---
title: "Torsional pendulum time period"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:torsional-pendulum-time-period"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/torsional-pendulum-time-period
site: BCT Engineering
---

# Torsional pendulum time period

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2071 Chaitra, 5 marks; 2068 Chaitra, 5 marks (asked 2 times)
- Video: [Torsional Pendulum | Time period equation derivation | Couple per unit Twist | Modulus of Rigidity](https://www.youtube.com/watch?v=52FRnYtPEwA) — Gautam Varde
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/torsional-pendulum-time-period
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

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.

## 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)).

## Answer

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.

```text
   ///////// 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:
```text
I d²θ/dt² = −Cθ
d²θ/dt² + (C/I) θ = 0
```
3. 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
4. T = 2π/ω:
```text
T = 2π √(I/C)
```
5. For a wire of modulus of rigidity η, C = πηr⁴/2l, so
```text
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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is a torsional pendulum?
   A. A rigid body hung from a wire that oscillates in rotation about the wire
   B. A mass on a spring
   C. A rigid body swinging about a horizontal axis
   D. A string with a bob
   Answer: A. It twists about the wire as axis. (Key point: Definition)

2. Which equation shows angular SHM?
   A. I d2theta/dt2 = -mgl theta
   B. I d2theta/dt2 = -C theta
   C. I d2theta/dt2 = C
   D. d2theta/dt2 = -g theta
   Answer: B. The restoring torque is proportional to and opposite to the twist. (Key point: Equation)

3. What is the period?
   A. T = 2pi sqrt(C/I)
   B. T = 2pi sqrt(l/g)
   C. T = 2pi sqrt(I/C)
   D. T = sqrt(I/C)
   Answer: C. With w^2 = C/I, T = 2pi/w. (Key point: Period)

4. Which statement about the period is correct?
   A. It increases with amplitude
   B. It depends on g
   C. It decreases with I
   D. It is independent of amplitude and of g
   Answer: D. The period depends on I and C only, within the elastic limit. (Key point: Dependence)

## Related in Engineering Physics

- [Bar vs torsional pendulum; minimum period](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Abar-torsional-similarities-minimum-period) — Engineering Physics, chapter 1
- [Torsional pendulum, modulus of rigidity](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Abar-torsional-pendulum-rigidity) — Engineering Physics, chapter 1, asked 2×
- [Ring pendulum minimum time period](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aring-pendulum-minimum-period) — Engineering Physics, chapter 1
- [Moment of inertia by torsional pendulum](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amoment-of-inertia-torsional-pendulum) — Engineering Physics, chapter 1
- [Compound pendulum centres and minimum period](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acompound-pendulum-centers-minimum) — Engineering Physics, chapter 1
- [Bar pendulum minimum period condition](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Abar-pendulum-minimum-period-gyration) — Engineering Physics, chapter 1

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