Calculate the period of a physical (compound) pendulum using T = 2π√(I/mgd). Common shapes with moment of inertia presets.
The **Physical Pendulum Calculator** determines the oscillation period of a real, extended body swinging about a fixed pivot — as opposed to an idealized point-mass simple pendulum. The period formula T = 2π√(I/mgd) accounts for how mass is distributed throughout the body by using the moment of inertia I about the pivot point, the total mass m, and the distance d from the pivot to the center of mass.
Different shapes have dramatically different periods even with the same mass and overall size, because their mass distributions yield different moments of inertia. A uniform rod pivoted at one end swings with a different period than a disk pivoted at its rim, even if they weigh the same and have the same characteristic dimension. This calculator includes built-in formulas for common shapes (rods, disks, rings, spheres) and a custom mode for entering your own moment of inertia.
The tool also calculates the equivalent simple pendulum length — the length a simple (point-mass) pendulum would need to have the same period. This comparison helps build intuition about how mass distribution affects oscillatory behavior. A comparison table shows periods for all built-in shapes with your parameters, making it easy to see which configurations oscillate faster or slower.
Real oscillating objects are never perfect point masses on massless strings. Swinging doors, metronome arms, wrecking balls, and pendulum clocks all behave as physical pendulums. Accurately predicting their period requires accounting for mass distribution through the moment of inertia.
This calculator is invaluable for engineering applications like designing precision pendulums, analyzing mechanical vibrations, and understanding oscillatory motion in robotics and biomechanics. The shape comparison feature lets you quickly evaluate how different geometries affect performance.
Physical pendulum period: T = 2π√(I/(mgd)) Equivalent simple pendulum length: L_eq = I/(md) Common moments of inertia (about pivot): Rod at end: I = mL²/3, d = L/2 Disk at rim: I = 3mR²/2, d = R Ring at rim: I = 2mR², d = R Sphere at surface: I = 7mR²/5, d = R Variables: I = moment of inertia about pivot, m = mass, g = gravity, d = distance from pivot to center of mass
Result: 1.6395 s period
A 2 kg, 1 m uniform rod pivoted at one end has I = mL²/3 = 2(1)²/3 = 0.667 kg·m², d = L/2 = 0.5 m. Period T = 2π√(0.667/(2×9.807×0.5)) = 2π√(0.0680) = 1.6395 s. The equivalent simple pendulum length is L_eq = I/(md) = 0.667 m.
The physical pendulum generalizes the simple pendulum to real, extended bodies. While a simple pendulum treats the bob as a point mass, every real swinging object — from a grandfather clock pendulum to a swinging gate — has mass distributed over its volume. This distribution is captured by the moment of inertia I about the pivot axis.
The equation of motion for a physical pendulum comes from Newton's second law for rotation: Iα = -mgd sin θ. For small angles (sin θ ≈ θ), this gives simple harmonic motion with period T = 2π√(I/mgd). The key insight is that increasing I (more mass far from the pivot) makes the pendulum swing more slowly, while increasing d (pivot farther from center of mass) makes it swing faster.
Each geometric shape has a characteristic moment of inertia formula. For a uniform rod of length L pivoted at one end: I = mL²/3. For a disk pivoted at its rim: I = 3mR²/2. These formulas, combined with the parallel axis theorem I_pivot = I_cm + md², allow calculation of the moment about any pivot point for any shape.
Physical pendulum analysis is used in clockmaking (designing pendulums with specific periods), automotive engineering (analyzing crankshaft oscillations), structural engineering (modeling building sway), and robotics (designing limbs and balance systems). The center of percussion — related to the center of oscillation — is important in sports equipment design (the "sweet spot" on a bat or racket).
A simple pendulum is an idealization: a point mass on a massless string. A physical pendulum is any real, extended body that can swing about a fixed axis. The physical pendulum formula uses the moment of inertia to account for mass distribution, while the simple pendulum formula only needs length.
Mass farther from the pivot has more rotational inertia (resistance to angular acceleration). A rod pivoted at its end has more inertia than the same rod pivoted at its center, so it swings more slowly. The moment of inertia captures this effect mathematically.
It is the length a simple pendulum would need to have the same period as the physical pendulum: L_eq = I/(md). This is always longer than the actual pivot-to-CoM distance d, which is why physical pendulums swing slower than you might expect.
Yes — select "Custom" and enter the moment of inertia about the pivot point and the distance from pivot to center of mass. You can calculate I using the parallel axis theorem: I_pivot = I_cm + md².
The period becomes infinite (the body does not oscillate). This makes physical sense — there is no gravitational torque when the center of mass is at the pivot point.
Like the simple pendulum formula, T = 2π√(I/mgd) assumes small oscillations. For large angles, the same elliptic integral corrections apply. For angles under 15°, the formula is accurate to within 0.5%.