# Newton's Laws of Motion

## Newton's First Law

A body in motion remains in motion unless it is acted on by an external force. If the body is at rest it remains at rest.

Newton's first law may seem counter intuitive to our everyday experience. If we move an object by pushing it, it will move and then come to a halt. To keep it moving at the same speed a force must be applied. However, most objects are slowed to a halt by friction or air-resistance. Newton's first law gives rise to the idea of inertia. It takes a large force to start to move an object with a large mass and it also takes a large force to stop it moving. A moving object has a reluctance to stop moving and a stationary object has a reluctance to move.

An external force is a force that is applied to the object. There may be other forces inside the object, for example, there are forces produced by the motion of the atoms that make up the object but on average, they cancel each other out and so do not effect the motion of the object.

Newton's first law applies in inertial reference frames, i.e. when the frame of reference is stationary or moving with constant velocity. It does not apply when the frame of reference is accelerating. Strictly speaking, the Earth is accelerating because it is rotating as well as orbiting the Sun, however the effect of this is small enough to consider the Earth as an inertial reference frame.

## Newton's Second Law

The force,Fon an object is equal to the rate of change of momentum,p, with time.

In mathematical terms.

**F** = d**p**d*t*(1)

Or since **p** = *m***v**, then

d**p**d*t* = d d*t* (*m***v**) = *m* d**v**d*t* + **v**d*m*d*t* (using the product rule)

For constant mass, d*m*/dt = 0 is zero therefore,

**F**= *m*d**v**dt

We recognise d**v**/d*t* as the acceleration, **a** and so an alternative form of Newton's 2nd law is

**F** = *m***a**(2)

and since, **a** = d**v**d*t* = d^{2}**r**d*t*^{2}

**F** = *m*d^{2}**r**d*t*^{2}(3)

For this relationship to be true, we must measure the force **F** in Newtons [N], the mass m in kilograms and the acceleration in ms^{-2} otherwise there will be a constant of proportionality will have a value other than 1.

The force in Newton's 2nd law refers to the resultant force. If there are many forces acting on an object, it is the vector sum of those forces is the force that produces any acceleration.

∑**F**_{ext} = *m***a**(4)

## Newton's Third Law of Motion

Newton's third law applies to pairs of bodies. If a body A exerts a force on a body B, then body B exerts an equal and opposite force on body A.

Newton's third law reminds us that forces occur in equal and opposite pairs.

There are many examples of Newton's third law all around us. Every object that is on the floor or on top of another object is exerting a weight force toward the center of the Earth. If the object is stationary, then whatever the object is on top of is exerting an equal force in the opposite direction.

A rocket is also a good example of Newton's third law, it works by accelerating a mass in the form of hot gases, which produces a force. This produces an equal and opposite fore that pushes the rocket in the opposite direction.

## The Importance of Newton's Laws

Newton's laws are extremely important not just in mechanics but in the whole of physics. When trying to understand a physical process, we often understand it by looking at the forces acting and working out the equations of motion. This is true of the motion of the planets to the flow of electrons in an electric or magnetic field.