---
title: "Energy of SHM, total energy constant"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:shm-energy-total-invariant"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/shm-energy-total-invariant
site: BCT Engineering
---

# Energy of SHM, total energy constant

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2074 Asoj, 5 marks
- Video: [Energy in Simple Harmonic Motion](https://www.youtube.com/watch?v=Hs_GeBx15Ws) — Physics with Professor Matt Anderson
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/shm-energy-total-invariant
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

Define SHM. Derive the expression for energy of SHM. Show that the KE and PE of a simple harmonically oscillating object change with time however the total energy is invariant.

## Short answer

SHM is periodic motion with restoring force proportional to displacement (F=-kx). Kinetic energy = ½mω²a²cos²(ωt+φ), potential energy = ½mω²a²sin²(ωt+φ). Sum is constant total energy E = ½mω²a² = ½k a². KE and PE oscillate out of phase, each varying between 0 and maximum, but total remains invariant.

## Answer

**Simple harmonic motion (SHM)** is periodic motion of a particle in which the restoring force (or acceleration) is proportional to the displacement from the equilibrium position and is directed towards it: F = −kx.

#### Equation and solution
m d²x/dt² = −kx gives x = a sin(ωt + φ), with ω = √(k/m), a = amplitude. Then v = dx/dt = aω cos(ωt + φ).

#### Potential energy
Work done against the restoring force to displace by x:
```text
PE = ∫₀ˣ kx dx = ½ k x² = ½ m ω² a² sin²(ωt + φ)
```

#### Kinetic energy
```text
KE = ½ m v² = ½ m ω² a² cos²(ωt + φ) = ½ m ω² (a² − x²)
```

#### Total energy
```text
E = KE + PE = ½ m ω² a² [cos²(ωt + φ) + sin²(ωt + φ)]
E = ½ m ω² a² = ½ k a²
```
This does not depend on time t.

#### Variation with time
- KE and PE both vary with time between 0 and ½mω²a², with frequency 2ω (twice that of the motion).
- At the mean position (x = 0): PE = 0, KE = ½mω²a² (maximum).
- At extreme positions (x = ±a): KE = 0, PE = ½mω²a² (maximum).
- When KE falls, PE rises by exactly the same amount.

```text
 E |-------------------------  total E
   |  \    /\    /\
   |   \  /  \  /     KE and PE are
   |    \/    \/      out of phase
   |__________________ t
```
So KE and PE change with time, but their sum ½ka² stays constant; no energy is lost in ideal SHM.

## Check yourself

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

1. SHM is motion where?
   A. Restoring force is proportional to displacement and directed towards equilibrium
   B. Force is constant
   C. Force is proportional to velocity
   D. Acceleration is constant
   Answer: A. F = -kx. (Key point: Definition)

2. PE at displacement x is?
   A. k x^2
   B. (1/2) k x^2
   C. (1/2) k x
   D. k x
   Answer: B. Work done against the restoring force: integral of kx dx. (Key point: Potential energy)

3. The total energy is?
   A. (1/2) k x^2
   B. (1/2) m v^2 only
   C. (1/2) k a^2 constant in time
   D. Varies with time
   Answer: C. KE + PE = (1/2)mw^2a^2 (cos^2 + sin^2). (Key point: Total energy)

4. KE and PE vary with what frequency?
   A. w
   B. w/2
   C. They do not vary
   D. 2w, twice that of the motion
   Answer: D. Because sin^2 and cos^2 have double frequency. (Key point: Variation)

## 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
- [SHM energy fractions at half amplitude](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ashm-half-amplitude-ke-pe) — Engineering Physics, chapter 1
- [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
- [Cyclotron working; semicircle time constant](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acyclotron-time-semicircle-independent) — Engineering Physics, chapter 8, asked 2×
- [Energy stored in coil, time constants](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acoil-energy-stored-time-constants) — Engineering Physics, chapter 8
- [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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