Simple Pendulum

The net restoring force is

Fnet = - m g sin

- m g sin = m a

a = - g sin

And this is not exactly what we had for our prototype,

Fnet = - k x = ma

a = - ( k / m ) x

Or is it?

Look more closely at this sin :

sin = opp / hyp

sin = opp / hyp

sin = x / l

And for small angles, this is nearly equal to

sin = s / l

Now we have

a = - g sin


a = - g (s / l)


a = - ( g / l ) s

where s is the displacement from equilibrium.

So this simple pendulum is a SHO for small angular displacements!

It is interesting and useful to look at the sine function as a power series expansion

sin x = x - (x3/3!) + (x5/5!) - (x7/7!) + . . .

That means, for small angles x,

sin x = x

and sin x deviates from that like x3 so we must go to very large angles x to see a difference from sin x = x. If we watch a simple pendulum with very large amplitude we find that it is not a simple harmonic oscillator (SHO). But it is a very close approximation for small amplitudes!

Vertical Spring

Torsional Pendulum
Return to ToC, Simple Harmonic Motion
(c) Doug Davis, 2001; all rights reserved