# Newton's laws of motion

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Newton's laws of motion are three laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:[1]

Isaac Newton (1643–1727), the physicist who formulated the laws

Law 1. A body continues in its state of rest, or in uniform motion in a straight line, unless acted upon by a force.

Law 2. A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.

Law 3. If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

The three laws of motion were first stated by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687.[2] Newton used them to explain and investigate the motion of many physical objects and systems, which laid the foundation for Newtonian mechanics.[3]

## Newton's first law

Newton's first law, also called the "law of inertia", states that an object at rest remains at rest, and an object that is moving will continue to move straight and with constant velocity, if and only if there is no net force acting on that object.[4]: 140

If any number of different external forces ${\displaystyle \mathbf {F} _{1},\mathbf {F} _{2},\ldots }$  are being applied to an object, then the net force ${\displaystyle F_{\text{Net}}}$  is the vector sum of those forces, so ${\displaystyle F_{\text{Net}}=\mathbf {F} _{1}+\mathbf {F} _{2}+\cdots }$ . If that net force is zero, then the object's velocity must not be changing. Conversely, if the object's velocity is not changing, then it must have a net force of zero.[4]: 140

Mathematically,[5]

${\displaystyle \mathbf {F} _{\text{Net}}=0\;\Leftrightarrow \;{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=0}$

Newton's first law describes objects that are in two different situations: objects that are stationary, and objects that are moving straight at a constant speed. Newton observed that objects in both situations will only change their speed if a net force is applied to them. An object which is undergoing a net force of zero is said to be at mechanical equilibrium, and Newton's first law suggests two different types of mechanical equilibrium: an object which has net forces of zero and which is not moving is at mechanical equilibrium, but an object that is moving in a straight line and with constant velocity is also at mechanical equilibrium.[4]: 140

Newton's first law is valid only in an inertial reference frame.[5]

## Newton's second law

Newton's second law describes a simple relationship between the acceleration of an object with mass ${\displaystyle m}$ , and the net force ${\displaystyle \mathbf {F} _{\text{net}}}$  acting on that object:[4]: 130

${\displaystyle \mathbf {F} _{\text{net}}=m\mathbf {a} }$

The net force and the object's acceleration are both vectors, and they point in the same direction.[4]: 130  This version of the law applies to an object with a fixed mass ${\displaystyle m}$ .[6][7][8] This relationship says that the net force applied to a body produces a proportional acceleration. It also means that if a body is accelerating, then a net force is being applied to it.

The law is also commonly stated in terms of the object's momentum ${\displaystyle \mathbf {p} }$ , since ${\displaystyle \mathbf {p} =m\mathbf {v} }$  and ${\displaystyle \mathbf {a} ={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}}$ . So, Newton's second law is also written as:[citation needed]

${\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}}$

Some textbooks use Newton's second law as a definition of force,[9][10][11] but this has been disparaged in other textbooks.[12]: 12–1 [13]: 59

### 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;[7][8] the 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 or accreted mass. The result is[6]

${\displaystyle \mathbf {F} +\mathbf {u} {\frac {\mathrm {d} m}{\mathrm {d} t}}=m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}}$

where u is the exhaust 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 ${\displaystyle \mathbf {u} {\frac {\mathrm {d} m}{\mathrm {d} t}}}$  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.[14] The third law means that all forces are interactions between different bodies,[15][16] 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 other.[14]

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 force.[17]

Newton used the third law to derive the law of conservation of momentum;[18] 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

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"—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.

## 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. For example, in the third volume of the Principia, Newton showed that his laws of motion, combined with the law of universal gravitation, explained Kepler's laws of planetary motion.

Newton's laws are applied to bodies which are idealised as single point masses,[19] in the sense that the size and shape of the body are neglected to focus on its motion more easily. This can be done when the line of action of the resultant of all the external forces acts through the center of mass of the body. 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 are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newton's laws of motion for rigid bodies called Euler's laws of motion, later applied as well for deformable bodies assumed as a continuum. If a body is represented as an assemblage of discrete particles, each governed by Newton's laws of motion, then Euler's laws can be derived from Newton's laws. Euler's laws can, however, be taken as axioms describing the laws of motion for extended bodies, independently of any particle structure.[20]

Newton's laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Some authors interpret the first law as defining what an inertial reference frame is; from this point of view, the second law holds only when the observation is made from an inertial reference frame, and therefore the first law cannot be proved as a special case of the second. Other authors do treat the first law as a corollary of the second.[21][22] The explicit concept of an inertial frame of reference was not developed until long after Newton's death.

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 special relativity, the second law holds in the original form F = dp/dt, where F and p are four-vectors. Special relativity reduces to Newtonian mechanics when the speeds involved are much less than the speed of light.

Some also describe a fourth law that is assumed but was never stated by Newton, which states that forces add like vectors, that is, that forces obey the principle of superposition.[23][24][25]

## References

1. ^ Thornton, Stephen T.; Marion, Jerry B. Classical Dynamics of Particles and Systems (5th ed.). Brooke Cole. p. 49. ISBN 0-534-40896-6.
2. ^ See the Principia on line at Andrew Motte Translation
3. ^ "Axioms, or Laws of Motion". gravitee.tripod.com. Retrieved 14 February 2021.
4. Knight, Randall D. (2008). Physics for scientists and engineers: A strategic approach (2 ed.). Addison-Wesley. ISBN 978-0805327366.
5. ^ a b Thornton, Marion (2004). Classical dynamics of particles and systems (5th ed.). Brooks/Cole. p. 53. ISBN 978-0-534-40896-1.
6. ^ a b 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): 227–232. Bibcode:1992CeMDA..53..227P. doi:10.1007/BF00052611. ISSN 0923-2958. S2CID 122212239. "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."
7. ^ a b Halliday; Resnick (1977). Physics. 1. p. 199. ISBN 978-0-471-03710-1. 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(M v) 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. [Emphasis as in the original]
8. ^ a b Kleppner, Daniel; Kolenkow, Robert (1973). An Introduction to Mechanics. McGraw-Hill. pp. 133–134. ISBN 978-0-07-035048-9 – via archive.org. 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.
9. ^ Landau, L.D.; Akhiezer, A.I.; Lifshitz, A.M. (1967). General Physics; mechanics and molecular physics (First English ed.). Oxford: Pergamon Press. ISBN 978-0-08-003304-4. Translated by: J.B. Sykes, A.D. Petford, and C.L. Petford. LCCN 67--30260. In section 7, pp. 12–14, this book defines force as dp/dt.
10. ^ Kibble, Tom W.B.; Berkshire, Frank H. (2004). Classical Mechanics (Fifth ed.). London: Imperial College Press. ISBN 1860944248. According to page 12, "[Force] can of course be introduced, by defining it through Newton's second law".
11. ^ de Lange, O. L.; Pierrus, J. (2010). Solved Problems in Classical Mechanics (First ed.). Oxford: Oxford University Press. ISBN 978-0-19-958252-5. According to page 3, "[Newton's second law of motion] can be regarded as defining force".
12. ^ Feynman Vol. 1
13. ^ Kleppner & Kolenkow 2010
14. ^ a b Resnick; Halliday; Krane (1992). Physics, Volume 1 (4th ed.). p. 83.
15. ^ C Hellingman (1992). "Newton's third law revisited". Phys. Educ. 27 (2): 112–115. Bibcode:1992PhyEd..27..112H. doi:10.1088/0031-9120/27/2/011. 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.
16. ^ Resnick & Halliday (1977). Physics (Third ed.). John Wiley & Sons. pp. 78–79. Any single force is only one aspect of a mutual interaction between two bodies.
17. ^ Hewitt (2006), p. 75
18. ^ Newton, Principia, Corollary III to the laws of motion
19. ^ 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 978-3-7643-1476-7. [...] 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;
20. ^ Lubliner, Jacob (2008). Plasticity Theory (PDF) (Revised ed.). Dover Publications. ISBN 978-0-486-46290-5. Archived from the original (PDF) on 31 March 2010.
21. ^ 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. doi:10.1023/A:1022632600805. S2CID 118508770.
22. ^ Benjamin Crowell (2001). "4. Force and Motion". Newtonian Physics. ISBN 978-0-9704670-1-0.
23. ^ Greiner, Walter (2003). Classical mechanics: point particles and relativity. New York: Springer. ISBN 978-0-387-21851-9.
24. ^ Zeidler, E. (1988). Nonlinear Functional Analysis and its Applications IV: Applications to Mathematical Physics. New York: Springer. ISBN 978-1-4612-4566-7.
25. ^ Wachter, Armin; Hoeber, Henning (2006). Compendium of theoretical physics. New York: Springer. ISBN 978-0-387-25799-0.

### Bibliography

Historical

For explanations of Newton's laws of motion by Newton in the early 18th century and by the physicist William Thomson (Lord Kelvin) in the mid-19th century, see the following: