Physical Pendulum Calculator

A physical pendulum is any rigid body that oscillates about a fixed horizontal pivot under the influence of gravity, where the center of mass does not coincide with the pivot point. Unlike the idealized simple pendulum — which assumes a point mass on a massless string — a physical pendulum accounts for the full mass distribution of the object via its moment of inertia. Examples include a swinging clock weight, a playground swing, or a hinged rod.

The period is given by T = 2π √(I / (mgd)), where I is the moment of inertia about the pivot axis (kg·m²), m is the total mass (kg), g is gravitational acceleration (m/s²), and d is the distance from the pivot to the center of mass (m). This formula assumes small-angle oscillations (amplitude ≤ 15°).

Because a physical pendulum is an extended rigid body, its resistance to rotational motion depends on how its mass is distributed around the pivot — not just how far the center of mass is from the pivot. A larger moment of inertia means more rotational inertia, which slows the oscillation and increases the period, even if the center of mass distance stays the same.

Angular frequency (ω) is measured in radians per second and equals 2π times the regular frequency (f). It describes how fast the pendulum sweeps through angles. Regular frequency (f) measured in Hertz tells you how many complete oscillations occur per second. Period (T) is simply the reciprocal of frequency: T = 1/f.

The small-angle approximation assumes sin(θ) ≈ θ (in radians), which simplifies the pendulum's equation of motion into simple harmonic motion. This approximation is accurate when the initial angle does not exceed about 15°. Beyond 15°, the true motion deviates noticeably from the formula, and the period becomes dependent on amplitude.

Yes, in principle. If the moment of inertia I is small relative to the product mgd — for instance, when most of the mass is concentrated very close to the pivot — the period can approach or even be shorter than a simple pendulum of the same center-of-mass distance. However, due to the parallel axis theorem, I is always at least md², so there is a physical lower bound on the period.

Temperature causes materials to expand or contract (thermal expansion), which changes the pendulum's dimensions and therefore its moment of inertia and center-of-mass distance. Even small dimensional changes can cause measurable period drift in precision instruments like pendulum clocks. High-precision clocks use temperature-compensating alloys (e.g., Invar) to minimize this effect.

For small oscillations, the total mechanical energy is E = ½ I ω² θ₀², where I is the moment of inertia, ω is the angular frequency, and θ₀ is the initial angle in radians. This equals the maximum potential energy at the extremes of the swing and also equals the maximum kinetic energy at the equilibrium position.