# Oscillations

> Edexcel International A-Level Physics · IAL Physics Unit 5
> Source: https://www.owlsprep.com/study/edexcel-ial-physics-u5-oscillations/

This guide covers all Edexcel IAL Physics Unit 5 oscillation content, including SHM conditions, formulae, graph interpretation, resonance, damping, and Core Practical 16 for WPH15 exams.

**Prerequisites:** Basic trigonometry (cosine, sine functions, radians); Graph gradient calculation skills

## Learning objectives

- Identify systems undergoing simple harmonic motion using the F=-kx condition
- Apply SHM equations to calculate displacement, velocity, acceleration, period and frequency
- Use mass-spring and simple pendulum period formulae to solve for unknown variables
- Interpret displacement, velocity and time graphs for oscillating systems
- Explain free, forced and damped oscillations and resonance effects qualitatively
- Carry out Core Practical 16 calculations to find unknown mass from resonant frequencies

## Simple Harmonic Motion (SHM) Fundamentals

**Simple Harmonic Motion (SHM)** — Oscillatory motion where restoring force and acceleration are directly proportional to displacement from equilibrium, and act opposite to displacement.

*Notation:* F = -kx, a = -ω²x

*Example:* A mass oscillating on a frictionless spring obeys SHM, as the spring force always pulls the mass back to equilibrium.

To confirm a system is undergoing SHM, you must verify that acceleration is always proportional to negative displacement. The constant of proportionality is equal to `ω²`, where ω is the angular frequency of the oscillation.

**Worked example:** A mass on a spring is displaced 0.02 m from equilibrium, and experiences an acceleration of 0.8 m s⁻² towards equilibrium. Show that the motion is SHM, and calculate the angular frequency ω.

1. Confirm acceleration is proportional to -x: $a = -(\frac{a}{x})x$. Substitute values: $a = -(\frac{0.8}{0.02})x = -40x$, which matches the SHM condition $a = -ω²x$, so motion is SHM.
2. Equate coefficients: $ω² = 40$
3. $$ω = \sqrt{40} \approx 6.3 \text{ rad s}^{-1}$$

> **Exam tip:** Always explicitly link back to the `a = -ω²x` condition when asked to prove a system undergoes SHM in exams.

## SHM Equations and Period Calculations

All SHM equations are provided on the exam formula sheet. Note that $x = A\cos{ωt}$ assumes $x = A$ (maximum displacement) at $t=0$; use $x = A\sin{ωt}$ if $x=0$ at $t=0$.

**Period of Oscillation** — Time taken for one complete oscillation, measured in seconds. For mass-spring systems $T = 2π\sqrt{\frac{m}{k}}$, for simple pendulums $T = 2π\sqrt{\frac{l}{g}}$.

*Notation:* T = \frac{1}{f} = \frac{2π}{ω}

> **tip**
>
> You do not need to memorise period formulae, but you must be able to rearrange them to solve for unknowns like mass, spring constant, or pendulum length.

**Worked example:** A simple pendulum has length 0.75 m. Calculate its period and frequency. Use $g = 9.81$ m s⁻².

1. Use the simple pendulum period formula: $T = 2π\sqrt{\frac{l}{g}}$
2. $$T = 2π \times \sqrt{\frac{0.75}{9.81}} \approx 2π \times 0.276 \approx 1.74 \text{ s}$$
3. Calculate frequency $f = \frac{1}{T}$
4. $$f = \frac{1}{1.74} \approx 0.57 \text{ Hz}$$

## Oscillation Graph Interpretation

Displacement-time (x-t) graphs for SHM are cosine or sine curves. The gradient of an x-t graph gives velocity, and the gradient of a velocity-time (v-t) graph gives acceleration. Maximum velocity occurs at $x=0$ (equilibrium), maximum acceleration occurs at $x=±A$ (maximum displacement).

**Worked example:** An SHM x-t graph has a maximum displacement of 0.1 m and period 2 s. At $t=0$, $x=0.1$ m. State the magnitude of velocity at $t=0.5$ s, and magnitude of acceleration at $t=1$ s.

1. At $t=0.5$ s (1/4 of the period), the mass is at equilibrium, so velocity is maximum. $ω = \frac{2π}{T} = π$ rad s⁻¹, so $|v_{max}| = Aω$
2. $$|v| = 0.1 \times π \approx 0.31 \text{ m s}^{-1}$$
3. At $t=1$ s (1/2 of the period), displacement is $x = -A = -0.1$ m, so acceleration is maximum: $|a_{max}| = Aω²$
4. $$|a| = 0.1 \times π² \approx 0.99 \text{ m s}^{-2}$$

## Free, Forced Oscillations, Resonance and Damping

**Resonance** — Occurs when the frequency of an external driving force equals the natural frequency of the oscillating system, leading to maximum amplitude of oscillation.

*Example:* A singer breaking a glass by singing at the natural frequency of the glass is an example of resonance.

Free oscillations have constant amplitude if undamped, as no energy is lost. Forced oscillations are driven by an external periodic force. Light damping reduces peak resonance amplitude slightly and shifts the peak slightly lower, while heavy damping reduces peak amplitude significantly and broadens the resonance curve. Plastic deformation also damps oscillations by dissipating energy as heat.

**Worked example:** A mass-spring system has a natural frequency of 2 Hz. It is driven by a periodic force of frequency 2 Hz, with light damping applied. Describe how the amplitude changes if (a) the driving frequency is increased to 3 Hz, and (b) damping is increased to heavy damping.

1. (a) When driving frequency is 2 Hz, the system is at resonance so amplitude is maximum. Increasing driving frequency to 3 Hz moves it away from natural frequency, so amplitude decreases significantly.
2. (b) Increasing damping to heavy damping will reduce the maximum resonance amplitude further, even when driving frequency equals natural frequency, and the amplitude will fall off more gradually as driving frequency moves away from natural frequency.

> **Exam tip:** Qualitative questions on damping and resonance are very common in Unit 5 papers: always link amplitude changes to energy loss from damping for full marks.

## Core Practical 16: Unknown Mass from Resonant Frequencies

In this practical, you suspend different known masses from a spring, apply a driving force (e.g. from a vibration generator connected to a signal generator), and find the resonant frequency for each mass. Plot a graph of $T²$ against $m$: the gradient is $\frac{4π²}{k}$, which you use to find the spring constant $k$. You then use the resonant frequency of an unknown mass to calculate its mass.

> **warning**
>
> Common errors in this practical include not waiting for amplitude to stabilise when finding resonant frequency, and plotting $T$ instead of $T²$ against $m$, which gives a non-linear graph.

**Worked example:** A student plots $T²$ against $m$ for a spring, and gets a gradient of 0.4 s² kg⁻¹. An unknown mass has a resonant period of 0.6 s. Calculate the unknown mass.

1. Square the mass-spring period formula: $T² = (\frac{4π²}{k})m$, so gradient of $T²$ vs $m$ = $\frac{4π²}{k} = 0.4$ s² kg⁻¹
2. Rearrange to solve for $m$: $m = \frac{T²}{0.4}$
3. $$m = \frac{0.6²}{0.4} = \frac{0.36}{0.4} = 0.9 \text{ kg}$$

## Common pitfalls

- **Wrong:** Using calculus to differentiate $x$ to get $v$ or $a$ when asked for velocity/acceleration from graphs
  - Why it fails: The Edexcel IAL Unit 5 specification explicitly forbids calculus for this topic; you will lose marks if you use differentiation instead of calculating gradients.
  - Correct: Calculate velocity as the tangent gradient of a displacement-time graph, and acceleration as the tangent gradient of a velocity-time graph.
- **Wrong:** Assuming amplitude of forced oscillation is maximum when driving frequency is higher than natural frequency
  - Why it fails: Resonance occurs exactly when driving frequency equals natural frequency for undamped systems, and only shifts slightly lower for very heavily damped systems.
  - Correct: State that maximum amplitude occurs when driving frequency equals the natural frequency of the system, unless explicitly told damping is very heavy.
- **Wrong:** Using $g = 10$ m s⁻² for pendulum period calculations
  - Why it fails: Edexcel specifies $g = 9.81$ m s⁻² for all calculations, so using 10 will lead to rounding errors and lost marks.
  - Correct: Always use $g = 9.81$ m s⁻² unless explicitly told otherwise in the question.
- **Wrong:** Confusing the period formula for mass-spring and pendulum systems
  - Why it fails: Mass-spring period depends on mass and spring constant, while pendulum period depends on length and $g$, so mixing them up gives completely wrong answers.
  - Correct: Check the system first: if it is a mass on a spring use $T = 2π\sqrt{\frac{m}{k}}$, if it is a pendulum use $T = 2π\sqrt{\frac{l}{g}}$. Both formulae are given on the exam formula sheet.
- **Wrong:** Stating that damping increases resonant frequency
  - Why it fails: Damping removes energy from the system, reducing maximum amplitude, but only shifts the resonant frequency very slightly downwards for heavy damping; it does not increase frequency.
  - Correct: Link damping to reduced amplitude of resonance, and only mention a small downward shift of the resonance peak if asked about heavy damping.

## Cheatsheet

| Concept | Formula/Key Rule |
| --- | --- |
| SHM Condition | $F = -kx$; $a = -ω²x$ |
| Angular Frequency | $ω = 2πf = \frac{2π}{T}$ |
| SHM Displacement ($x=A$ at $t=0$) | $x = A\cos{ωt}$ |
| SHM Velocity | $v = -Aω\sin{ωt}$; $\|v_{max}\| = Aω$ |
| SHM Acceleration | $a = -Aω²\cos{ωt}$; $\|a_{max}\| = Aω²$ |
| Mass-Spring Period | $T = 2π\sqrt{\frac{m}{k}}$ |
| Simple Pendulum Period | $T = 2π\sqrt{\frac{l}{g}}$ |
| Resonance Condition | Driving frequency = natural frequency of system |
| Damping Effect | Reduces amplitude of oscillations; lowers resonance peak, broadens curve |

## What's next

Now that you have mastered oscillations for Edexcel IAL Physics Unit 5, you are ready to move on to the remaining topics in the unit: nuclear radiation, thermodynamics, and cosmology. Oscillations concepts, particularly resonance, are sometimes tested alongside astrophysics content in synoptic questions, so make sure you revise the key formulae and qualitative rules regularly before your WPH15 exam. Practice as many past paper questions on SHM graph interpretation and resonance as possible, as these are high-frequency question types that often carry 4-6 marks each. Make sure you can complete Core Practical 16 calculations quickly and accurately, as practical-based questions make up ~15% of the Unit 5 paper.

- [Thermodynamics (Unit 5)](https://www.owlsprep.com/study/edexcel-ial-physics-u5-thermodynamics/)

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