The Pendulum

The pendulum is another example of a simple harmonic oscillator, at least for small oscillations. Suppose we have a mass $m$ attached to a string of length $\ell$. We swing it up so that the stretched string makes a (small) angle $\theta_0$ with the vertical and release it. What happens?

We write Newton's Second Law for the force component tangent to the arc of the circle of the swing as:

$$F_t = - m g \sin(\theta) = m a_t = m \ell \frac{d^2\theta}{dt^2}$$ (70)

where the latter follows from $a_t = \ell \alpha$ (the angular acceleration). Then we rearrange to get:

$$\frac{d^2\theta}{dt^2} + \frac{g}{\ell}\sin(\theta) = 0$$ (71)

This is almost a simple harmonic equation with $\omega^2 = \frac{g}{\ell}$. To make it one, we have to use the small angle approximation $\sin(\theta) \approx \theta$. Then

$$\frac{d^2\theta}{dt^2} + \frac{g}{\ell} \theta = 0$$ (72)

and we can just read off the solution:

$$\theta(t) = \theta_0 \cos(\omega t + \phi)$$ (73)

If you compute the gravitational potential energy for the pendulum for arbitrary angle $\theta$, you get:

$$U(\theta) = m g \ell \left(1 - \cos(\theta)\right)$$ (74)

Somehow, this doesn't look like the form we might expect from blindly substituting into our solution for the SHO above:

$$U(t) = \frac{1}{2} m g \ell \theta_0^2 \sin^2(\omega t + \phi)$$ (75)

As an interesting and fun exercise (that really isn't too difficult) see if you can prove that these two forms are really the same, if you make the small angle approximation for $\theta$ in the first form! This shows you pretty much where the approximation will break down as $\theta_0$ is gradually increased. For large enough $\theta$, the period of a pendulum clock does depend on the amplitude of the swing. This might explain grandfather clocks - clocks with very long penduli that can swing very slowly through very small angles - and why they were so accurate for their day.