Fly By Night

Dimensional Analysis

Pendulum Period

Simple Dimensional Analysis

Time \(T\) have the dimension of time \([T] = T\). If we want ot express time in terms of length \(l\) and acceleration \(a\), with \([l] = L\) and \([a] = L/T^2\). Thus the period must be:

\[ T \sim \sqrt{l/a} \]

The exact equation should be \(T = 2 \pi \sqrt{l/a}\). The \(2 \pi\) can be determined by experiment.

Or if we consider angular speed and with energy conservation equation, we can get the direct equation for \(\omega\) and \(T\):

\[ E = \frac{1}{2} m \dot{\theta}^2 + m g l (1 - \cos \theta) \simeq \frac{1}{2} m \dot \theta^2 + \frac{1}{2} m g l \theta^2 \]

The \(\simeq\) was obtained from the small angle approximation (with taylor series). Plug in \(\theta = \theta_0 \cos{\omega t}\), we get:

\[ E = \frac{1}{2} m \omega^2 \theta_0^2 \sin^2{\omega t} + \frac{1}{2} m g l \theta_0^2 \cos^2{\omega t} \]

In order for \(E\) to be a constant, \(m \omega^2 \theta_0^2 = m g l \theta_0^2\). Thus \(\omega = \sqrt{g/l}\), and \(T = 2 \pi / \omega = 2 \pi \sqrt{l/g}\).