Physical Pendulum Calculator
Enter moment of inertia, mass, gravity and pivot distance to compute the period of a physical (rigid-body) pendulum.
Input Data
Results
At a glance:A physical (compound) pendulum is any rigid body swinging under gravity about a fixed horizontal axis. Its small-angle period is T = 2π·√(I/(m·g·d)), with I the moment of inertia about the pivot, m the mass, g gravity and d the distance from the pivot to the center of mass. Frequency f = 1/T and angular frequency ω = 2πf = √(m·g·d/I). It reduces to the simple pendulum when I = m·L² (point mass at distance L, d=L). The parallel-axis theorem gives I = I_cm + m·d². This tool returns T, f and ω from I, m, g, d.
Formula
T = 2π·√(I/(m·g·d))
ω = √(m·g·d/I)
I = I_cm + m·d²
$$T = 2\pi\sqrt{\frac{I}{mgd}}, \quad L_{\text{eq}} = \frac{I}{md}$$How to Use
- Enter the moment of inertia I about the pivot.
- Enter mass m, gravity g and pivot-to-CM distance d.
- The calculator returns T, f and ω.
Case Studies
Rod about one end
I = mL²/3, d = L/2.
T = 2π√((mL²/3)/(mg·L/2)) = 2π√(2L/3g).
Slower than a point mass at L (T=2π√(L/g)).
FAQ
How is this different from a simple pendulum?
A simple pendulum assumes a point mass on a massless string (I = mL²). The physical pendulum uses the real I of an extended body about its pivot.
What is the pivot distance d?
The distance from the rotation axis to the center of mass; it sets the gravitational torque m·g·d that drives the oscillation.
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.