Period and amplitude

The period of a body undergoing simple harmonic motion can be shown to be independent of the amplitude of the motion.

We will start by assuming an equation for T that depends on the force on the body F, its displacement x and its mass m. This can be written as:

T = KF^px^qm^r where K is a constant. Using the method of dimensions to solve for p, q and r

T = K(mx/F)

and therefore if the period is to be independent of amplitude then x/F must be a constant.

Therefore x is proportional to F, and since m is constant x is proportional to the acceleration. This is the definition of simple harmonic motion.

Therefore for s.h.m. the period is independent of the amplitude, providing that the motion is not damped (see below). This motion is also known as isochronous motion.

If the displacement at a time t is x₁, then x₁ is given by the formula

x₁ = r sin θt

and the displacement at a time (t + 2p/θ) is x₂, where

x₂ = rsinθ(t + 2π/q) = rsin (θt + 2π) = r sin θt cos 2p +r cos θt sin 2π = r sin θt = x₁

That is, the motion repeats itself after a time T where T = 2π/q, and T is therefore the period of the motion:

Phase shift

In both these proofs we have assumed that timing was started when the displacement of the body was zero, that is, that t = 0 when x = 0. If this is not the case then we have to introduce a phase shift (e) into the equations giving:

This means that t = 0 when x = r sine. The phase shift can clearly be seen from Figure 1.