---
title: "Physical pendulum and interchangeability"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:compound-pendulum-interchangeability"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/compound-pendulum-interchangeability
site: BCT Engineering
---

# Physical pendulum and interchangeability

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2070 Asar, 5 marks; 2068 Shrawan, 5 marks (asked 2 times)
- Video: [Compound Pendulum: Four Co-linear Points about which Time Period is same | Part-2](https://www.youtube.com/watch?v=59Zxkj9TBHI) — MechE Design Course
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/compound-pendulum-interchangeability
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

Derive an expression for the time period of a physical (compound) pendulum and establish the interchangeability of the center of oscillation and the point of suspension.

## Short answer

For a compound pendulum of mass m, distance l from pivot to CG, radius of gyration k, T = 2π√((k²+l²)/(gl)). The centre of oscillation O′ is at distance L = l + k²/l from pivot. Suspending at O′ gives same period: T′ = 2π√((k²+l′²)/(gl′)) = T, proving interchangeability.

## Answer

A **compound (physical) pendulum** is any rigid body free to oscillate in a vertical plane about a fixed horizontal axis that does not pass through its centre of gravity.

```text
      O  <- point of suspension
      |\
      | \  l
      |  \
      |   G  (centre of gravity)  mg
      |
```

#### Time period
Let the mass be m, O the pivot, G the centre of gravity at distance *l* from O, and k the radius of gyration about G. Displace the body by a small angle θ.
1. Restoring torque = −mg l sinθ ≈ −mg l θ.
2. Moment of inertia about O (parallel-axis theorem): I = mk² + ml² = m(k² + l²).
3. I d²θ/dt² = −mgl θ gives d²θ/dt² + [gl/(k² + l²)] θ = 0, which is SHM with ω² = gl/(k² + l²).
```text
T = 2π √((k² + l²)/(g l)) = 2π √(L/g) ,   L = (k² + l²)/l = l + k²/l
```
L is the length of the equivalent simple pendulum.

#### Interchangeability
4. The centre of oscillation O′ lies on the line OG extended, at distance L from O. Its distance from G is
```text
l′ = L − l = k²/l
```
5. Now suspend the body at O′. Its distance from G is l′, so the new period is
```text
T′ = 2π √((k² + l′²)/(g l′))
```
6. Substitute l′ = k²/l: k² + l′² = k² + k⁴/l² = k²(l² + k²)/l². Dividing by l′ = k²/l gives (l² + k²)/l.
```text
T′ = 2π √((k² + l²)/(g l)) = T
```
7. So the period about O′ equals the period about O: the centre of suspension and the centre of oscillation are interchangeable. Also, the new equivalent length is L′ = l′ + k²/l′ = k²/l + l = L, the same as before.

## Check yourself

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

1. The moment of inertia about the pivot O is found using?
   A. I = m l^2 only
   B. I = m k^2 only
   C. Parallel-axis theorem: I = m(k^2 + l^2)
   D. Perpendicular-axis theorem
   Answer: C. I about G is mk^2; shifting by l adds ml^2. (Key point: Moment of inertia)

2. The period of a compound pendulum is?
   A. 2pi sqrt(g l/(k^2 + l^2))
   B. 2pi sqrt(l/g)
   C. 2pi sqrt((k^2 - l^2)/g l)
   D. 2pi sqrt((k^2 + l^2)/(g l))
   Answer: D. From I theta'' = -mgl theta, w^2 = gl/(k^2+l^2). (Key point: Time period)

3. Where is the centre of oscillation from G?
   A. On the opposite side at l' = k^2/l
   B. On the same side at l
   C. At G
   D. At l' = k l
   Answer: A. L - l = (l + k^2/l) - l = k^2/l. (Key point: Centre of oscillation)

4. Why is the period the same about O'?
   A. Because O' has the same mass
   B. Substituting l' = k^2/l gives (k^2 + l'^2)/l' = (k^2 + l^2)/l
   C. Because the period is independent of l
   D. Because k = l
   Answer: B. The equivalent length is unchanged so T' = T. (Key point: Interchangeability)

## Related in Engineering Physics

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- [Ring pendulum minimum time period](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aring-pendulum-minimum-period) — Engineering Physics, chapter 1

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