Period of a Simple Pendulum

We derive the period equation by comparing the pendulum's angular acceleration to the standard form for simple harmonic motion.

Angular Acceleration of the Pendulum

From the small angle approximation, a simple pendulum displaced by angle $\theta$ experiences angular acceleration:

$$\alpha = -\frac{g}{L}\theta$$

where $L$ is the pendulum length and $g$ is the gravitational acceleration.

Identifying Angular Frequency

The defining equation for SHM states that acceleration is proportional to displacement with a negative constant:

$$\alpha = -\omega^2 \theta$$

Comparing these two expressions:

$$-\omega^2 \theta = -\frac{g}{L}\theta$$

This gives:

$$\omega^2 = \frac{g}{L}$$ $$\omega = \sqrt{\frac{g}{L}}$$

Finding the Period

Using the relationship between period and angular frequency, $T = 2\pi/\omega$:

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

This is the period equation for a simple pendulum:

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