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Newton's laws of motion
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{{redirect|Laws of Motion||Laws of motion}}{{redirect|Newton's Law|the 2017 Australian TV series|Newton's Law (TV series)}}{{short description|Classical formulation of three laws of Mechanics by Isaac Newton}}{{EngvarB|date=July 2019}}{{Use dmy dates|date=July 2019}}{{pp-semi-vandalism|small=yes}}{{Classical mechanics}}Newton's 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 quantitative measure of the force, and the third asserts that a single isolated force doesn't exist. These three laws have been expressed in several ways, over nearly three centuries,{{efn|For explanations of Newton's laws of motion by Newton in the early 18th century, by the physicist William Thomson (Lord Kelvin) in the mid-19th century, and by a modern text of the early 21st century, see:
• Newton's "Axioms or Laws of Motion"BOOK

, 19
, 1
, 1729 translation
, Principia
,
• {{harvp|Thomson|Tait|1867|at=Section 242, Newton's laws of motion}} and
• {{harvp|Crowell|2011|at=Newtonian Physics}}.
}} and can be summarised as follows:----{|
 {{nowrap|First law:}} inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.BROWNE TITLE =SCHAUM'S OUTLINE OF THEORY AND PROBLEMS OF PHYSICS FOR ENGINEERING AND SCIENCE DATE =JULY 1999 PAGE =58 ISBN =978-0-07-008498-8, HOLZNER> FIRST = STEVEN PUBLISHER =WILEY, JOHN & SONS, INCORPORATED PAGE =64 ISBN =978-0-7645-5433-9, 2005pfd..book.....H,
 {{nowrap|Second law:}} vector sum>sum of the forces {{math F}} on an object is equal to the mass {{math>m}} of that object multiplied by the acceleration {{math a}} of the object: {{math>F {{=}} ma}}. (It is assumed here that the mass m is constant â€“ see below.)
 {{nowrap|Third law:}}|When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

The three laws of motion were first compiled by Isaac Newton in his PhilosophiÃ¦ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687.See the Principia on line at Andrew Motte Translation Newton used them to explain and investigate the motion of many physical objects and systems.Andrew Motte translation of Newton's Principia (1687) Axioms or Laws of Motion For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion.Some also describe a fourth law which states that forces add up like vectors, that is, that forces obey the principle of superposition.BOOK, Greiner, Walter, Classical mechanics: point particles and relativity, 2003, Springer, New York, 978-0-387-21851-9, BOOK, Zeidler, E., Nonlinear Functional Analysis and its Applications IV: Applications to Mathematical Physics, 1988, Springer, New York, 978-1-4612-4566-7, BOOK, Wachter, Armin, Hoeber, Henning, Compendium of theoretical physics, 2006, Springer, New York, 978-0-387-25799-0,

## Overview

(File:Sir Isaac Newton (1643-1727).jpg|thumb|upright|Isaac Newton (1643â€“1727), the physicist who formulated the laws)Newton's laws are applied to objects which are idealised as single point masses,BOOK
, Truesdell, Clifford A., Becchi, Antonio, Benvenuto, Edoardo
, Essays on the history of mechanics: in memory of Clifford Ambrose Truesdell and Edoardo Benvenuto
, BirkhÃ¤user, 2003, New York, 207
, [...] while Newton had used the word 'body' vaguely and in at least three different meanings, Euler realized that the statements of Newton are generally correct only when applied to masses concentrated at isolated points;

## Laws

Newton's laws read:Law I: Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.Isaac Newton, The Principia, A new translation by I.B. Cohen and A. Whitman, University of California press, Berkeley 1999.{{efn|Newton's original Latin reads: Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.}}Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.Motte's 1729 translation{{efn|The original Latin reads: Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.}}Law III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.{{efn|In Latin:Lex III: Actioni contrariam semper et Ã¦qualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse Ã¦quales et in partes contrarias dirigi.}}{{efn|Newton's Scholium (explanatory comment) to this law:Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinges upon another, and by its force changes the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, as the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.This translation of the third law and the commentary following it can be found in the Principia on p. 20 of volume 1 of the 1729 translation.}}

### Newton's first law

The first law states that if the net force (the vector sum of all forces acting on an object) is zero, then the velocity of the object is constant. Velocity is a vector quantity which expresses both the object's speed and the direction of its motion; therefore, the statement that the object's velocity is constant is a statement that both its speed and the direction of its motion are constant.The first law can be stated mathematically when the mass is a non-zero constant, as,
sum mathbf{F} = 0; Leftrightarrow; frac{mathrm{d} mathbf{v} }{mathrm{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 (demonstrated when a tablecloth is skilfully whipped from under dishes on a tabletop and the dishes remain in their initial state of rest). If an object is moving, it continues to move without turning or changing its speed. This is evident in space probes that continuously move in outer space. Changes in motion must be imposed against the tendency of an object to retain its state of motion. In the absence of net forces, a moving object tends to move along a straight line path indefinitely.Newton placed the first law of motion to establish frames of reference for which the other laws are applicable. However, Newton implicitly referred to the absolute co-ordinate of cosmos for this frame. Since we cannot precisely measure our velocity relative to a far star, Newton's frame is based on a pure imagination, not based on measurable physics. In current physics, an observer defines himself as in inertial frame by preparing one stone hooked by a spring, and rotating the spring to any direction, and observing the stone static and the length of that spring unchanged. By Einstein's equivalence principle, if there was one such observer A and another observer B moving in a constant velocity related to A, then A and B will both observe the same physics phenomena. if A verified the first law, then B will verify it too. In this way, the definition of inertial can get rid of absolute space or far star, and only refer to the objects locally reachable and measurable.A particle not subject to forces moves (related to inertial frame) in a straight line at a constant speed.BOOK,weblink Special relativity, 6, NMJ Woodhouse, Springer, 2003, 978-1-85233-426-0, London/Berlin, Newton's first law is often referred to as the law of inertia. Thus, a condition necessary for the uniform motion of a particle relative to an inertial reference frame is that the total net force acting on it is zero. In this sense, the first law can be restated as:}}Newton's first and second laws are valid only in an inertial reference frame. Any reference frame that is in uniform motion with respect to an inertial frame is also an inertial frame, i.e. Galilean invariance or the principle of Newtonian relativity.BOOK, Thornton, Marion, 2004, Classical dynamics of particles and systems, 53, Brooks/Cole, 5th, 978-0-534-40896-1,weblink

### Newton's second law

The second law states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.
mathbf{F} = frac{mathrm{d}mathbf{p}}{mathrm{d}t} = frac{mathrm{d}(mmathbf v)}{mathrm{d}t}.
The second law can also be stated in terms of an object's acceleration. Since Newton's second law is valid only for constant-mass systems,JOURNAL, Plastino, Angel R., Muzzio, Juan C., 1992, On the use and abuse of Newton's second law for variable mass problems, Celestial Mechanics and Dynamical Astronomy, 53, 3, 227â€“232, 0923-2958, 1992CeMDA..53..227P, 10.1007/BF00052611, "We may conclude emphasizing that Newton's second law is valid for constant mass only. When the mass varies due to accretion or ablation, [an alternate equation explicitly accounting for the changing mass] should be used."BOOK, Halliday, Resnick, Physics, 1, 199, It is important to note that we cannot derive a general expression for Newton's second law for variable mass systems by treating the mass in F = dP/dt = d(Mv) as a variable. [...] We can use F = dP/dt to analyze variable mass systems only if we apply it to an entire system of constant mass having parts among which there is an interchange of mass., 978-0-471-03710-1, [Emphasis as in the original]BOOK, Kleppner, Daniel, Robert Kolenkow, An Introduction to Mechanics, McGraw-Hill, 1973, 133â€“134, 978-0-07-035048-9, Recall that F = dP/dt was established for a system composed of a certain set of particles[. ... I]t is essential to deal with the same set of particles throughout the time interval[. ...] Consequently, the mass of the system can not change during the time of interest.,weblink {{math|m}} can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus,
mathbf{F} = m,frac{mathrm{d}mathbf{v}}{mathrm{d}t} = mmathbf{a},
where F is the net force applied, {{math|m}} is the mass of the body, and a is the body's acceleration. Thus, the net force applied to a body produces a proportional acceleration. In other words, if a body is accelerating, then there is a force on it. An application of this notation is the derivation of G Subscript C.The above statements hint that the second law is merely a definition of mathbf{F}, not a precious observation of nature. However, current physics restate the second law in measurable steps: (1)defining the term 'one unit of mass' by a specified stone, (2)defining the term 'one unit of force' by a specified spring with specified length, (3)measuring by experiment or proving by theory (with a principle that every direction of space are equivalent), that force can be added as a mathematical vector, (4)finally conclude that mathbf{F} = mmathbf{a}. These steps hint the second law is a precious feature of nature.The second law also implies the conservation of momentum: when the net force on the body is zero, the momentum of the body is constant. Any net force is equal to the rate of change of the momentum.Any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems (see below).Newton's second law is an approximation that is increasingly worse at high speeds because of relativistic effects.According to modern ideas of how Newton was using his terminology,According to Maxwell in Matter and Motion, Newton meant by motion "the quantity of matter moved as well as the rate at which it travels" and by impressed force he meant "the time during which the force acts as well as the intensity of the force". See Harman and Shapiro, cited below. the law is understood, in modern terms, as an equivalent of:This may be expressed by the formula F = p', where p' is the time derivative of the momentum p. This equation can be seen clearly in the Wren Library of Trinity College, Cambridge, in a glass case in which Newton's manuscript is open to the relevant page.Motte's 1729 translation of Newton's Latin continued with Newton's commentary on the second law of motion, reading:The sense or senses in which Newton used his terminology, and how he understood the second law and intended it to be understood, have been extensively discussed by historians of science, along with the relations between Newton's formulation and modern formulations.See for example (1) I Bernard Cohen, "Newton's Second Law and the Concept of Force in the Principia", in "The Annus Mirabilis of Sir Isaac Newton 1666â€“1966" (Cambridge, Massachusetts: The MIT Press, 1967), pp. 143â€“185; (2) Stuart Pierson, "'Corpore cadente...': Historians Discuss Newtonâ€™s Second Law", Perspectives on Science, 1 (1993), pp. 627â€“658; and (3) Bruce Pourciau, "Newton's Interpretation of Newton's Second Law", Archive for History of Exact Sciences, vol.60 (2006), pp. 157â€“207; also an online discussion by G E Smith, in 5. Newton's Laws of Motion, s. 5 of "Newton's PhilosophiÃ¦ Naturalis Principia Mathematica" in (online) Stanford Encyclopedia of Philosophy, 2007.

#### Impulse

An impulse J occurs when a force F acts over an interval of time Î”t, and it is given byHannah, J., Hillier, M.J., Applied Mechanics, p221, Pitman Paperbacks, 1971BOOK, College Physics, 161, Raymond A. Serway, Jerry S. Faughn,weblink 978-0-534-99724-3, 2006, Thompson-Brooks/Cole, Pacific Grove CA,
mathbf{J} = int_{Delta t} mathbf F ,mathrm{d}t .
Since force is the time derivative of momentum, it follows that
mathbf{J} = Deltamathbf{p} = mDeltamathbf{v}.
This relation between impulse and momentum is closer to Newton's wording of the second law.BOOK, The investigation of difficult things: essays on Newton and the history of the exact sciences in honour of D.T. Whiteside, 353, I. Bernard Cohen (Peter M. Harman & Alan E. Shapiro, Eds),weblink 978-0-521-89266-7, 2002, Cambridge University Press, Cambridge, Impulse is a concept frequently used in the analysis of collisions and impacts.BOOK, Impact mechanics, 12 ff, Cambridge University Press, 2004, Cambridge, W.J. Stronge,weblink 978-0-521-60289-1,

#### Variable-mass systems

Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed and cannot be directly treated by making mass a function of time in the second law; that is, the following formula is wrong:
mathbf{F}_mathrm{net} = frac{mathrm{d}}{mathrm{d}t}big[m(t)mathbf{v}(t)big] = m(t) frac{mathrm{d}mathbf{v}}{mathrm{d}t} + mathbf{v}(t) frac{mathrm{d}m}{mathrm{d}t}. qquad mathrm{(wrong)}
The falsehood of this formula can be seen by noting that it does not respect Galilean invariance: a variable-mass object with F = 0 in one frame will be seen to have F â‰  0 in another frame.The correct equation of motion for a body whose mass m varies with time by either ejecting or accreting mass is obtained by applying the second law to the entire, constant-mass system consisting of the body and its ejected/accreted mass; the result is
mathbf F + mathbf{u} frac{mathrm{d} m}{mathrm{d}t} = m {mathrm{d} mathbf v over mathrm{d}t}
where u is the velocity of the escaping or incoming mass relative to the body. From this equation one can derive the equation of motion for a varying mass system, for example, the Tsiolkovsky rocket equation.Under some conventions, the quantity u dm/dt on the left-hand side, which represents the advection of momentum, is defined as a force (the force exerted on the body by the changing mass, such as rocket exhaust) and is included in the quantity F. Then, by substituting the definition of acceleration, the equation becomes F = ma.

### Newton's third law

File:Skaters showing newtons third law.svg|thumb|An illustration of Newton's third law in which two skaters push against each other. The first skater on the left exerts a normal forcenormal forceThe third law states that all forces between two objects exist in equal magnitude and opposite direction: if one object A exerts a force FA on a second object B, then B simultaneously exerts a force FB on A, and the two forces are equal in magnitude and opposite in direction: FA = âˆ’FB.BOOK, Resnick, Halliday, Krane, Physics, Volume 1, 4th, 83, 1992, The third law means that all forces are interactions between different bodies,JOURNAL, Newton's third law revisited, C Hellingman, Phys. Educ., 27, 1992, 2, 112â€“115, Quoting Newton in the Principia: It is not one action by which the Sun attracts Jupiter, and another by which Jupiter attracts the Sun; but it is one action by which the Sun and Jupiter mutually endeavour to come nearer together.bibcode = 1992PhyEd..27..112H, BOOK, Physics, Resnick, Halliday, yes, Third, John Wiley & Sons, 1977, 78â€“79two bodies., or different regions within one body, and thus that there is no such thing as a force that is not accompanied by an equal and opposite force. In some situations, the magnitude and direction of the forces are determined entirely by one of the two bodies, say Body A; the force exerted by Body A on Body B is called the "action", and the force exerted by Body B on Body A is called the "reaction". This law is sometimes referred to as the Reaction (physics)>action-reaction law, with FA called the "action" and FB the "reaction". In other situations the magnitude and directions of the forces are determined jointly by both bodies and it isn't necessary to identify one force as the "action" and the other as the "reaction". The action and the reaction are simultaneous, and it does not matter which is called the action and which is called reaction; both forces are part of a single interaction, and neither force exists without the other.The two forces in Newton's third law are of the same type (e.g., if the road exerts a forward frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third law predicts for the tires pushing backward on the road).From a conceptual standpoint, Newton's third law is seen when a person walks: they push against the floor, and the floor pushes against the person. Similarly, the tires of a car push against the road while the road pushes back on the tiresâ€”the tires and road simultaneously push against each other. In swimming, a person interacts with the water, pushing the water backward, while the water simultaneously pushes the person forwardâ€”both the person and the water push against each other. The reaction forces account for the motion in these examples. These forces depend on friction; a person or car on ice, for example, may be unable to exert the action force to produce the needed reaction force.Hewitt (2006), p. 75Newton used the third law to derive the law of conservation of momentum;Newton, Principia, Corollary III to the laws of motion from a deeper perspective, however, conservation of momentum is the more fundamental idea (derived via Noether's theorem from Galilean invariance), and holds in cases where Newton's third law appears to fail, for instance when force fields as well as particles carry momentum, and in quantum mechanics.

## History

File:Newtons laws in latin.jpg|thumb|Newton's First and Second laws, in Latin, from the original 1687 Principia Mathematica]]The ancient Greek philosopher Aristotle had the view that all objects have a natural place in the universe: that heavy objects (such as rocks) wanted to be at rest on the Earth and that light objects like smoke wanted to be at rest in the sky and the stars wanted to remain in the heavens. He thought that a body was in its natural state when it was at rest, and for the body to move in a straight line at a constant speed an external agent was needed continually to propel it, otherwise it would stop moving. Galileo Galilei, however, realised that a force is necessary to change the velocity of a body, i.e., acceleration, but no force is needed to maintain its velocity. In other words, Galileo stated that, in the absence of a force, a moving object will continue moving. (The tendency of objects to resist changes in motion was what Johannes Kepler had called inertia.) This insight was refined by Newton, who made it into his first law, also known as the "law of inertia"â€”no force means no acceleration, and hence the body will maintain its velocity. As Newton's first law is a restatement of the law of inertia which Galileo had already described, Newton appropriately gave credit to Galileo.Leonardo da Vinci understood that "An object offers as much resistance to the air as the air does to the object".BOOK, Fairlie, Gerard, Cayley, Elizabeth, The life of a genius, Hodder and Stoughton, 1965, 163, The law of inertia apparently occurred to several different natural philosophers and scientists independently, including Thomas Hobbes in his Leviathan (1651).{{efn|Hobbes wrote in Leviathan: }} The 17th-century philosopher and mathematician RenÃ© Descartes also formulated the law, although he did not perform any experiments to confirm it.BOOK, Cohen, I.B., I. Bernard Cohen, Science and the Founding Fathers: Science in the Political Thought of Jefferson, Franklin, Adams and Madison, New York, W.W. Norton, 117, 1995,weblink 978-0-393-24715-2, BOOK, Cohen, I.B., I. Bernard Cohen, The Newtonian Revolution: With Illustrations of the Transformation of Scientific Ideas, Cambridge, England, Cambridge University Press, 1980, 183â€“184,weblink 978-0-521-27380-0,

## Importance and range of validity

Newton's laws were verified by experiment and observation for over 200 years, and they are excellent approximations at the scales and speeds of everyday life. Newton's laws of motion, together with his law of universal gravitation and the mathematical techniques of calculus, provided for the first time a unified quantitative explanation for a wide range of physical phenomena.These three laws hold to a good approximation for macroscopic objects under everyday conditions. However, Newton's laws (combined with universal gravitation and classical electrodynamics) are inappropriate for use in certain circumstances, most notably at very small scales, at very high speeds, or in very strong gravitational fields. Therefore, the laws cannot be used to explain phenomena such as conduction of electricity in a semiconductor, optical properties of substances, errors in non-relativistically corrected GPS systems and superconductivity. Explanation of these phenomena requires more sophisticated physical theories, including general relativity and quantum field theory.In quantum mechanics, concepts such as force, momentum, and position are defined by linear operators that operate on the quantum state; at speeds that are much lower than the speed of light, Newton's laws are just as exact for these operators as they are for classical objects. At speeds comparable to the speed of light, the second law holds in the original form F = dp/dt, where F and p are four-vectors.

## Relationship to the conservation laws

In modern physics, the laws of conservation of momentum, energy, and angular momentum are of more general validity than Newton's laws, since they apply to both light and matter, and to both classical and non-classical physics.This can be stated simply, "Momentum, energy and angular momentum cannot be created or destroyed."Because force is the time derivative of momentum, the concept of force is redundant and subordinate to the conservation of momentum, and is not used in fundamental theories (e.g., quantum mechanics, quantum electrodynamics, general relativity, etc.). The standard model explains in detail how the three fundamental forces known as gauge forces originate out of exchange by virtual particles. Other forces, such as gravity and fermionic degeneracy pressure, also arise from the momentum conservation. Indeed, the conservation of 4-momentum in inertial motion via curved space-time results in what we call gravitational force in general relativity theory. The application of the space derivative (which is a momentum operator in quantum mechanics) to the overlapping wave functions of a pair of fermions (particles with half-integer spin) results in shifts of maxima of compound wavefunction away from each other, which is observable as the "repulsion" of the fermions.Newton stated the third law within a world-view that assumed instantaneous action at a distance between material particles. However, he was prepared for philosophical criticism of this action at a distance, and it was in this context that he stated the famous phrase "I feign no hypotheses". In modern physics, action at a distance has been completely eliminated, except for subtle effects involving quantum entanglement. (In particular, this refers to Bell's theoremâ€”that no local model can reproduce the predictions of quantum theory.) Despite only being an approximation, in modern engineering and all practical applications involving the motion of vehicles and satellites, the concept of action at a distance is used extensively.The discovery of the second law of thermodynamics by Carnot in the 19th century showed that not every physical quantity is conserved over time, thus disproving the validity of inducing the opposite metaphysical view from Newton's laws. Hence, a "steady-state" worldview based solely on Newton's laws and the conservation laws does not take entropy into account.

{{Wikipedia books|Isaac Newton}}{{Commons category|Newton's laws of motion}}{{div col|colwidth=30em}} {{div col end}}

{{Notelist}}

## References

{{Reflist}}

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• BOOK, Fowles, G.R., Cassiday, G.L., 1999, Analytical Mechanics, 6th, Saunders College Publishing, 978-0-03-022317-4, fow1999,
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{{Isaac Newton}}{{Authority control}}

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