Torsion pendulum
A disk suspended by a wire oscillates rotationally. The restoring torque is proportional to angular displacement: τ = −κθ. Add damping and it spirals to rest. Add a periodic driving torque and watch resonance amplify the motion when the drive frequency matches the natural frequency.
A torsion pendulum consists of a body (here a disk) suspended from a wire or fiber that provides a restoring torque proportional to the angular displacement. The equation of motion is Iθ'' + bθ' + κθ = τdrive, which is the rotational analog of a mass-spring-damper system.
Without damping or driving, the disk oscillates at its natural frequency ω0 = √(κ/I), where κ is the torsion constant of the wire and I = ½mR² is the moment of inertia of the disk. The period is T = 2π/ω0.
Adding damping causes the amplitude to decay exponentially. The oscillation frequency shifts slightly to ωd = √(ω0² − γ²), where γ = b/(2I) is the damping rate. If γ ≥ ω0, the system is overdamped and returns to rest without oscillating.
When driven with a periodic torque at frequency ωf, the steady-state amplitude peaks sharply when ωf ≈ ω0 — this is resonance. The amplitude at resonance is limited only by damping. Try setting the drive frequency equal to the natural frequency and watch the amplitude grow.
Torsion pendulums are used in Cavendish-style gravitational experiments, seismometers, mechanical watches (balance wheels), and the measurement of material shear modulus.