Motion of Simple Pendulum

Using the small angle approximation

$$\tag{1} \sin\theta \approx \theta$$

equation of motion of a simple pendulum is reduced to

$$\tag{2} \frac{d^2 \theta}{dt^2} + \frac{g}{L} \theta = 0,$$

which gives the simple harmonic solution

$$\tag{3} \theta(t) = \theta_A \sin (\omega t + \varphi_0)$$

that describes the angular motion of the pendulum, where

$$\tag{4} \omega = \sqrt{\frac{g}{L}}$$

is the angular frequency of the oscillation.

• Daniel A. Russel, “Oscillation of a Simple Pendulum”, in Acoustics and Vibration Animations, Graduate Program in Acoustics, Pennsylvania State University, 18 Jun 2018, url https://www.acs.psu.edu/drussell/Demos/Pendulum/Pendulum.html [20221125].