---
title: "SHM energy fractions at half amplitude"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:shm-half-amplitude-ke-pe"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/shm-half-amplitude-ke-pe
site: BCT Engineering
---

# SHM energy fractions at half amplitude

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2072 Chaitra, 5 marks
- Video: [When the displacement is one-half the amplitude, what fraction of the total energy is kinetic and](https://www.youtube.com/watch?v=8ibAP3BYaso) — Doubtnut
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/shm-half-amplitude-ke-pe
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

In simple harmonic motion, when the displacement is one-half the amplitude, what fraction of the total energy is KE and what fraction is PE? At what displacement is the energy half KE and half PE?

## Short answer

At x = a/2, PE/E = (a/2)²/a² = 1/4, so KE/E = 3/4. Energy is half KE and half PE when ½kx² = ½E → x² = a²/2, so x = ±a/√2 ≈ ±0.707a.

## Answer

For SHM with amplitude a and displacement x:
```text
Total energy  E = ½ k a²
PE = ½ k x²
KE = ½ k (a² − x²)
```

#### Energy fractions at x = a/2
```text
PE/E = x²/a² = (a/2)²/a² = 1/4
KE/E = 1 − PE/E = 1 − 1/4 = 3/4
```
**Answer: KE = 3/4 (75%) of the total energy and PE = 1/4 (25%).**

#### Displacement at which KE = PE
Equal halves means PE = E/2:
```text
½ k x² = ½ (½ k a²)
x² = a²/2
x = ± a/√2 = ± 0.707 a
```
**Answer: the energy is half KE and half PE at x = ± a/√2 ≈ ± 0.707 a.**

Check: at x = a/√2, KE = ½k(a² − a²/2) = ¼ka², PE = ¼ka²; each is ½E.

## Check yourself

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

1. KE at displacement x is?
   A. (1/2)k x^2
   B. (1/2)k a^2
   C. (1/2)k(a^2 - x^2)
   D. k a x
   Answer: C. E - PE. (Key point: Energy relations)

2. At x = a/2, KE fraction is?
   A. 1/4
   B. 1/2
   C. 1/8
   D. 3/4
   Answer: D. PE/E = 1/4. (Key point: At a/2)

3. At x = a/2, PE fraction is?
   A. 1/4
   B. 3/4
   C. 1/2
   D. 1/2^(1/2)
   Answer: A. x^2/a^2. (Key point: PE)

4. KE = PE at x =?
   A. a/2
   B. a/sqrt(2)
   C. a/4
   D. a/sqrt(3)
   Answer: B. x^2 = a^2/2. (Key point: Equal energies)

## Related in Engineering Physics

- [Average KE is half total energy](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ashm-average-ke-half-total) — Engineering Physics, chapter 1
- [Energy of SHM, total energy constant](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ashm-energy-total-invariant) — Engineering Physics, chapter 1
- [SHM amplitude from two speeds](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ashm-amplitude-from-speeds) — Engineering Physics, chapter 1

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