Newton's laws

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Newton's laws were verified by experiment and observation for over 200 years, until 1916, when they were superseded by Einstein's theory of relativity.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.

Newton's first law

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), unless acted upon by a net outside force.

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.

This means that a stationary object will remain stationary, and a moving object will continue to move (in a straight line and at a constant speed), unless a force acts upon it. In everyday life, the force of friction usually acts upon moving objects. Newton's law indicates that some force (gravity) must be acting upon the planets, as they do not travel in a straight line.

Newton's second law

Alternative formulations:

• The time rate of change in momentum is proportional to the net force acting on the object and takes place in the direction of the force.: F= dp/dt
• The acceleration of an object of constant mass is proportional to the force acting upon it.: F = ma

This is expressed by the equation:

• F = ma
  • F = force
  • m = mass
  • a = acceleration.

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 = 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.

The most general form of Newton's second law is given in terms of the momentum p which is given by p=mv:

<math>\mathbf{F}=\frac{d\mathbf{p}}{dt}<math>

In general the mass of the object and its velocity are variable, so that

<math>\mathbf{F}=\frac{d\mathbf{p}}{dt} = \frac{d}{dt}(m\mathbf{v}) = m\frac{d\mathbf{v}}{dt} + \mathbf{v}\frac{dm}{dt} = m\mathbf{a} + \mathbf{v}\frac{dm}{dt} <math>

This equation works in cases when the mass is variable, unlike F=ma, which is just an approximation when the mass is constant. This equation is also valid in special relativity if we express the momentum as <math>\mathbf{p}=\gamma m \mathbf{v}<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. We can very easily arrive to F=ma from this equation. If we take the mass as constant the second term of the preceding equation is zero and we have

<math>\mathbf{F}=\frac{d\mathbf{p}}{dt}= \frac{d}{dt}(m\mathbf{v}) =m\frac{d\mathbf{v}}{dt} =m\mathbf{a}<math>

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

Newton's third law

Alternative formulations:

• Whenever one body exerts force upon a second body, the second body exerts an equal and opposite force upon the first body.
• For every action, there is an equal and opposite reaction.

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 does a bullet exert force upon a target; but, the target exerts equal force upon the bullet. Not only do planets accelerate toward stars; but, stars accelerate toward planets. The reaction force has the same line of action, and is of the same type and magnitude as the original force.

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 becomes correct if the momentum of the field is included in the calculations..

Also see:

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 and electrostatic forces satisfy the strong form, while moving electric charges only satisfy the weak form.

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.