A physical pendulum is a rigid body that undergoes oscillation about a fixed axis — a swinging door, a metronome arm, a rod pivoted at one end. Unlike the idealized simple pendulum, its mass isn't concentrated at a single point, so its period formula has to account for how that mass is actually distributed:
Here, I is the rigid body's rotational inertia about the pivot, m is its total mass, g is gravitational acceleration, and d is the distance from the pivot to the body's center of mass. For small amplitudes of motion, this period comes directly from applying Newton's second law in rotational form — which is exactly where the next section picks up.
Adjust a rigid body's rotational inertia, mass, and the distance from its pivot to its center of mass, and watch the period respond.
When a physical pendulum is displaced from equilibrium, the gravitational force acting at its center of mass produces a restoring torque:
As written, τ = −mgd sinθ isn't linear in θ, so it doesn't immediately look like SHM. But for small amplitudes of motion, the small-angle approximation applies:
Substituting that in, combined with Newton's second law in rotational form (τ = Iα), gives:
Slide the displacement angle and compare the exact restoring torque to the small-angle approximation — notice how well they agree at small angles, and how they drift apart as θ grows.
Rearranging that equation into the form α = d²θ/dt² produces exactly the second-order differential equation that defines SHM — now written in terms of angle instead of linear displacement:
A simple pendulum — the hanging point mass on a string you met back in Lesson 7.2 — is just a physical pendulum where all the mass is modeled as a single point at distance l from the pivot. Nothing new is needed; the general formula already contains this case.
Starting from the physical pendulum period formula, show that modeling a simple pendulum as a point mass at distance l reduces it exactly to the formula from Lesson 7.2.
Not every pendulum swings under gravity. A torsion pendulum is a case of SHM where the restoring torque is proportional to the angular displacement of a rotating system — the classic example being a horizontal disk suspended from a wire attached to its center of mass, which twists back and forth in the horizontal plane as the wire resists being twisted:
(This is often written with the symbol k, as in the CED — but it's a torsional constant with units N·m/rad, not the same kind of quantity as a linear spring constant.) The mathematical form is identical to a mass on a spring or a small-angle pendulum: a restoring effect proportional to displacement, producing the exact same SHM differential equation once again.
A horizontal disk hangs from a wire and twists back and forth. Adjust the disk's rotational inertia and the wire's torsion constant to see how the period responds — the same math as a spring, with angle standing in for displacement.
A horizontal disk (I = 0.02 kg·m²) hangs from a wire with torsion constant κ = 0.8 N·m/rad — the same basic setup used in a Cavendish-style torsion balance, the classic instrument for measuring the gravitational constant. Find its period of oscillation.
That's Lesson 7.5, Unit 7, and the entire AP Physics C: Mechanics course, complete — from kinematics and Newton's laws all the way through orbits, rotating systems, and oscillations. Congratulations on making it through all seven units.