saint Helena, while there he observed a transit of Mercury, and realised that a similar transit of Venus could be used to determine the absolute size of the Solar System. He returned to England in May 1678, in the following year he went to Danzig on behalf of the Royal Society to help resolve a dispute. Because astronomer Johannes Hevelius did not use a telescope, his observations had been questioned by Robert Hooke, Halley stayed with Hevelius and he observed and verified the quality of Hevelius observations. In 1679, Halley published the results from his observations on St. Helena as Catalogus Stellarum Australium which included details of 341 southern stars and these additions to contemporary star maps earned him comparison with Tycho Brahe, e. g. the southern Tycho as described by Flamsteed. Halley was awarded his M. A. degree at Oxford, in 1686, Halley published the second part of the results from his Helenian expedition, being a paper and chart on trade winds and monsoons. The symbols he used to represent trailing winds still exist in most modern day weather chart representations, in this article he identified solar heating as the cause of atmospheric motions. He also established the relationship between pressure and height above sea level. His charts were an important contribution to the field of information visualisation. Halley spent most of his time on lunar observations, but was interested in the problems of gravity. One problem that attracted his attention was the proof of Keplers laws of planetary motion, Halleys first calculations with comets were thereby for the orbit of comet Kirch, based on Flamsteeds observations in 1680-1. Although he was to calculate the orbit of the comet of 1682. They indicated a periodicity of 575 years, thus appearing in the years 531 and 1106 and it is now known to have an orbital period of circa 10,000 years. In 1691, Halley built a bell, a device in which the atmosphere was replenished by way of weighted barrels of air sent down from the surface
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Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
61.
Jean le Rond d'Alembert
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Jean-Baptiste le Rond dAlembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was also co-editor with Denis Diderot of the Encyclopédie, DAlemberts formula for obtaining solutions to the wave equation is named after him. The wave equation is referred to as dAlemberts equation. Born in Paris, dAlembert was the son of the writer Claudine Guérin de Tencin and the chevalier Louis-Camus Destouches. Destouches was abroad at the time of dAlemberts birth, days after birth his mother left him on the steps of the Saint-Jean-le-Rond de Paris church. According to custom, he was named after the saint of the church. DAlembert was placed in an orphanage for foundling children, but his father found him and placed him with the wife of a glazier, Madame Rousseau, Destouches secretly paid for the education of Jean le Rond, but did not want his paternity officially recognized. DAlembert first attended a private school, the chevalier Destouches left dAlembert an annuity of 1200 livres on his death in 1726. Under the influence of the Destouches family, at the age of twelve entered the Jansenist Collège des Quatre-Nations. Here he studied philosophy, law, and the arts, graduating as baccalauréat en arts in 1735, in his later life, DAlembert scorned the Cartesian principles he had been taught by the Jansenists, physical promotion, innate ideas and the vortices. The Jansenists steered DAlembert toward a career, attempting to deter him from pursuits such as poetry. Theology was, however, rather unsubstantial fodder for dAlembert and he entered law school for two years, and was nominated avocat in 1738. He was also interested in medicine and mathematics, Jean was first registered under the name Daremberg, but later changed it to dAlembert. The name dAlembert was proposed by Johann Heinrich Lambert for a moon of Venus. In July 1739 he made his first contribution to the field of mathematics, at the time Lanalyse démontrée was a standard work, which dAlembert himself had used to study the foundations of mathematics. DAlembert was also a Latin scholar of note and worked in the latter part of his life on a superb translation of Tacitus. In 1740, he submitted his second scientific work from the field of fluid mechanics Mémoire sur la réfraction des corps solides, in this work dAlembert theoretically explained refraction. In 1741, after failed attempts, dAlembert was elected into the Académie des Sciences
62.
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
63.
Joseph-Louis Lagrange
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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
64.
Pierre-Simon Laplace
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Pierre-Simon, marquis de Laplace was an influential French scholar whose work was important to the development of mathematics, statistics, physics and astronomy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste and this work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace, Laplace formulated Laplaces equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is named after him. Laplace is remembered as one of the greatest scientists of all time, sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries. Laplace became a count of the Empire in 1806 and was named a marquis in 1817, Laplace was born in Beaumont-en-Auge, Normandy on 23 March 1749, a village four miles west of Pont lEveque in Normandy. According to W. W. Rouse Ball, His father, Pierre de Laplace and his great-uncle, Maitre Oliver de Laplace, had held the title of Chirurgien Royal. It would seem that from a pupil he became an usher in the school at Beaumont, however, Karl Pearson is scathing about the inaccuracies in Rouse Balls account and states, Indeed Caen was probably in Laplaces day the most intellectually active of all the towns of Normandy. It was here that Laplace was educated and was provisionally a professor and it was here he wrote his first paper published in the Mélanges of the Royal Society of Turin, Tome iv. 1766–1769, at least two years before he went at 22 or 23 to Paris in 1771, thus before he was 20 he was in touch with Lagrange in Turin. He did not go to Paris a raw self-taught country lad with only a peasant background, the École Militaire of Beaumont did not replace the old school until 1776. His parents were from comfortable families and his father was Pierre Laplace, and his mother was Marie-Anne Sochon. The Laplace family was involved in agriculture until at least 1750, Pierre Simon Laplace attended a school in the village run at a Benedictine priory, his father intending that he be ordained in the Roman Catholic Church. At sixteen, to further his fathers intention, he was sent to the University of Caen to read theology, at the university, he was mentored by two enthusiastic teachers of mathematics, Christophe Gadbled and Pierre Le Canu, who awoke his zeal for the subject. Here Laplaces brilliance as a mathematician was recognised and while still at Caen he wrote a memoir Sur le Calcul integral aux differences infiniment petites et aux differences finies. About this time, recognizing that he had no vocation for the priesthood, in this connection reference may perhaps be made to the statement, which has appeared in some notices of him, that he broke altogether with the church and became an atheist. Laplace did not graduate in theology but left for Paris with a letter of introduction from Le Canu to Jean le Rond dAlembert who at time was supreme in scientific circles. According to his great-great-grandson, dAlembert received him rather poorly, and to get rid of him gave him a mathematics book
65.
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
66.
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
67.
Johann Bernoulli
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Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to calculus and educating Leonhard Euler in the pupils youth. Johann was born in Basel, the son of Nicolaus Bernoulli, an apothecary, however, Johann Bernoulli did not enjoy medicine either and began studying mathematics on the side with his older brother Jacob. Throughout Johann Bernoulli’s education at Basel University the Bernoulli brothers worked together spending much of their time studying the newly discovered infinitesimal calculus and they were among the first mathematicians to not only study and understand calculus but to apply it to various problems. After graduating from Basel University Johann Bernoulli moved to teach differential equations, later, in 1694, he married Dorothea Falkner and soon after accepted a position as the professor of mathematics at the University of Groningen. At the request of Johann Bernoulli’s father-in-law, Johann Bernoulli began the voyage back to his town of Basel in 1705. Just after setting out on the journey he learned of his brother’s death to tuberculosis, Johann Bernoulli had planned on becoming the professor of Greek at Basel University upon returning but instead was able to take over as professor of mathematics, his older brother’s former position. As a student of Leibniz’s calculus, Johann Bernoulli sided with him in 1713 in the Newton–Leibniz debate over who deserved credit for the discovery of calculus, Johann Bernoulli defended Leibniz by showing that he had solved certain problems with his methods that Newton had failed to solve. Johann Bernoulli also promoted Descartes’ vortex theory over Newton’s theory of gravitation and this ultimately delayed acceptance of Newton’s theory in continental Europe. In consequence he was disqualified for the prize, which was won by Maclaurin, however, Bernoullis paper was subsequently accepted in 1726 when the Académie considered papers regarding elastic bodies, for which the prize was awarded to Pierre Mazière. Bernoulli received a mention in both competitions. Although Jacob and Johann worked together before Johann graduated from Basel University, shortly after this, Johann was jealous of Jacobs position and the two often attempted to outdo each other. After Jacobs death Johanns jealousy shifted toward his own talented son, in 1738 the father–son duo nearly simultaneously published separate works on hydrodynamics. Johann Bernoulli attempted to take precedence over his son by purposely predating his work two prior to his son’s. Johann married Dorothea Falkner, daughter of an Alderman of Basel and he was the father of Nicolaus II Bernoulli, Daniel Bernoulli and Johann II Bernoulli and uncle of Nicolaus I Bernoulli. The Bernoulli brothers often worked on the problems, but not without friction. In 1697 Jacob offered a reward for its solution, a protracted, bitter dispute then arose when Jacob challenged the solution and proposed his own. The dispute marked the origin of a new discipline, the calculus of variations, Bernoulli was hired by Guillaume de lHôpital for tutoring in mathematics
68.
Augustin-Louis Cauchy
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Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician who made pioneering contributions to analysis. He was one of the first to state and prove theorems of calculus rigorously and he almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had an influence over his contemporaries. His writings range widely in mathematics and mathematical physics, more concepts and theorems have been named for Cauchy than for any other mathematician. Cauchy was a writer, he wrote approximately eight hundred research articles. Cauchy was the son of Louis François Cauchy and Marie-Madeleine Desestre, Cauchy married Aloise de Bure in 1818. She was a relative of the publisher who published most of Cauchys works. By her he had two daughters, Marie Françoise Alicia and Marie Mathilde, Cauchys father was a high official in the Parisian Police of the New Régime. He lost his position because of the French Revolution that broke out one month before Augustin-Louis was born, the Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris, there Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, the famous mathematician Lagrange was also a friend of the Cauchy family. On Lagranges advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, most of the curriculum consisted of classical languages, the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of successes, Augustin-Louis chose an engineering career. In 1805 he placed second out of 293 applicants on this exam, one of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young, nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées. He graduated in engineering, with the highest honors. After finishing school in 1810, Cauchy accepted a job as an engineer in Cherbourg. Cauchys first two manuscripts were accepted, the one was rejected
69.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
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Coordinate system
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The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space, depending on the direction and order of the coordinate axis the system may be a right-hand or a left-hand system. This is one of many coordinate systems, another common coordinate system for the plane is the polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis, for a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the point