Newton's laws of motion
First law: In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.
Second law: In an inertial reference frame, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma. (It is assumed here that the mass m is constant – see below.)
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
1 Overview 2 Laws
2.1 Newton's first law 2.2 Newton's second law
2.2.1 Impulse 2.2.2 Variable-mass systems
2.3 Newton's third law
3.1 Newton's 1st Law 3.2 Newton's 2nd Law 3.3 Newton's 3rd Law
4 Importance and range of validity 5 Relationship to the conservation laws 6 See also 7 References and notes 8 Further reading and works cited 9 External links
Newton's laws are applied to objects which are idealised as single
point masses, in the sense that the size and shape of the object's
body are neglected to focus on its motion more easily. 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,
Newton's laws of motion
displaystyle sum mathbf F =0;Leftrightarrow ; frac mathrm d mathbf v mathrm d t =0.
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. The first law of motion postulates the existence of at least one frame of reference called a Newtonian or inertial reference frame, relative to which the motion of a particle not subject to forces is a straight line at a constant speed. 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:
In every material universe, the motion of a particle in a preferential reference frame Φ is determined by the action of forces whose total vanished for all times when and only when the velocity of the particle is constant in Φ. That is, a particle initially at rest or in uniform motion in the preferential frame Φ continues in that state unless compelled by forces to change it.
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. 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.
displaystyle 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, m can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus,
displaystyle mathbf F =m, frac mathrm d mathbf v mathrm d t =mmathbf a ,
where F is the net force applied, 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. Consistent with the first law, the time derivative of the momentum is non-zero when the momentum changes direction, even if there is no change in its magnitude; such is the case with uniform circular motion. The relationship 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. Impulse An impulse J occurs when a force F acts over an interval of time Δt, and it is given by
displaystyle mathbf J =int _ Delta t mathbf F ,mathrm d t.
Since force is the time derivative of momentum, it follows that
= m Δ
displaystyle mathbf J =Delta mathbf p =mDelta mathbf v .
This relation between impulse and momentum is closer to Newton's wording of the second law. Impulse is a concept frequently used in the analysis of collisions and impacts. Variable-mass systems Main article: Variable-mass system 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:
n e t
m ( t )
( t )
= m ( t )
( t )
( w r o n g )
displaystyle 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 ≠ 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
displaystyle 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
An illustration of Newton's third law in which two skaters push against each other. The first skater on the left exerts a normal force N12 on the second skater directed towards the right, and the second skater exerts a normal force N21 on the first skater directed towards the left. The magnitudes of both forces are equal, but they have opposite directions, as dictated by Newton's third law.
The 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 = −FB. The third law means that all forces are interactions
between different 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
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
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 tires—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 forward—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
Newton's 1st Law
From the original
“ Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare. ”
Translated to English, this reads:
“ 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. ”
The ancient Greek philosopher
“ Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur. ”
This was translated quite closely in Motte's 1729 translation as:
“ 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. ”
According to modern ideas of how Newton was using his terminology, this is understood, in modern terms, as an equivalent of:
The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.
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
If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.
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. Newton's 3rd Law
“ Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi. ”
Translated to English, this reads:
“ 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. ”
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.
In the above, as usual, motion is Newton's name for momentum, hence
his careful distinction between motion and velocity.
Newton used the third law to derive the law of conservation of
momentum; 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.
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,
very high speeds (in special relativity, the
Book: Isaac Newton
Wikimedia Commons has media related to Newton's laws of motion.
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References and notes
^ For explanations of
Newton's laws of motion
Newton's "Axioms or Laws of Motion" starting on page 19 of volume 1 of
the 1729 translation Archived 28 September 2015 at the Wayback
Machine. of the Principia;
Newton's laws of motion
^ Browne, Michael E. (July 1999). Schaum's outline of theory and
problems of physics for engineering and science (Series: Schaum's
Outline Series). McGraw-Hill Companies. p. 58.
^ Holzner, Steven (December 2005). Physics for Dummies. Wiley, John
& Sons, Incorporated. p. 64.
^ See the Principia on line at Andrew Motte Translation
^ Andrew Motte translation of Newton's Principia (1687) Axioms or Laws
^ Greiner, Walter (2003). Classical mechanics: point particles and
relativity. New York: Springer. ISBN 978-0-387-21851-9.
^ Zeidler, E. (1988). Nonlinear Functional Analysis and its
Applications IV: Applications to Mathematical Physics. New York, NY:
Springer New York. ISBN 978-1-4612-4566-7.
^ Wachter, Armin; Hoeber, Henning (2006). Compendium of theoretical
physics. New York, NY: Springer. ISBN 0-387-25799-3.
^ [...]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;Truesdell, Clifford A.; Becchi, Antonio; Benvenuto,
Edoardo (2003). Essays on the history of mechanics: in memory of
Clifford Ambrose Truesdell and Edoardo Benvenuto. New York:
Birkhäuser. p. 207. ISBN 3-7643-1476-1.
^ Lubliner, Jacob (2008). Plasticity Theory (Revised Edition) (PDF).
Dover Publications. ISBN 0-486-46290-0. Archived from the
original (PDF) on 31 March 2010.
^ a b Galili, I.; Tseitlin, M. (2003). "Newton's First Law: Text,
Translations, Interpretations and Physics Education". Science &
Education. 12 (1): 45–73. Bibcode:2003Sc&Ed..12...45G.
^ Benjamin Crowell. "4.
That when a thing lies still, unless somewhat else stir it, it will lie still forever, is a truth that no man doubts. But [the proposition] that when a thing is in motion it will eternally be in motion unless somewhat else stay it, though the reason be the same (namely that nothing can change itself), is not so easily assented to. For men measure not only other men but all other things by themselves. And because they find themselves subject after motion to pain and lassitude, [they] think every thing else grows weary of motion and seeks repose of its own accord, little considering whether it be not some other motion wherein that desire of rest they find in themselves, consists.
^ Cohen, I. B. (1995). Science and the Founding Fathers: Science in
the Political Thought of Jefferson, Franklin, Adams and Madison. New
York: W.W. Norton. p. 117. ISBN 978-0393315103. Archived
from the original on 22 March 2017.
^ Cohen, I. B. (1980). The Newtonian Revolution: With Illustrations of
the Transformation of Scientific Ideas. Cambridge, England: Cambridge
University Press. pp. 183–4. ISBN 978-0521273800.
^ 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
^ See for example (1) I Bernard Cohen, "Newton's Second Law and the
Further reading and works cited
Crowell, Benjamin (2011), Light and Matter (2011, Light and Matter),
especially at Section 4.2, Newton's First Law, Section 4.3, Newton's
Second Law, and Section 5.1, Newton's Third Law.
Feynman, R. P.; Leighton, R. B.; Sands, M. (2005). The Feynman
Lectures on Physics. Vol. 1 (2nd ed.). Pearson/Addison-Wesley.
Fowles, G. R.; Cassiday, G. L. (1999). Analytical Mechanics (6th ed.).
Saunders College Publishing. ISBN 0-03-022317-2.
Likins, Peter W. (1973). Elements of Engineering Mechanics.
Newton, Isaac, "Mathematical Principles of Natural Philosophy", 1729
English translation based on 3rd
MIT Physics video lecture on Newton's three laws Light and Matter – an on-line textbook Simulation on Newton's first law of motion "Newton's Second Law" by Enrique Zeleny, Wolfram Demonstrations Project. Newton's 3rd Law demonstrated in a vacuum on YouTube
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