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

# Damped oscillator time for 1/e amplitude

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2071 Chaitra, 5 marks
- Video: [Lecture 11 RELAXATION  TIME  ( FOR DAMPED ELECTRICAL  AND MECHANICAL OSCILLATOR) //With Notes](https://www.youtube.com/watch?v=IJfHGjp2hwI) — Sarbdeep Kaur Sidhu
- 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-sixty
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

If the relaxation time of a damped harmonic oscillator is 60 sec, find the time for which (a) the amplitude falls to 1/e times the initial value and (b) the energy falls to 1/e² times the initial value.

## Short answer

With τ=60 s, amplitude a = a0 e^(-t/2τ) so for 1/e, t=2τ=120 s. Energy E = E0 e^(-t/τ) so for 1/e², t=2τ=120 s. Both times are 120 s because energy ∝ amplitude².

## Answer

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

Given τ = 60 s.

#### (a) Amplitude falls to 1/e
```text
a/a0 = e^(−t/2τ) = e^(−1)
t = 2τ = 2 × 60 = 120 s
```

#### (b) Energy falls to 1/e²
```text
E/E0 = e^(−t/τ) = e^(−2)
t = 2τ = 2 × 60 = 120 s
```

**Answer: (a) 120 s (2 min); (b) 120 s.**

Both times are the same, as they must be: energy ∝ (amplitude)², so when the amplitude is 1/e the energy is 1/e².

## Check yourself

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

1. Relaxation time is the time energy falls to?
   A. 1/e^2
   B. 1/2
   C. 1/e
   D. Zero
   Answer: C. E = E0 e^(-t/tau). (Key point: Convention)

2. With tau = 60 s, amplitude falls to 1/e at?
   A. 60 s
   B. 30 s
   C. 240 s
   D. 120 s
   Answer: D. t = 2 tau. (Key point: Amplitude 1/e)

3. Energy falls to 1/e^2 at?
   A. 120 s
   B. 60 s
   C. 180 s
   D. 240 s
   Answer: A. t = 2 tau. (Key point: Energy 1/e^2)

4. Why are both times equal?
   A. Amplitude and energy are independent
   B. Energy is proportional to amplitude squared
   C. Energy decays slower
   D. tau is the same as amplitude time
   Answer: B. (1/e)^2 = 1/e^2. (Key point: Why equal)

## Close variants set in other exams

- [Damped oscillator relaxation time numerical](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adamped-relaxation-amplitude-energy-fifty) — 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