The Torsion Pendulum and Real Oscillators

Twist a disk hanging from a wire and let go. The wire resists the twist and pulls the disk back the way a spring pulls a stretched mass back. That restoring torque is proportional to the angle of twist, not the angle squared or anything messier, so the equation of motion looks exactly like the one for a mass on a spring, but with rotational quantities standing in for linear ones. Moment of inertia I replaces mass m, torsion constant kappa (the torque per radian of twist) replaces spring constant k, and angle theta replaces position x.

Newton's second law for rotation gives I times the angular acceleration equals minus kappa times theta. That is the same differential equation as x'' = -(k/m)x, just relabeled. Its solution is theta(t) = theta_max cos(omega t + phi), with angular frequency omega = √(kappa/I) and period T = 2 π √(I/kappa). Bigger I means more rotational inertia to drag around, so the period grows; bigger kappa means a stiffer wire fighting the twist harder, so the period shrinks.

No real wire is frictionless and no real disk swings in a vacuum. Air resistance and internal friction in the wire bleed energy out of the system every cycle. For light damping, this does not change the frequency much, but it shrinks the amplitude cycle after cycle, usually following an exponential decay envelope. This is the same story told by every oscillator in the unit: a spring losing energy to air drag, a pendulum losing energy to friction at its pivot, all obeying the same underlying differential equation of motion plus a damping term, with the undriven, undamped version being the idealized case this topic builds from.