# Forced oscillations and resonance

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

This module covers forced oscillations, where an external periodic force drives a system, and the phenomenon of resonance, which occurs when driving frequency matches a system's natural frequency. We explore resonance curves, damping effects, and real-world applications.

**Prerequisites:** [Free and damped oscillations](https://www.owlsprep.com/study/cie-9702-u18-free-damped-oscillations/)

## Learning objectives

- Distinguish between free, damped and forced oscillations
- Explain the phenomenon of resonance and its causes
- Sketch and interpret resonance curves for different levels of damping
- Give examples of useful and harmful resonance in real systems

## Forced vs Free Oscillations

A free oscillation occurs when a system is given an initial disturbance and then left to oscillate on its own, with no external driving force. Free oscillations always occur at the system's **natural frequency ($f_0$)**, and amplitude decreases over time due to damping.

A forced oscillation occurs when a continuous external periodic driving force acts on the system, continuously supplying energy. For any driving frequency, the system will eventually settle into oscillating *at the driving frequency*, not its natural frequency.

**Forced Oscillation** — Oscillation of a system driven by a continuous external periodic force, resulting in steady oscillation at the driving frequency

*Example:* A child on a swing pushed repeatedly at regular intervals by an adult

**Worked example:** A radio tuner circuit is adjusted to receive a station broadcasting at 98 MHz. State the driving frequency and the natural frequency of the tuner when correctly tuned.

1. The driving force comes from the incoming radio broadcast signal, so the driving frequency equals the broadcast frequency:
2. $$f_d = 98\ \text{MHz}$$
3. For the tuner to detect the signal, it must resonate, meaning its natural frequency must match the driving frequency:
4. $$f_0 = f_d = 98\ \text{MHz}$$

> **Exam tip:** Multiple choice questions often trap candidates by asking for the frequency of a forced oscillation — remember it is always the driving frequency, not natural frequency.

## The Phenomenon of Resonance

Resonance occurs when the driving frequency is equal (or very close) to the natural frequency of the driven system. At this point, the driving force is always in phase with the system's motion, so maximum energy is transferred to the system every cycle, leading to maximum amplitude of oscillation.

**Resonance** — The maximum amplitude response of a forced oscillating system, occurring when driving frequency matches the system's natural frequency (for light damping)

*Example:* Pushing a swing at its natural frequency to make it reach maximum height

**Worked example:** Explain why soldiers are ordered to break step when crossing a suspension bridge with natural frequency 1.5 Hz.

1. When soldiers march in step, they produce a periodic driving force with frequency equal to their marching step frequency.
2. If the marching frequency equals the bridge's natural frequency of 1.5 Hz, resonance will occur.
3. At resonance, the amplitude of the bridge's oscillation becomes very large, which can cause catastrophic structural damage.
4. Breaking step means the driving forces from each soldier are out of sync, so resonance cannot occur.

> **tip**
>
> At resonance, not just amplitude is maximum: energy transfer from the driving force to the system is also maximum.

## Resonance Curves and Damping Effects

A resonance curve plots the amplitude of a forced oscillation against driving frequency. The shape of the curve depends heavily on the level of damping in the system.

| Damping Level | Maximum Amplitude | Peak Position | Peak Width |
| --- | --- | --- | --- |
| Zero/Light | Very large | Very close to $f_0$ | Narrow |
| Moderate | Medium | Slightly below $f_0$ | Medium width |
| Heavy | Small | Significantly below $f_0$ | Broad |

**Worked example:** Compare the resonance curves for a lightly damped and heavily damped mass-spring system with the same natural frequency $f_0$.

1. 1. Label the y-axis *Amplitude* and the x-axis *Driving Frequency ($f_d$)*, mark $f_0$ on the x-axis.
2. 2. For the lightly damped system: draw a sharp, high peak centered almost exactly at $f_0$. Amplitude drops off rapidly for $f_d$ far from $f_0$.
3. 3. For the heavily damped system: draw a lower, broader peak shifted slightly to the left of $f_0$. Amplitude is lower at all driving frequencies than the lightly damped system.

> **Exam tip:** CIE commonly asks for comparisons of resonance curves — always remember higher damping gives a lower, broader peak.

## Real-World Examples of Resonance

- **Useful resonance**: Radio tuners (adjust natural frequency to match broadcast frequency to select a station), microwave ovens (match frequency to natural frequency of water molecules for heating), musical instruments (soundboards resonate to amplify produced sound)
- **Harmful resonance**: Bridge vibration from wind or traffic, building oscillation during earthquakes (if earthquake frequency matches building natural frequency), aircraft wing vibration leading to structural failure

**Check your understanding**

Test your understanding of key concepts:

1. A microwave oven heats food primarily because:

   - Microwaves are hotter than the surrounding air
   - The microwave frequency matches the natural frequency of water molecules

   *Why:* Correct! This is an example of useful resonance, where maximum energy is transferred to water molecules to heat food.

2. What frequency does a forced oscillating system oscillate at when not at resonance?

   - Natural frequency $f_0$
   - Driving frequency $f_d$

   *Why:* Correct! Forced oscillations always oscillate at the driving frequency, regardless of whether resonance occurs.

## Common pitfalls

- **Wrong:** Assuming forced oscillations always occur at the system's natural frequency.
  - Why it fails: Confuses free and forced oscillations; only free oscillations occur at natural frequency.
  - Correct: Forced oscillations always occur at the frequency of the external driving force, regardless of natural frequency.
- **Wrong:** Drawing resonance peaks for higher damping as higher and sharper than for lower damping.
  - Why it fails: Reverses the effect of damping on amplitude and peak width.
  - Correct: Higher damping reduces the maximum amplitude at resonance and broadens the resonance peak.
- **Wrong:** Claims maximum amplitude always occurs exactly at $f_d = f_0$ for all levels of damping.
  - Why it fails: Ignores the shift of the resonance peak for heavily damped systems.
  - Correct: For heavy damping, the peak resonance frequency shifts slightly below the natural frequency $f_0$.
- **Wrong:** Assumes all resonance is harmful.
  - Why it fails: Only considers structural examples of damaging resonance, not technological applications.
  - Correct: Resonance is intentionally used in many technologies including radio tuners, microwave ovens and musical instruments.

## Cheatsheet

| Concept | Key Property | Exam Note |
| --- | --- | --- |
| Free oscillation | Oscillates at $f_0$ (natural frequency) | No external driving force |
| Forced oscillation | Oscillates at $f_d$ (driving frequency) | Continuous external driving force |
| Resonance | Maximum amplitude of oscillation | Occurs when $f_d \approx f_0$ |
| Light damping | High, sharp peak at $f_0$ | Small energy loss per cycle |
| Heavy damping | Low, broad peak below $f_0$ | Large energy loss per cycle |

## What's next

Forced oscillations and resonance are foundational for many other topics in CIE A-Level Physics. Standing waves, which you will encounter in wave mechanics, are formed by resonance between incident and reflected waves. Resonance also explains how buildings respond to seismic waves during earthquakes, a common topic in AS and A2 Level questions. Mastery of resonance curves and damping effects is critical for both multiple choice and structured exam questions, and this concept appears frequently in CIE past papers.

- [Ideal gases](https://www.owlsprep.com/study/cie-9702-u19-overview/)
- [Gas laws](https://www.owlsprep.com/study/cie-9702-u19-gas-laws/)
- [Ideal gas equation of state](https://www.owlsprep.com/study/cie-9702-u19-ideal-gas-equation-of-state/)

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