Physical Pendulum Angular Frequency

Angular frequency of a physical pendulum is

$$\tag{1} \omega = \sqrt{\frac{mgl}{I}},$$

where $m$ is mass of the pendulum, $g$ is acceleration due to gravity, $l$ is distance from center of mass to pivot point, and $I$ is moment of inertia with respect to the pivot point

$$\tag{2} I = I_0 + ml^2,$$

where $I_0$ is moment of inertia at the center of mass of the pendulum.

If a physical pendulum consists of $N$ objects, where distance between center of mass of object $i$ and center of mass of pendulum is $h_i$ then

$$\tag{3} I_0 = \sum_{i = 1}^N I_{0i} + m_i h_i^2,$$

where $I_{0i}$ is moment of inertia of object $i$ at its center of mass and $m_i$ is mass of object $i$, that contributes to $m$

$$\tag{4} m = \sum_{i = 1}^N m_i,$$

mass of the pendulum.

• Jan Awrejcewicz, “Mathematical and Physical Pendulum”, in Classical Mechanics, Advances in Mechanics and Mathematics, vol 29, May 2012, pp 69-102, Springer, New York, NY, url https://doi.org/10.1007/978-1-4614-3740-6_2.