vik dhillon: phy105 - celestial mechanics

newton's laws of motion

You will all be familiar with Newton's laws of motion, so we will just summarize them here, using the symbols F for force (which is a vector quantity), v for velocity (also a vector quantity) and m for mass (which is a scalar).

law I

Unless a resultant force acts on a body, its velocity will not change.

i.e. if F = 0, v = 0

This gives us an intuitive meaning of force: a resultant force is that agent which changes the velocity (and momentum) of a body. Law I is a special case of law II.

law II

The rate of change of momentum of a body is proportional to the resultant force that acts on it.

i.e. F

d(mv)/dt

or        F = k d(mv)/dt

Hence F = km dv/dt + kv dm/dt

              = km dv/dt   (since dm/dt = 0 in most problems in classical mechanics)

              = kma

We then choose k=1, and in so doing we also define our unit for force.

F[N] = m[kg]a[ms^-2]

1 newton (N) is that force which accelerates a mass of 1 kg at 1 ms^-2.
F = ma is one form of Newton's second law.

law III

If body A exerts a force F on body B, then body B exerts a force F on body A of the same size and along the same line, but in the opposite direction.

i.e. F_AB = - F_BA

Law III refers to a pair of forces which must always act on two different bodies. These two forces have the same size at every instant of time.