Newton's laws of motion

Newton's laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. These laws are fundamental to classical mechanics.

Newton first published these laws in Philosophiae Naturalis Principia Mathematica (1687) and used them to prove many results concerning the motion of physical objects. In the third volume (of the text), he showed how, combined with his law of universal gravitation, the laws of motion would explain Kepler's laws of planetary motion.

Importance of Newton's laws of motion

Nature and Nature's laws lay hid in night;

God said, Let Newton be! And all was light.

— Alexander Pope

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 such as: the motion of spinning bodies, motion of bodies in fluids; projectiles; motion on an inclined plane; motion of a pendulum; the tides; the orbits of the Moon and the planets. The law of conservation of momentum, which Newton derived as a corollary of his second and third laws, was the first conservation law to be discovered.

Newton's laws were verified by experiment and observation for over 200 years, until 1916, when they were superseded by Einstein's theory of relativity. Newton's laws still provide a completely adequate approximation for the behaviour of objects in "everyday" situations, i.e. situations where objects do not move near the speed of light.

Newton's First Law : Law of Inertia

This law is also called the Law of Inertia or Galileo's Principle.

Alternative formulations:

Every body continues in its state of rest, or of uniform motion in a right [straight] line, unless it is compelled to change that state by forces impressed upon it.
A body remains at rest, or moves in a straight line (at a constant velocity, v), unless acted upon by a net outside force.

In full calculus notation, this may be expressed as: <math>\frac{d}{dt}v = 0<math>

Despite the fact that Newton's First Law appears to be a special case of Newton's Second Law (see below), the First Law defines the reference frames in which the other two laws are valid. These reference frames are called inertial reference frames or Galilean reference frames, and are moving at constant speed, that is to say, without acceleration.

In less formal terms, Aristotle thought that things stood still if you left them alone; that to be at rest was natural; and that movement needed a cause. But Newton (and Galileo) taught us that "Things travel naturally at a steady speed (which may or may not be zero), if left alone"; it is acceleration that requires a cause - and we call this cause a force.

Moving from Aristotle's "A body's natural state is at rest" to Newton's First Law was one of the most profound and important discoveries in physics. In everyday life, the force of friction usually acts upon moving objects, slowing them down and eventually bringing them to rest. At the time of Newton and Aristotle, this was far from obvious. It is a tribute to Newton's genius that he was able to see this.

Newton's Second Law : Fundamental law of dynamics

Alternative formulations:

The rate of change in momentum is proportional to the net force acting on the object and takes place in the direction of the force.
The acceleration of an object of constant mass is proportional to the resultant force acting upon it.

These formulations may be expressed mathematically in the following ways:

<math> F \propto \frac{d}{dt}mv \textrm{\ or\ } F = m\frac{dv}{dt} = ma<math>

where

<math> F <math> is the force acting,
<math> m <math> is the mass of the object in question,
<math> a <math> is the object's acceleration,
<math> v <math> is the object's velocity, and
<math> mv <math> collectively is called the object's momentum.

This equation expresses that the more net force acts on an object, the greater its acceleration will be. The quantity m, or mass, in the above equation is the constant of proportionality, and is a characteristic of the object. This equation, therefore, indirectly defines the concept of mass.

In the equation, <>F</b> = ma, a is directly measurable but F is not. The second law only has meaning if we are able to assert, in advance, the value of F. Rules for calculating force include Newton's law of universal gravitation.

If we were to generalize the above equations further such that both the mass of the object and its velocity are variable, we arrive at:

This equation works in cases when the mass is variable, unlike <math> \mathbf{F} = ma<math>, which is only valid when the mass is constant. This equation is also valid in special relativity if we express the momentum as <math>\mathbf{p}=\gamma mv<math>. The physical meaning behind this equation is important as it implies that objects interact by exchanging momentum, and they do this via a force.

Taken together with Newton's Third Law of Motion, Newton's Second Law implies the Law of Conservation of Momentum.

Newton's Third Law : Law of reciprocal actions

Alternative formulations:

Whenever one body exerts force upon a second body, the second body exerts an equal and opposite force upon the first body.
Momentum is conserved.

The very common formulation "for every action there is an equal and opposite reaction" should be avoided, as it is, at best, ambiguous and confusing. A better formulation would be that when there exists a force acting on a body A, due to another body B, there exists also a reciprocal force, acting on body B, due to the existence of body A.

These formulations imply that if you strike an object with a force of 200 N, then the object also strikes you (with a force of 200 N). Not only do planets accelerate toward stars; but, stars accelerate toward planets. The reaction force has the opposite direction of action, and is of the same type and magnitude as the original force. However, it doesn't necessarily "line up" in space with the action. One example of this is a force on an electric dipole due to a point charge, when the dipole points in a direction perpendicular to the line connecting the point charge and the dipole. The force on the dipole due to the point charge is perpendicular to the line connecting them, so there is a reaction force on the point charge in the opposite direction, but these two force vectors are parallel and, even when extended to a line, they never cross each other in space.

It is often contended that Newton's third law is incorrect when electromagnetic forces are included: if a body A exerts a force on body B, then body B will in general exert a different force on body A (the force considered is the Lorentz force, generated by electric and magnetic fields). Modern theory predicts that the electromagnetic field generated by such interactions itself transports momentum via electromagnetic radiation. Newton's third law is valid if the momentum of the field is included in the calculations.

Also see: Physics Study Guide (http://wikibooks.org/wiki/Force_(Physics_Study_Guide))

Weak and strong forms of Newton's third law

The so-called "weak form" of Newton's Third Law applies for classical physical forces. In a system of particles, let <math>\mathbf{F}_{ab}<math> represent the force exerted on particle <math>a<math> due to particle <math>b<math>. The weak form requires that:

<math>\mathbf{F}_{ab} = -\mathbf{F}_{ba}<math>

All classical physical forces satisfy this condition.

The "strong form" of Newton's Third Law requires that, in addition to being equal and opposite, the forces must be directed along the line connecting the two particles. Gravitational force satisfies the strong form, while electromagnetic forces satisfy the weak form. For an example in electrostatics where the strong form is not obeyed, consider the interaction between a point charge and a perfect dipole aligned in a direction perpendicular to the line connecting the charge and the dipole.

The weak form is a valuable mathematical abstraction, because it allows one to study concepts such as the center of mass in the presence of arbitrary forces.

Newton's laws of motion

Importance of Newton's laws of motion

Newton's First Law : Law of Inertia

Newton's Second Law : Fundamental law of dynamics

Newton's Third Law : Law of reciprocal actions

Weak and strong forms of Newton's third law

See also