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Torsional pendulum time period

Short answer

A torsional pendulum is a disc suspended by a wire that twists. Restoring torque τ = −Cθ yields I d²θ/dt² + Cθ = 0, angular SHM. Time period T = 2π√(I/C). For a wire of length l, radius r, C = πηr⁴/(2l) so T = 2π√(2Il/(πηr⁴)).

What is a torsional pendulum? Show that its motion is angular harmonic motion and derive the expression for the time period of oscillation for this pendulum.

A torsional pendulum is a rigid body (usually a disc or bar) suspended by a thin wire fixed at its upper end. When twisted and released, it oscillates in rotation about the wire as axis.

   ///////// support
      |
      |  wire (length l, radius r)
      |
   ===O===  disc, moment of inertia I
   <- θ ->  twist

Angular harmonic motion

  1. Twist the disc by a small angle θ. The wire is sheared and exerts a restoring torque proportional to θ (within the elastic limit): τ = −Cθ. C is the torsional rigidity (torque per unit twist).
  2. Equation of rotational motion, τ = I α:
I d²θ/dt² = −Cθ
d²θ/dt² + (C/I) θ = 0
  1. This has the form d²θ/dt² + ω²θ = 0 with ω² = C/I. The angular acceleration is proportional to the angular displacement and opposite to it, so the motion is angular SHM: θ = θ0 sin(ωt + φ).

Time period

  1. T = 2π/ω:
T = 2π √(I/C)
  1. For a wire of modulus of rigidity η, C = πηr⁴/2l, so
T = 2π √(2 I l / (π η r⁴))

The period depends on I and the wire (C), not on the angular amplitude, so it is isochronous, and it does not depend on g.

✓ Answer checked by Claude Opus 5.5 (high effort)

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