# Damped oscillations

> CIE A-Level Physics · Unit 18: Oscillations
> Source: https://www.owlsprep.com/study/cie-9702-u18-damped-oscillations/

This sub-topic covers damping, the reduction in amplitude of oscillations caused by dissipative forces that remove energy from oscillating systems. You will learn the three main types of damping and their effects on motion and graphs.

**Prerequisites:** [Simple Harmonic Motion (SHM) fundamentals](https://www.owlsprep.com/study/cie-9702-u18-simple-harmonic-motion/); [Energy in oscillations](https://www.owlsprep.com/study/cie-9702-u18-energy-in-oscillations/)

## Learning objectives

- Distinguish between light, critical and heavy damping
- Describe the effect of damping on amplitude, frequency and energy of oscillations
- Apply the exponential decay formula for amplitude in light damping
- Identify practical applications of different damping types

## What is Damping?

**Damping** — A process where the amplitude of an oscillation decreases over time due to resistive (dissipative) forces that remove energy from the oscillating system. Free damped oscillations have no external energy input.

*Example:* A swinging pendulum slowing down due to air resistance, a mass on a spring moving through water

All real-world oscillations experience damping, because there are always resistive forces like friction or air resistance. An undamped oscillation is an idealization where no energy is lost, so amplitude stays constant forever.

**Worked example:** A student releases a pendulum and records its maximum displacement over time. The maximum displacement decreases from 10 cm to 5 cm over 10 oscillations. Explain why this is an example of damped motion.

1. Recall that damping is caused by dissipative forces that remove energy from the system. For a pendulum, air resistance acts continuously against the motion.
2. The total energy of a pendulum is proportional to the square of its amplitude: $E \propto A^2$.
3. As energy is lost to the surroundings as heat from air resistance, total energy decreases, so maximum amplitude decreases over time. This matches the definition of damped motion.

## Types of Damping

Damping is classified by how quickly amplitude decays and whether the system overshoots equilibrium when returning to rest. There are three core types you need to know for 9702:

| Damping Type | Description | Behaviour |
| --- | --- | --- |
| Light (underdamping) | Amplitude decreases gradually over many oscillations | Repeated oscillations around equilibrium with decreasing peak displacement |
| Critical damping | System returns to equilibrium in the shortest time, no overshoot | No oscillations, displacement decays to zero without crossing equilibrium |
| Heavy (overdamping) | System returns to equilibrium slower than critical damping, no overshoot | No oscillations, displacement decays very gradually to zero |

**Worked example:** After a car drives over a bump, the car body returns to its original ride height quickly without any oscillation. What type of damping is this? Justify your answer.

1. Identify the key observations: there are no oscillations past equilibrium, and the system returns to equilibrium in the fastest possible time.
2. Compare to damping definitions: light damping produces multiple oscillations, heavy damping returns to equilibrium much slower than needed, critical damping is defined as the fastest return to equilibrium without overshoot.
3. Conclusion: Car suspension is intentionally designed for critical damping, so this is critical damping.

> **Exam tip:** CIE examiners regularly ask you to sketch or identify each damping type from a displacement-time graph, so memorise the key features of each category.

## Amplitude and Energy Decay in Light Damping

For light damping, amplitude decays exponentially with time, following the relationship where $A_0$ is initial amplitude, $A(t)$ is amplitude at time $t$, and $\gamma$ is the damping constant:

$$A(t) = A_0 e^{-\gamma t}$$

> **info**
>
> For CIE 9702, you can assume that damping has a negligible effect on the frequency of light damping. The period of lightly damped oscillations is equal to the natural period of the undamped system.

Since total energy is proportional to the square of amplitude, energy also decays exponentially with double the decay constant of amplitude:

$$E(t) = E_0 A^2 / A_0^2 = E_0 e^{-2\gamma t}$$

**Worked example:** A lightly damped oscillator has an initial amplitude of 8.0 cm and damping constant $\gamma = 0.02 \text{ s}^{-1}$. Calculate the amplitude after 50 s and the ratio of final energy to initial energy.

1. Substitute values into the amplitude decay formula:
2. $$A = 8.0 \times e^{-(0.02)(50)} = 8.0 e^{-1} \approx 2.9 \text{ cm (2 s.f.)}$$
3. Energy ratio equals the square of the amplitude ratio:
4. $$\frac{E}{E_0} = \left(\frac{A}{A_0}\right)^2 = (e^{-1})^2 = e^{-2} \approx 0.14$$

*Calculator:* allowed

## Practical Applications of Damping

Damping is intentionally designed into many everyday systems to control unwanted oscillations:

- Car suspension: Critical damping removes oscillations after bumps quickly for a smooth ride
- Moving coil galvanometers: Critical damping makes the needle settle to the correct reading quickly
- Door closers: Overdamping prevents doors from slamming by slowing their motion
- Seismometers: Damping is adjusted to record earthquake waves clearly

**Check your understanding**

Test your understanding:

1. Which type of damping returns to equilibrium the fastest without overshoot?

   - Light damping
   - Critical damping
   - Heavy damping

   *Answer:* Critical damping

   *Why:* Correct! Critical damping is specifically designed for the fastest return to equilibrium without oscillation.

2. For lightly damped oscillations, what is the effect of damping on frequency?

   - Frequency halves
   - Frequency doubles
   - Frequency is almost unchanged

   *Answer:* Frequency is almost unchanged

   *Why:* Correct! For CIE purposes, light damping has no significant effect on frequency.

## Common pitfalls

- **Wrong:** Claiming that light damping significantly changes the frequency of oscillations for exam questions
  - Why it fails: CIE 9702 expects you to assume frequency change is negligible for light damping
  - Correct: State that frequency remains approximately equal to the natural undamped frequency for light damping
- **Wrong:** Confusing critical damping and heavy damping, claiming heavy damping returns to equilibrium faster
  - Why it fails: Critical damping is defined as the fastest possible return to equilibrium without overshoot
  - Correct: Memorise the order of return time: critical < heavy < light
- **Wrong:** Using the same exponential decay constant for energy and amplitude
  - Why it fails: Energy is proportional to amplitude squared, so it decays faster than amplitude
  - Correct: Use $E \propto A^2$ to get decay constant $2\gamma$ for energy when amplitude has decay constant $\gamma$
- **Wrong:** Drawing critical/heavy damping graphs that cross the equilibrium line
  - Why it fails: Only underdamped (light) oscillations overshoot and cross equilibrium
  - Correct: Draw critical/heavy damping as displacement decaying to zero from the initial position, never crossing the equilibrium axis

## Cheatsheet

| Damping Type | Key Features | Common Examples |
| --- | --- | --- |
| Light (Under) | Gradual exponential amplitude decay, many oscillations, frequency unchanged | Swinging pendulum, mass on spring in air |
| Critical | No oscillation, fastest return to equilibrium, no overshoot | Car suspension, analog meters |
| Heavy (Over) | No oscillation, slow return to equilibrium, no overshoot | Door closers, mass in viscous oil |

## What's next

Damped oscillations are the foundation for understanding forced oscillations and resonance, the next core sub-topic in CIE A-Level Oscillations. Resonance occurs when a system is driven at its natural frequency, and damping directly determines the sharpness of the resonance peak. Exam questions regularly ask you to explain how damping is used to prevent dangerous resonance in structures like bridges and buildings. This topic also reinforces your understanding of exponential decay, which you will apply again in radioactive decay and capacitor discharge later in the course.

- [Forced Oscillations and Resonance](https://www.owlsprep.com/study/cie-9702-u18-forced-oscillations-and-resonance/)
- [Simple Harmonic Motion](https://www.owlsprep.com/study/cie-9702-u18-simple-harmonic-motion/)
- [Ideal gases](https://www.owlsprep.com/study/cie-9702-u19-overview/)

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