---
title: "Torsional pendulum, modulus of rigidity"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:bar-torsional-pendulum-rigidity"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/bar-torsional-pendulum-rigidity
site: BCT Engineering
---

# Torsional pendulum, modulus of rigidity

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2073 Chaitra, 5 marks; 2072 Kartik, 5 marks (asked 2 times)
- Video: [Torsion Pendulum- Dynamic method | Jeya P | Department of Physics](https://www.youtube.com/watch?v=1yvhjjgLxhs) — CMS College Kottayam
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/bar-torsional-pendulum-rigidity
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

Differentiate between bar (compound) pendulum and torsional pendulum. Using a torsional pendulum, derive a relation for the modulus of rigidity of the metallic wire. (Show that the motion of a torsional pendulum is angular SHM.)

## Short answer

Bar pendulum swings about horizontal axis under gravity; torsional pendulum twists about vertical wire with elastic torque. Motion is angular SHM: d²θ/dt² + (C/I)θ = 0. With C = πηr⁴/(2l) and T = 2π√(I/C), modulus of rigidity η = 8πl I/(r⁴ T²).

## Answer

#### Difference between bar pendulum and torsional pendulum
| Bar (compound) pendulum | Torsional pendulum |
|---|---|
| Rigid body swings about a horizontal axis | Body twists about a vertical wire |
| Restoring torque is due to gravity: −mgl sinθ | Restoring torque is due to elasticity of wire: −Cθ |
| T = 2π√((k² + l²)/gl) | T = 2π√(I/C) |
| Depends on g | Independent of g |
| Small angles needed (sinθ ≈ θ) | Twist must stay within elastic limit |
| Used to find g | Used to find η and moment of inertia I |

#### Angular SHM of a torsional pendulum
A disc of moment of inertia I is hung from a wire of length *l* and radius *r*. Twist the disc by angle θ.

1. Wire exerts restoring torque proportional to the twist: τ = −Cθ, where C = torsional constant (torque per unit twist).
2. Newton's law for rotation: I d²θ/dt² = −Cθ.
3. d²θ/dt² + (C/I) θ = 0, which is of the form of SHM, d²θ/dt² + ω²θ = 0, with ω = √(C/I).
4. So the motion is angular SHM with period T = 2π√(I/C).

#### Modulus of rigidity
5. Consider a thin cylindrical shell of radius x, thickness dx, length *l*. Twisting by θ shears its surface through angle φ = xθ/l.
6. Shear stress = ηφ, force on the shell's cross-section = ηφ · 2πx dx, torque = force × x:
```text
dτ = η (xθ/l)(2πx dx) x = (2πηθ/l) x³ dx
τ = (2πηθ/l) ∫₀ʳ x³ dx = πηr⁴ θ / 2l
```
7. Compare with τ = Cθ: C = πηr⁴/2l.
8. Substitute in T = 2π√(I/C): T² = 4π²I/C = 8π l I /(η r⁴).

```text
η = 8π l I / (r⁴ T²)
```
Measure *l*, *r*, T and know I (e.g. I = ½MR² for a disc) to get η of the wire.

## Check yourself

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

1. What provides the restoring torque in a torsional pendulum?
   A. Gravity acting on the disc
   B. Elasticity of the twisted wire, tau = -C theta
   C. Air resistance
   D. The tension in the wire acting at the centre
   Answer: B. The sheared wire exerts a torque proportional to the twist, unlike the bar pendulum where gravity does. (Key point: Restoring torque)

2. Why is the torsional pendulum's motion angular SHM?
   A. Because T depends on g
   B. Because the disc moves in a straight line
   C. I d2theta/dt2 = -C theta, so angular acceleration is proportional and opposite to angular displacement
   D. Because the torque is constant
   Answer: C. The equation has the SHM form d2theta/dt2 + (C/I) theta = 0 with w = sqrt(C/I). (Key point: Angular SHM)

3. What is the torsional constant of a wire of length l and radius r?
   A. C = pi eta r^2 / l
   B. C = 2 pi eta l / r^4
   C. C = eta r^4 / l
   D. C = pi eta r^4 / 2l
   Answer: D. Integrating torques over thin cylindrical shells gives tau = pi eta r^4 theta / 2l. (Key point: Torsional constant)

4. Which relation gives eta from a torsional pendulum?
   A. eta = 8 pi l I / (r^4 T^2)
   B. eta = 4 pi^2 I / T^2
   C. eta = 8 pi l I T^2 / r^4
   D. eta = 2 pi l I / (r^2 T^2)
   Answer: A. From T^2 = 4pi^2 I/C with C = pi eta r^4/2l we get eta = 8 pi l I/(r^4 T^2). (Key point: Modulus of rigidity)

## 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 time period](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atorsional-pendulum-time-period) — Engineering Physics, chapter 1, asked 2×
- [Four collinear points of compound pendulum](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Abar-torsional-four-collinear-points) — Engineering Physics, chapter 1, asked 2×
- [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
- [Compound pendulum with added mass](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acompound-pendulum-added-mass) — Engineering Physics, chapter 1

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