Physical Pendulum Calculator
Find the oscillation period of any rigid body swinging about a fixed pivot: enter the moment of inertia about the pivot, the mass, the pivot-to-centre-of-mass distance, and gravitational acceleration.
kg·m²
kg
m
m/s²
Time for one complete oscillation — small-angle (< 15°) approximation
- 1
Restoring torque m·g·d
1 × 9.81 × 0.5 = 4.905 - 2
Ratio I ÷ (m·g·d)
0.333 ÷ 4.905 = 0.06789 - 3
√(I / m·g·d)
√0.06789 = 0.260557 - 4
Period T = 2π × √(I / m·g·d)
2π × 0.260557 = 1.6371
How does this calculator work?
A physical pendulum has period T = 2π√(I/mgd), where I is the moment of inertia about the pivot, m the mass, g gravitational acceleration, and d the pivot-to-CoM distance. Frequency f = 1/T; angular frequency ω = 2πf. Valid for small oscillations (< 15°) of any rigid body.
Formula
How this is calculated
A physical (compound) pendulum is any rigid body free to rotate about a fixed pivot that is not at its centre of mass. Unlike the idealised simple pendulum (all mass at a point, massless string), the physical pendulum accounts for the object's full mass distribution through its moment of inertia I (kg·m²) about the pivot axis.
For small angular displacements (typically below about 15°), the restoring torque is approximately linear and the motion is simple harmonic. The period is T = 2π√(I / mgd), where m is the total mass, g is gravitational acceleration, and d is the straight-line distance from the pivot to the centre of mass. A useful shorthand is the equivalent simple-pendulum length L_eq = I / (md): a massless-string pendulum of that length has exactly the same period.
To find I about the pivot, use the parallel-axis theorem: I_pivot = I_cm + md², where I_cm is the moment of inertia about the centre of mass (e.g. mL²/12 for a uniform rod, mR²/2 for a disk). The formula assumes a rigid body, frictionless pivot, and small-angle motion. For large swings the true period exceeds this prediction (≈ 0.5% error at 15°, 18% at 90°); air drag and pivot friction cause amplitude decay over time.
Frequently asked questions
A simple pendulum concentrates all mass at one point on a massless string, giving T = 2π√(L/g). A physical pendulum is a real rigid body; you need the moment of inertia I about the pivot and the pivot-to-CoM distance d to compute T = 2π√(I/mgd). Every simple pendulum is also a physical pendulum, but not vice versa.
Apply the parallel-axis theorem: I_pivot = I_cm + md². Common values of I_cm: uniform rod (about CoM) = mL²/12; disk (about centre) = mR²/2; solid sphere = 2mR²/5. For a rod pivoted at one end, I = mL²/12 + m(L/2)² = mL²/3.
For swings below 15° the error is less than 0.5% and the formula is excellent for most lab and engineering purposes. Above 30° the period grows noticeably; at 90° the true period is roughly 18% longer than T = 2π√(I/mgd). For large-angle accuracy an elliptic-integral correction is needed.
Also known as
TG we-Calculate Editorial Team. (2026). Physical Pendulum Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/physical-pendulum-calculator
TG we-Calculate Editorial Team. "Physical Pendulum Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/physical-pendulum-calculator.
TG we-Calculate Editorial Team, "Physical Pendulum Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/physical-pendulum-calculator
@misc{wecalculate_physical_pendulum_calculator, title = {Physical Pendulum Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/physical-pendulum-calculator}}, year = {2026}, note = {TG we-Calculate} }
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