Newton's laws of motion

"Newton's laws" redirects here. For other uses, see Newton's law.

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

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:

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. Newton used them to explain and investigate the motion of many physical objects and systems, which laid the foundation for Newtonian mechanics.

Laws

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.

If any number of different external forces $\mathbf {F} _{1},\mathbf {F} _{2},\ldots$ are being applied to an object, then the net force $F_{\text{Net}}$ is the vector sum of those forces, so $F_{\text{Net}}=\mathbf {F} _{1}+\mathbf {F} _{2}+\cdots$. 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.

Mathematically,

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

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.

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

Second law

Newton's second law describes a simple relationship between the acceleration of an object with mass m, and the net force F_net acting on that object:

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

The net force and the object's acceleration are both vectors, and they point in the same direction. This version of the law applies to an object with a fixed mass $m$. 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 p, since p = mv and a = ⁠dv/dt⁠. So, Newton's second law is also written as:

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

Some textbooks use Newton's second law as a definition of force, but this has been disparaged in other textbooks.

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; 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

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

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 u⁠dm/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.

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 F_A on a second object B, then B simultaneously exerts a force F_B on A, and the two forces are equal in magnitude and opposite in direction: F_A = −F_B. 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 F_A called the "action" and F_B 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 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.

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.

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 but no force is needed to maintain its velocity. 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, 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.

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. 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, which states that forces add like vectors, that is, that forces obey the principle of superposition. Newton gave the law geometrically as the parallelogram of force. In the Principia, it is stated immediately after the laws of motion as Corollary I.

See also

• Euler's laws of motion
• Hamiltonian mechanics
• Lagrangian mechanics
• List of scientific laws named after people
• Orbit of Mercury
• Modified Newtonian dynamics
• Newton's law of universal gravitation
• Principle of least action
• Principle of relativity
• Reaction (physics)

Sources

Citations

• Crowell, Benjamin (2011). Light and Matter. 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. ISBN 978-0-8053-9049-0.
• Fowles, G. R.; Cassiday, G. L. (1999). Analytical Mechanics (6th ed.). Saunders College Publishing. ISBN 978-0-03-022317-4.
• Likins, Peter W. (1973). Elements of Engineering Mechanics. McGraw-Hill Book Company. ISBN 978-0-07-037852-0.
• Marion, Jerry; Thornton, Stephen (1995). Classical Dynamics of Particles and Systems. Harcourt College Publishers. ISBN 978-0-03-097302-4.
• Woodhouse, N. M. J. (2003). Special Relativity. London/Berlin: Springer. p. 6. ISBN 978-1-85233-426-0.

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:

• Newton, Isaac. "Axioms or Laws of Motion". Mathematical Principles of Natural Philosophy. Vol. 1, containing Book 1 (1729 English translation based on 3rd Latin edition (1726) ed.). p. 19.
• Newton, Isaac. "Axioms or Laws of Motion". Mathematical Principles of Natural Philosophy. Vol. 2, containing Books 2 & 3 (1729 English translation based on 3rd Latin edition (1726) ed.). p. 19.
• Thomson, W.; Tait, P. G. (1867). "242, Newton's laws of motion". Treatise on natural philosophy. Vol. 1.

External links

• MIT Physics video lecture on Newton's three laws
• Simulation on Newton's first law of motion
• "Newton's Second Law" by Enrique Zeleny, Wolfram Demonstrations Project.
• The Laws of Motion, BBC Radio 4 discussion with Simon Schaffer, Raymond Flood & Rob Iliffe (In Our Time, 3 April 2008)