Newton's laws of motion
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Newton's laws of motion are three physical laws which provide relationships between the forces acting on a body and the motion of the body, first compiled by Sir Isaac Newton. Newton's laws were first published together in his work Philosophiae Naturalis Principia Mathematica (1687). The laws form the basis for classical mechanics. Newton used them to explain many results concerning the motion of physical objects. In the third volume of the text, he showed that the laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion.
Crudely stated, the three laws are:
- An object in motion will remain in motion unless acted upon by another force.
- Force equals mass multiplied by acceleration.
- For every action there is an equal but opposite reaction.
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The statements of the laws
Newton's laws of motion describe the acceleration of massive objects. The modern understanding of Newton's three laws of motion is:
- First Law
- If no external force acts on a particle, then it is possible to select a set of reference frames, called inertial reference frames, observed from which the particle moves without any change in velocity.
- Second Law
- Observed from an inertial reference frame, the net force on a particle is proportional to the time rate of change of its linear momentum. Momentum is the product of mass and velocity. This law is often stated as F = ma (the force on an object is equal to its mass multiplied by its acceleration).
- Third Law
- Whenever A exerts a force on B, B simultaneously exerts a force on A with the same magnitude in the opposite direction. The strong form of the law further postulates that these two forces act along the same line.
In the given interpretation mass, acceleration and most importantly force are assumed to be externally defined quantities. This is the most common, but not the only interpretation: one can consider the laws to be definitions of these quantities. Notice that the second law only holds when the observation is made from an inertial reference frame, and since an inertial reference frame is defined by the first law, asking a proof of the first law from the second law is a logical fallacy.
The three laws—original formulation
Newton's first law: law of inertia
Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.
An object at rest will remain at rest unless acted upon by an external and unbalanced force. An object in motion will remain in motion unless acted upon by an external and unbalanced force.
This law is also called the law of inertia.
The net force on an object is the vector sum of all the forces acting on the object. Newton's first law says that if this sum is zero, the state of motion of the object does not change. Essentially, it makes the following two points:
- An object that is not moving will not move until a net force acts upon it.
- An object that is in motion will not change its velocity (accelerate) until a net force acts upon it.
The first point seems relatively obvious to most people, but the second may take some thinking through, because we have no experience in every-day life of things that keep moving forever (except celestial bodies). If one slides a hockey puck along a table, it doesn't move forever, it slows and eventually comes to a stop. But according to Newton's laws, this is because a force is acting on the hockey puck and, sure enough, there is frictional force between the table and the puck, and that frictional force is in the direction opposite the movement. It is this force which causes the object to slow to a stop. In the absence of such a force, as approximated by an air hockey table or ice rink, the puck's motion would not slow. Newton's first law is just a restatement of what Galileo had already described and Newton gave credit to Galileo. It differs from Aristotle's view that all objects have a natural place in the universe. Aristotle believed that heavy objects like 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.
However, a key difference between Galileo's idea from Aristotle's is that Galileo realized that force acting on a body determines acceleration, not velocity. This insight leads to Newton's First Law - no force means no acceleration, and hence the body will continue to maintain its velocity.
The Law of Inertia apparently occurred to many different natural philosophers independently. Inertia of motion was described in the third century BCE in the Mo Tzu, a collection of Chinese philosophical texts, and the 17th century philosopher René Descartes also formulated the law, although he did not perform any experiments to confirm it.
There are no perfect demonstrations of the law, as friction usually causes a force to act on a moving body, and even in outer space gravitational forces act and cannot be shielded against, but the law serves to emphasize the elementary causes of changes in an object's state of motion: forces.
Newton's second law: law of acceleration
Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.
The rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the same direction.
In an exact original 1792 translation (from Latin) Newton's Second Law of Motion reads:
LAW II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. — 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.
Here Newton is saying that the rate of change in the momentum of an object is directly proportional to the amount of force exerted upon the object. He also states that the change in direction of momentum is determined by the angle from which the force is applied.
However, it must be remembered that for Newton, mass was constant and independent of velocity. To take "motion" (motu) as meaning momentum gives a false impression of what Newton believed. Since he took mass as constant (part of the constant of proportionality) it can, in modern notation, be taken to the left of the derivative as 
. If m is dependent on velocity (and thus indirectly upon time) as we would now hold, then m has to be included in the derivative, giving 
or 
.
Using momentum in the terminology (which would never have occurred to Newton) is a latter-day revision of the law to bring it into correspondence with special relativity.
Interestingly, Newton is restating in his further explanation another prior idea of Galileo, what we call today the Galilean transformation or the addition of velocities.
An interesting fact when studying Newton's Laws of Motion from the Principia is that Newton himself does not explicitly write formulae for his laws which was common in scientific writings of that time period. In fact, it is today commonly added when stating Newton's second law that Newton has said, "and inversely proportional to the mass of the object." This however is not found in Newton's second law as directly translated above. In fact, the idea of mass is not introduced until the third law.
In mathematical terms, the differential equation can be written as:
where:
is 
is The product of the mass and velocity is the momentum of the object.
If mass of an object in question is known to be constant and using the definition of acceleration, this differential equation can be rewritten as:
where:
is the acceleration.
It has been a common convention to describe Newton's second law in the mathematical formula 
where 
is Force, is acceleration and 
is mass. This formula in this form did not even begin to be used until the 18th century, after Newton's death, but it is implicit in his laws.
Newton's third law: law of reciprocal actions
Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi.
All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.
This law of motion is most commonly paraphrased as: "For every action force there is an equal, but opposite, reaction force".
A more direct translation is:
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. -- 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 impinge upon another, and by its force change 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, because 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.
The explanation of mass is expressed here for the first time in the words "reciprocally proportional to the bodies" which have now been traditionally added to Law 2 as "inversely proportional to the mass of the object." This is because Newton in his definition 1 had already stated that when he said "body" he meant "mass".
The third law follows mathematically from the law of conservation of momentum.
As shown in the diagram opposite, the skaters' forces on each other are equal in magnitude, and opposite in direction. Although the forces are equal, the accelerations are not: the less massive skater will have a greater acceleration due to Newton's second law. It is important to note that the action/reaction pair act on different objects and do not cancel each other out.
If a basketball hits the ground, the basketball's force on the Earth is the same as Earth's force on the basketball. However, due to the ball's much smaller mass, Newton's second law predicts that its acceleration will be much greater than that of the Earth. Not only do planets accelerate toward stars, but stars also accelerate toward planets. If a star gravitationally attracts a planet, then the planet will gravitationally attract the star. The planet is less massive than the star and thus displays greater changes in its state of motion. Similarly, if a falling ball is pulled towards the Earth, then the reaction force is that the Earth is pulled toward the ball. We can not detect any change in the Earth's motion because it is much more massive than the ball.
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.
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. Together with Newton's law of universal gravitation, they are often adequate to calculate the motion of celestial bodies, but sometimes the exclusion of general relativistic effects make the calculations too inaccurate to be useful (see Mercury). 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 Lorentz factor must be included in the expression for momentum along with mass and velocity) or 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 theory, including General Relativity and Relativistic Quantum Mechanics.
According to the principle of relativity, there is no preferred frame of reference. The laws of physics are equally valid in all frames of reference. Motion can only be measured relative to a frame of reference. According to the equivalence principle, an observer on the surface of the Earth could not find any difference between the gravitational attraction of earth and the inertial force that he feels when he is in a rocket in outer space that accelerates upwards (from the standpoint of the observer) at g. In other words, he may regard any inertial force as a gravitational force. Consequently, Newton's laws of motion are only valid in an inertial frame of reference. Notice that the surface of the Earth does not define an inertial frame of reference because it is rotating and orbiting and because of Earth's gravity. However, since the speed of rotation and revolution change relatively slowly, the inertial force is tiny. Therefore, Newton's laws of motion remain a good approximation on earth. In a non-inertial frame of reference, inertial forces must be considered for Newton's laws to remain valid.
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, which says that the force is the derivative of the momentum of the object with respect to time, but some of the newer versions of the second law (such as the constant mass approximation above) do not hold at relativistic velocities.
Relationship to the conservation laws
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.
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 arise from conditions in the equations of motion in the underlying theories.
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 frame no hypotheses". In modern physics, action at a distance has been completely eliminated, except for subtle effects involving quantum entanglement.
Conservation of energy was discovered nearly two centuries after Newton's lifetime, the long delay occurring because of the difficulty in understanding the role of microscopic and invisible forms of energy such as heat and infra-red light.
See also
- Scientific laws named after people
- Mercury, orbit of
- Galilean invariance
- General relativity
- Modified Newtonian dynamics
- Lagrangian mechanics
- Principle of least action
References
- Marion, Jerry and Thornton, Stephen. Classical Dynamics of Particles and Systems. Harcourt College Publishers, 1995. ISBN 0-03-097302-3
- Fowles, G. R. and Cassiday, G. L. Analytical Mechanics (6ed). Saunders College Publishing, 1999. ISBN 0-03-022317-2
External links
- Science aid: Newton's laws of motion
- Newtonian Physics - an on-line textbook
- Motion Mountain - an on-line textbook
- Trajectory Video - video clip showing exchange of momentum
- Newtonian attraction for three Planets (Mathcad Application Server)
- Gravity - Newton's Law for Kids
- Simulation on Newton's first law of motion
