---
title: "Damped oscillation energy decreases with time"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:forced-oscillation-damped-energy-decrease"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/forced-oscillation-damped-energy-decrease
site: BCT Engineering
---

# Damped oscillation energy decreases with time

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2067 Asoj, 5 marks
- Video: [Damped Oscillations | Resistive Force & Energy Loss Explained | Class 12 Physics Ch 14 (2026)](https://www.youtube.com/watch?v=m-mJoU9dXFM) — SIR ATTA
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/forced-oscillation-damped-energy-decrease
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

Define forced oscillation. Show that the total energy of the damped oscillation decreases with increasing time.

## Short answer

Forced oscillation is motion driven by a perodic external force. In damped free oscillation, amplitude decays as a0 e^(−γt), so total energy E = ½kA² decreases exponentially as E0 e^(−2γt) = E0 e^(−bt/m). Also, dE/dt = –b v² ≤ 0 confirms continuous energy loss.

## Answer

#### Forced oscillation
A forced oscillation is the oscillation of a system under the continuous action of an external periodic force. After the initial transient, the system vibrates at the frequency of the applied force (not its natural frequency), with constant amplitude. Example: a mass on a spring driven by F0 cos pt, or an LCR circuit driven by an AC source.

#### Energy of damped oscillation decreases with time
For free damped oscillation, m x'' + b x' + kx = 0.

**Method 1 (from the solution).** x = a0 e^(−γt) cos(ω′t + φ), γ = b/2m. For light damping, the amplitude is a0 e^(−γt) and the energy is
```text
E = ½ k a² = ½ k a0² e^(−2γt) = E0 e^(−2γt) = E0 e^(−bt/m)
```
Since the exponent is negative, E falls exponentially with time.

**Method 2 (rate of loss of energy).**
1. E = ½ m v² + ½ k x².
2. dE/dt = m v v' + k x v = v (m x'' + k x).
3. From the equation of motion, m x'' + kx = −b x' = −b v:
```text
dE/dt = −b v²
```
4. Since b > 0 and v² ≥ 0, dE/dt ≤ 0 always. The energy decreases continuously, and the lost energy is converted into heat by the damping force. The rate of loss is largest when the speed is maximum (at the mean position).

```text
 E |\
   | \__
   |     \___
   |__________\____ t     E = E0 e^(−t/τ)
```
Relaxation time τ = m/b is the time for the energy to fall to 1/e of E0. In a forced oscillation, the external force supplies energy at the same average rate as is dissipated, so the amplitude stays constant.

## Check yourself

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

1. In steady state, a forced oscillation vibrates at the
   A. natural frequency of the system
   B. frequency of the applied periodic force
   C. damped frequency
   D. half the driving frequency
   Answer: B. After the transient dies out, the system follows the frequency of the external force, with constant amplitude. (Key point: Definition of forced oscillation)

2. For a damped oscillator, dE/dt equals
   A. +b v²
   B. -b v
   C. -b v²
   D. -k x²
   Answer: C. dE/dt = v(mx'' + kx) = v(-bv) = -bv², which is never positive. (Key point: Rate of energy loss)

3. For light damping, how does the energy vary with time?
   A. E = E0 e^(-bt/m)
   B. E = E0 e^(bt/m)
   C. E = E0 - bt
   D. E = E0 cos(bt/m)
   Answer: A. Amplitude falls as e^(-γt), so E ∝ a² gives E0 e^(-2γt) = E0 e^(-bt/m). (Key point: Exponential decay of energy)

4. The relaxation time τ = m/b is the time for
   A. the amplitude to become zero
   B. the energy to fall to 1/e of E0
   C. the energy to halve
   D. one full oscillation
   Answer: B. At t = τ, E = E0 e^(-1), so it falls to 1/e of the initial value. (Key point: Relaxation time)

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