---
title: "Damped oscillator relaxation time numerical"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:damped-relaxation-amplitude-energy-fifty"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/damped-relaxation-amplitude-energy-fifty
site: BCT Engineering
---

# Damped oscillator relaxation time numerical

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2073 Chaitra, 5 marks
- Video: [Decay Constant/Relaxation Time, Logarithmic  Decrement and Quality Factor for damped oscillation](https://www.youtube.com/watch?v=fNwxOOeTVqk) — PhysicsClassForYou
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/damped-relaxation-amplitude-energy-fifty
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

If the relaxation time of a damped harmonic oscillator is 50 sec, find the time in which the amplitude falls to 1/e³ times the initial value and the energy of the system falls to 1/e⁴ of its initial value.

## Short answer

Using relaxation time τ=50 s, amplitude a = a0 e^(-t/2τ) and energy E = E0 e^(-t/τ). For amplitude to 1/e³, t=6τ=300 s; for energy to 1/e⁴, t=4τ=200 s.

## Answer

**Convention used:** for a damped oscillator with damping force −bv, amplitude a = a0 e^(−bt/2m) and energy E = E0 e^(−bt/m). The relaxation time is τ = m/b, the time in which the energy falls to 1/e of its initial value. Hence a = a0 e^(−t/2τ) and E = E0 e^(−t/τ).

Given τ = 50 s.

#### (i) Amplitude falls to 1/e³
```text
a/a0 = e^(−t/2τ) = e^(−3)
t/2τ = 3
t = 6τ = 6 × 50 = 300 s
```

#### (ii) Energy falls to 1/e⁴
```text
E/E0 = e^(−t/τ) = e^(−4)
t = 4τ = 4 × 50 = 200 s
```

**Answer: the amplitude falls to 1/e³ in 300 s (5 min); the energy falls to 1/e⁴ in 200 s.**

Check: at t = 300 s the energy is E0 e^(−6), consistent with E ∝ a²: (e^(−3))² = e^(−6). At t = 200 s the amplitude is a0 e^(−2).

## Check yourself

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

1. Relaxation time tau is the time in which?
   A. Amplitude falls to 1/e
   B. Energy falls to zero
   C. Energy falls to 1/e
   D. Amplitude halves
   Answer: C. E = E0 e^(-t/tau). (Key point: Convention)

2. Amplitude behaves as?
   A. a0 e^(-t/tau)
   B. a0 e^(-2t/tau)
   C. a0 e^(-t^2/tau)
   D. a0 e^(-t/2tau)
   Answer: D. Amplitude falls at half the energy rate. (Key point: Amplitude)

3. Time for amplitude to fall to 1/e^3 with tau = 50 s?
   A. 300 s
   B. 150 s
   C. 100 s
   D. 200 s
   Answer: A. t = 6 tau. (Key point: Time for 1/e^3)

4. Time for energy to fall to 1/e^4?
   A. 300 s
   B. 200 s
   C. 100 s
   D. 400 s
   Answer: B. t = 4 tau = 200 s. (Key point: Time for energy)

## Close variants set in other exams

- [Damped oscillator time for 1/e amplitude](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adamped-relaxation-amplitude-energy-sixty) — Engineering Physics, chapter 1, asked 2×

## Related in Engineering Physics

- [Damped oscillation energy decreases with time](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-oscillation-damped-energy-decrease) — Engineering Physics, chapter 1

## Related in other subjects

- [Barkhausen criteria and relaxation oscillator](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Abarkhausen-relaxation-oscillator) — Fundamental of Electrical and Electronics Engineering, chapter 6