---
title: "Fractional change in damped frequency"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:damped-fractional-frequency-change"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/damped-fractional-frequency-change
site: BCT Engineering
---

# Fractional change in damped frequency

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2070 Asar, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/damped-fractional-frequency-change
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

Show that the fractional change in frequency of damped oscillation is 1/8Q² where Q is the quality factor.

## Short answer

Damped frequency ω' = √(ω0²−γ²) ≈ ω0(1−γ²/2ω0²). Fractional change (ω0−ω')/ω0 = γ²/(2ω0²). With Q = ω0/2γ, γ/ω0 = 1/2Q, so fractional change = 1/(8Q²).

## Answer

For a damped oscillator x'' + 2γx' + ω0²x = 0, the natural (undamped) angular frequency is ω0 and the damped frequency is
```text
ω′ = √(ω0² − γ²) ,    γ = b/2m
```
Quality factor: Q = ω0/2γ.

#### Derivation
1. Write ω′ = ω0 √(1 − γ²/ω0²).
2. For light damping γ << ω0, use the binomial expansion (1 − ε)^(1/2) ≈ 1 − ε/2:
```text
ω′ ≈ ω0 (1 − γ²/2ω0²)
```
3. Change in frequency:
```text
ω0 − ω′ = γ²/(2ω0)
(ω0 − ω′)/ω0 = γ²/(2ω0²)
```
4. From Q = ω0/2γ: γ/ω0 = 1/2Q, so γ²/ω0² = 1/4Q².
5. Substitute:
```text
(ω0 − ω′)/ω0 = (1/2)(1/4Q²) = 1/(8Q²)
```
Hence the fractional change in frequency due to damping is 1/8Q². For Q = 10 it is only 0.125%, so a high-Q oscillator oscillates at nearly its natural frequency.

## Check yourself

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

1. w' =?
   A. sqrt(w0^2 - gamma^2)
   B. sqrt(w0^2 + gamma^2)
   C. w0 - gamma
   D. w0 gamma
   Answer: A. Underdamped. (Key point: Damped frequency)

2. For gamma << w0, w' approx?
   A. w0 (1 - gamma/w0)
   B. w0 (1 - gamma^2/2w0^2)
   C. w0 (1 + gamma^2/2w0^2)
   D. w0 (1 - gamma^2/w0^2)
   Answer: B. (1-e)^(1/2) approx 1 - e/2. (Key point: Binomial)

3. Q = w0/2gamma gives gamma^2/w0^2 =?
   A. 1/2Q
   B. 1/Q^2
   C. 1/4Q^2
   D. 4Q^2
   Answer: C. gamma/w0 = 1/2Q. (Key point: Q relation)

4. Fractional change =?
   A. 1/4Q^2
   B. 1/2Q
   C. 1/Q^2
   D. 1/8Q^2
   Answer: D. (1/2)(1/4Q^2). (Key point: Result)

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