Physical Pendulum Calculator

Enter the mass, moment of inertia about the center of mass, pivot-to-center-of-mass distance, and local gravity to find the oscillation period (T = 2π√(I / (mgd))), frequency, moment of inertia about the pivot, and the equivalent simple-pendulum length.

Quick Facts

Period formula
T = 2π√(I / (mgd))
I is the moment of inertia about the pivot, not the center of mass — mass and distance alone are not enough.
Parallel axis theorem
I_pivot = I_cm + m·d²
Converts a tabulated center-of-mass moment of inertia into the pivot-axis value the period formula needs.
Simple-pendulum limit
I_cm = 0 → T = 2π√(d/g)
If all mass were concentrated at distance d, the physical pendulum formula reduces to the familiar simple-pendulum formula.

Your Results

Calculated
Period
-
T = 2π√(I / (mgd)), in seconds (s)
Frequency
-
f = 1 / T, in hertz (Hz)
Moment of Inertia about Pivot
-
I = I_cm + m·d² (parallel axis theorem), in kg·m²
Equivalent Simple-Pendulum Length
-
L = I / (m·d), in meters (m)

Ready

Enter the mass, moment of inertia, pivot distance, and gravity, then press Calculate.

Formula and Method for the Physical Pendulum

A physical (or compound) pendulum is any rigid body free to swing about a fixed horizontal pivot that does not pass through its center of mass — a swinging door, a metronome arm, a meter stick pivoted near one end, or a leg swinging from the hip. Unlike a simple pendulum, which idealizes all the mass as a single point on a massless string, a physical pendulum's mass is spread out, so its moment of inertia — not just its mass and length — controls how fast it swings. For small oscillations, the period is T = 2π√(I / (mgd)), where I is the moment of inertia about the pivot (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).

Deriving the period from the moment of inertia

Applying Newton's second law for rotation about the pivot, the restoring gravitational torque is -mgd·sin(θ), so I·θ'' = -mgd·sin(θ). For small angular displacements, sin(θ) ≈ θ, giving the simple-harmonic equation θ'' + (mgd / I)θ = 0 with angular frequency ω = √(mgd / I) and period T = 2π/ω = 2π√(I / (mgd)). Because reference tables almost always list the moment of inertia about an object's own center of mass (I_cm) rather than about an arbitrary pivot, this calculator applies the parallel axis theorem, I = I_cm + md², to shift that value to the pivot axis before computing the period.

Working with moment of inertia and distance

  • Moment of inertia must be entered in kg·m² and must already be the value about the object's center of mass, not about the pivot — the calculator adds the m·d² term for you.
  • The distance d is measured from the pivot axis straight to the center of mass, not along the body's surface or edge; a longer d generally shortens the period, but only if I does not grow faster than d² (it often does for extended bodies).
  • Setting I_cm = 0 reduces the formula to T = 2π√(d/g), recovering the familiar simple-pendulum result, since a point mass has no moment of inertia about its own center.

Knowing the limits

This formula assumes small-angle oscillations (roughly under 15-20°), a rigid body (no internal flexing or fluid sloshing), a frictionless pivot with no air resistance, and constant g near the object. Large-amplitude swings require an elliptic-integral correction and no longer have a perfectly constant period; a damped or driven pendulum needs additional terms for friction and any external forcing. Always keep mass, length, and moment of inertia in one consistent unit system (SI: kg, m, s) before comparing results.

Frequently Asked Questions

What is a physical pendulum?
A physical (or compound) pendulum is any rigid body that swings about a fixed pivot that does not pass through its center of mass. Unlike an idealized simple pendulum, which assumes all the mass sits at one point on a massless string, a physical pendulum's mass is distributed, so its moment of inertia — not just its mass and length — determines how fast it swings.
What is the formula for the period of a physical pendulum?
For small oscillations, the period is T = 2π√(I / (mgd)), where I is the moment of inertia about the pivot (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 comes from applying Newton's second law for rotation, I·θ'' = -mgd·sin(θ), and using sin(θ) ≈ θ for small angles.
What is the parallel axis theorem, and why is it needed here?
The parallel axis theorem converts a moment of inertia known about the center of mass into one about a parallel axis through the pivot: I_pivot = I_cm + m·d², where d is the distance between the two axes. Almost every reference table (rods, disks, spheres) lists I_cm, so this step is required before plugging into the period formula.
What is the equivalent simple pendulum length?
It is the length L = I / (m·d) that a simple pendulum (a point mass on a massless string) would need in order to swing with exactly the same period as the physical pendulum. It is useful for building intuition about how "long" a rigid body behaves, even though its actual size may be very different from L.

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