Pendulum

A simple pendulum is a particle-like bob of mass $m$ suspended by a light string of length $L$ that is fixed at the upper end.

Its position can be given by the angular position $\theta.$

The forces acting on the pendulum bob are gravity and the tension exerted by the string. The tension, opposed by the radial component of the force of gravity, provides the radial acceleration, whilst the tangential component of gravity acts as a restoring force, always urging the bob towards $\theta = 0.$

Applying Newton's second law in the tangential direction:

$$F_{\mathrm{t}} = m a_{\mathrm{t}},$$

$$-m g \sin\theta = m \frac{\mathrm{d}^2s}{\mathrm{d}t^2},$$

where $s$ is the position of the bob along the arc and the negative sign indicates that the tangential force acts towards the equilibrium position $\theta = 0.$

The arc can be expressed using the angular position $\theta$:

$$s = L \theta,$$

so the tangential acceleration is

$$a_{\mathrm{t}} = \frac{\mathrm{d}^2}{\mathrm{d} t^2}\left(L \theta\right) = L \frac{\mathrm{d}^2\theta}{\mathrm{d}t^2},$$

and the dynamical equation can be written as

$$\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} = -\frac{g}{L} \sin\theta.$$

We can see that the motion is not simple harmonic as the acceleration is not proportional to the position but the sine of the position. However, for small angles ($\theta < 5°$), $\sin\theta$ can be approximated with $\theta$, so

$$\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} \approx -\frac{g}{L} \cdot \theta$$

This is now a simple harmonic oscillator equation, whose solution, as we have seen, can be given in the form

$$\theta(t) = \theta_{\mathrm{max}} \cdot \sin\left(\omega t + \phi_0\right),$$

where $\theta_{\mathrm{max}}$ is the amplitude of the oscillation.

The parameter $\omega$ can be obtained from a comparison with the harmonic oscillator equation:

$$\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} = -\omega^2 \theta = -\frac{g}{L} \cdot\theta,$$

thus

$$\omega = \sqrt{\frac{g}{L}},$$

and the period is

$$T = \frac{2 \pi}{\omega} = 2 \pi \sqrt{\frac{L}{g}}.$$