Deriving a Pendulum Period from Rotational Dynamics

Start with a rigid body free to rotate about a fixed pivot, displaced by angle theta from vertical. Gravity acts at the center of mass, a distance d from the pivot, and creates a restoring torque of magnitude mgd sin(theta). Newton's second law for rotation says this torque equals I alpha, where alpha is the angular acceleration theta double-dot. That gives I theta'' = -mgd sin(theta), an exact but nasty equation: sine of theta mixes with theta itself, and there is no clean formula for theta(t).

The small-angle trick rescues it. For theta measured in radians and small, sin(theta) is almost exactly theta. Swap it in, and the equation becomes I theta'' = -mgd theta, or theta'' = -(mgd/I) theta. That is the harmonic oscillator equation, theta'' = -omega² theta, with omega² = mgd/I. Its solution is sinusoidal motion with period T = 2 π √(I / (mgd)). The physics content is still all there, torque and rotational inertia, but now it is wrapped in a form you already know how to solve because it looks exactly like a mass on a spring.

This is the rotational twin of the spring equation from earlier in the unit: same math, different physical ingredients, I and mgd standing in for mass and spring constant. It also contains the simple pendulum as a special case: treat the bob as a point mass at distance L, so I = mL² and d = L, and T = 2 π √(I/(mgd)) collapses to the familiar T = 2 π √(L/g). Everything downstream in oscillations, energy exchange and resonance, uses this same small-angle linearization.