# A pendulum is simple or not?

1. Apr 21, 2015

### Calpalned

1. The problem statement, all variables and given/known data
My textbook states that for oscillations of pendulums, the restoring force is $F = -mgsin(\theta)$. "Because F is proportional to the sine of $\theta$ and not $\theta$ itself, the motion is not SHM (simple harmonic motion)". I don't understand the last sentence.

2. Relevant equations
For small angles, $sin(\theta) ≈ \theta$.

3. The attempt at a solution
Why is it that if something is proportional to $\theta$ it is SHM, but $sin(\theta)$ is not SHM? What's the difference?

2. Apr 21, 2015

### SammyS

Staff Emeritus
How simple, or how complex an answer do you want?

One feature of simple harmonic motion is that the motion is sinusoidal as a function of time. It's also true that the period id independent of the amplitude.

Neither of those is exactly true if $\ F = -mg\sin(\theta) \$ .

3. Apr 22, 2015

### andrevdh

The derivation of the SHM theory usually starts out with something like

F = -kx .....

that is the restoring force is directly proportional to the displacement.
If the force is directly proportional to θ, and θ is small, the SHM theoretical equations
can again be derived, but not if F is proportional to the sine of theta. So in summary
the pendulum motion can not be descibed by the SHM equations for large amplitudes
and it is only an approximation for small amplitudes.

Last edited: Apr 22, 2015