Physical Pendulum Calculator

Unlike a simple pendulum that assumes all mass is concentrated at a single point, a physical pendulum accounts for the actual mass distribution of a rigid object. The math relies on three foundational relationships. First, the radius of oscillations (also known as the equivalent length, L) is defined as L = I / (m * d), where I represents the moment of inertia, m is the mass, and d is the distance from the pivot to the center of mass. Second, the period (T) of the oscillation is calculated using the formula T = 2 * pi * sqrt(L / g), where g is the acceleration due to gravity. Finally, the frequency (f) is simply the inverse of the period: f = 1 / T.

Imagine a uniform rod functioning as a physical pendulum with a mass (m) of 2 kg, a distance to the pivot (d) of 0.5 meters, and a known moment of inertia (I) of 0.6 kg·m². Using standard Earth gravity (g = 9.81 m/s²), we first find the radius of oscillations: L = 0.6 / (2 * 0.5) = 0.6 meters. Next, we calculate the period: T = 2 * 3.14159 * sqrt(0.6 / 9.81) = 1.55 seconds. Finally, we find the frequency: f = 1 / 1.55 = 0.65 Hz. This means the pendulum completes 0.65 cycles every second.

To ensure accurate results, always use SI units: kilograms for mass, meters for distance and length, and seconds for time. When determining the moment of inertia, make sure it is calculated relative to the exact pivot axis, not the center of mass, unless you are applying the parallel axis theorem. Additionally, keep angular displacements small (typically under 15 degrees) to maintain consistency with the small-angle approximation underlying standard pendulum formulas.