H2 Physics Oscillations & SHM Notes | A-Level 9478

Q: What does A-Level Physics: 9) Oscillations Guide cover?
A: From free vibrations to damping and resonance, this post unpacks Section I Topic 9 of the 2026 H2 Physics syllabus for IP students and parents.

TL;DR
Simple harmonic motion (SHM) is the “all-purpose grammar” behind pendulums, springs and even the car's shock absorbers. Nail the defining equation $a = -\omega^2 x$, practise turning graphs into equations, and treat resonance with the same respect you give gravity - it can launch bridges or break wine glasses.

Concrete example: how to use this page

For a mass on a spring, check whether acceleration is proportional to displacement and opposite in direction. Once that is true, use SHM equations. If the force is not restoring toward equilibrium, do not force the SHM model.

Decision map - choose the oscillation route first

Most SHM mistakes happen before the first calculation: the student sees a sine graph and reaches for a formula before identifying what the question is testing.

Misconception check: SHM is not just "motion that repeats". It needs a restoring acceleration proportional to displacement and directed towards equilibrium.

Need the rest of the mechanics + waves refresh? Hop back to the H2 Physics notes hub for the circular motion, gravitation, and wave-motion companions so your SHM practice stays connected to earlier topics.

1 Where oscillations sit in the syllabus

Section I Topic 9 condenses four big ideas - free, damped, forced oscillations and energy - into 12 learning outcomes. They are reproduced verbatim in the SEAB H2 Physics 9478 document for 2026.

1.1 Why parents should care

Oscillations show up in WA graph-plotting, Paper 1 MCQs and the practical. One clean diagram of a resonance curve can rescue three marks in under a minute.

2 Free oscillations - the “default setting”

A free oscillator moves under its own restoring force with no external input or loss. Examples include a frictionless mass-spring or a pendulum in vacuum.

Key checkpoint: the motion always occurs at the natural frequency $f_0$.

3 The mathematics of SHM

3.1 Defining equation

The motion is SHM if and only if

$$a = -\omega^2 x,$$

where $a$ is acceleration, $x$ displacement from equilibrium and $\omega = 2\pi f$ the angular frequency.

3.2 Solution set

Using the calculus of the syllabus, two complementary solutions arise:

$$x = x_0 \sin(\omega t + \phi), \qquad v = \dfrac{\mathrm{d}x}{\mathrm{d}t} = \omega x_0 \cos(\omega t + \phi), \qquad a = -\omega^2 x.$$

These forms translate directly into exam graph questions.

3.3 Graph trio (displacement-velocity-acceleration)

All three quantities share the same frequency but differ by phase shifts of $\pi/2$ rad. Visualising them side-by-side is a scoring trick when time is tight.

3.4 Phase relationships - the mark-saving drill

This is the single most common source of dropped marks in Paper 3 MCQs. Students under time pressure confuse which quantity is at its maximum when another is at zero.

The three key relationships follow directly from the solution set in 3.2:

• Velocity leads displacement by $\pi/2$ - when displacement reaches its positive maximum (turning point), velocity is instantaneously zero; when displacement passes through zero, velocity is at its maximum magnitude.
• Acceleration is in antiphase with displacement - because $a = -\omega^2 x$, a positive maximum displacement produces a maximum negative acceleration, and vice versa. There is a phase difference of $\pi$ rad between them.
• Acceleration leads velocity by $\pi/2$

Quick-reference table

Commit this table to memory by tracing one full cycle on a sine graph. Once the pattern is automatic, phase-relationship MCQs take under 30 seconds.

4 Experimental & graphical skills

Plotting $x$ vs $t$ for a mass-spring reveals a sine curve whose gradient gives $v$. A second derivative check confirms the SHM criterion.

IP booster drill: import Logger Pro data and add a d²x/dt² column - students see $a +\omega^2 x = 0$ line up to within experimental uncertainty.

5 Energy interchange

Total mechanical energy in ideal SHM is constant:

$$E = E_k + E_p = \tfrac{1}{2} m v^{2} + \tfrac{1}{2} k x^{2} = \tfrac{1}{2} k x_0^{2}.$$

Kinetic energy peaks at equilibrium; potential dominates at the extremes. Graphs of $E_k$ and $E_p$ both stay positive and oscillate at twice the SHM frequency - when one peaks the other is zero, and their sum equals total energy at all times.

Energy-displacement shortcut for phase relationships. SHM involves a continuous exchange between kinetic and potential energy; total energy is constant for undamped oscillations. KE is maximum when displacement is zero; PE is maximum when displacement equals the amplitude. If you can sketch the energy-displacement graph - a parabola for PE, an inverted parabola for KE - the phase relationships in section 3.4 follow automatically without memorising each case separately.

Energy route checkpoint

When a question asks for speed at a stated displacement, use energy before choosing a sign.

Worked check: at $x=0.6x_0$,

$$v^2=\omega^2(x_0^2-0.36x_0^2)=0.64\omega^2x_0^2,$$

so the speed is $0.8\omega x_0$. The velocity is positive or negative only after the question tells you which way the oscillator is moving.

Misconception check: energy tells you how fast the oscillator is moving at that displacement. It does not, by itself, tell you whether it is moving left or right.

6 Damped oscillations

Real systems lose energy to friction or drag. Three regimes matter:

The sharper the damping, the wider and lower the resonance peak becomes.

7 Forced oscillations & resonance

Applying a periodic driving force of frequency $f_d$ generates a steady-state amplitude that depends on how close $f_d$ is to $f_0$

Resonance-curve checkpoint

When a question gives an amplitude-frequency graph, read the curve before naming the phenomenon. The graph is usually testing how the oscillator responds as the driving frequency changes.

Worked check: if curve A has a high sharp peak and curve B has a lower broad peak for the same oscillator, curve B represents greater damping. Do not infer that curve B has a larger driving force unless the question explicitly changes the driver.

7.1 Good vs bad resonance

Useful: MRI radio-frequency coils, microwave ovens, quartz watches.
Disastrous: Tacoma Narrows bridge (1940), wine-glass shattering in a loud note.

8 Velocity relations worth memorising

$$v = \pm \omega \sqrt{x_0^{2} - x^{2}}, \qquad v_{\text{max}} = \omega x_0.$$

Expect these in “find speed at $x = 0.6x_0$” type MCQs. They drop straight out of the energy equation by equating $E_k$ and $E_p$

9 Three WA timing hacks

1. Sketch the energy bar first. It organises numbers before you reach for a calculator.
2. State phase differences ($x\to v\to a$) instead of redrawing every graph.
3. Quote damping type whenever you see a decaying sinusoid - it is a free descriptor mark.

Need structured practice on Oscillations? Our H2 Physics tuition programme covers this topic with weekly problem sets and Paper 4 practical drills.

Comprehensive revision pack

9478 Section II, Topic 9 Syllabus outcomes

Candidates should be able to:

• (a) describe simple examples of free oscillations, where particles periodically return to an equilibrium position without gaining energy from or losing energy to the environment.
• (b) investigate the motion of an oscillator using experimental and graphical methods.
• (c) show an understanding of and use the terms amplitude, period, frequency, angular frequency, phase and phase difference, and express the period in terms of both frequency and angular frequency.
• (d) show an understanding that $a = -\omega^2 x$ is the defining equation of simple harmonic motion, where acceleration is (directly) proportional to displacement from an equilibrium position and acceleration is always directed towards the equilibrium position.
• (e) recognise and use $x = x_0 \sin\omega t$

Concept map (in words)

Identify the restoring force and show it is proportional to displacement → SHM. Use $x(t) = x_0 \sin(\omega t + \phi)$ to generate $v(t)$ and $a(t)$

Key relations

Derivations & reasoning to master

1. Mass-spring SHM: apply Hooke's law and Newton's 2nd law to derive $\ddot{x} + \dfrac{k}{m} x = 0$.
2. Pendulum SHM approximation: linearise $\sin\theta \approx \theta$

Worked example 1 - mass-spring system

A $0.30 \pu{kg}$ mass oscillates on a spring with $k = 18 \pu{N.m-1}$. It is pulled $5.0 \pu{cm}$ from equilibrium and released from rest. Find the period, maximum speed, and displacement after $0.75 \pu{s}$

Solution path: $\omega = \sqrt{\dfrac{k}{m}}, \space T = \dfrac{2\pi}{\omega}$

Taking $x_0 = \pu{0.050 m}$,

$$\omega = \sqrt{\dfrac{18}{0.30}} = 7.75\,\pu{rad.s-1},\qquad T = \dfrac{2\pi}{\omega} = \pu{0.811 s}.$$

$$v_{\text{max}} = \omega x_0 = 7.75(0.050) = \pu{0.387 m.s-1}.$$

$$x(0.75) = 0.050\cos(7.75\times 0.75) \approx \pu{0.0445 m} \;\text{(about 4.45 cm from equilibrium)}.$$

Worked example 2 - driven oscillation

A lightly damped oscillator ($f_0 = 2.5 \pu{Hz}$) is driven by a motor whose frequency increases from $1 \pu{Hz}$ to $5 \pu{Hz}$. Sketch the amplitude response and estimate the frequency at which amplitude peaks if light damping lowers the resonance to $2.3 \pu{Hz}$

Discussion: below resonance → in phase, at resonance → 90° lag, above resonance → 180° out of phase.

Practical & data tasks

• Use a data logger to record displacement vs time for a mass-spring; fit a sine curve to extract $\omega$.
• Investigate damping by immersing the oscillator in water; measure exponential decay constant.
• Build a resonance board with driving speaker; map amplitude vs frequency to see sharpness changes.

Common misconceptions & exam traps

• Confusing frequency with angular frequency (mixing up $\pu{Hz}$ and $\pu{rad.s-1}$).
• Thinking damping changes natural frequency significantly (light damping only slightly reduces it).
• Forgetting the phase relation when sketching velocity/acceleration graphs.
• Assuming resonance occurs at any high driving frequency; it is specific to near $f_0$

Quick self-check quiz

1. What condition must a motion satisfy to be classified as SHM? - Acceleration proportional to displacement and directed towards equilibrium.
2. For SHM, when is potential energy maximum? - At maximum displacement (turning points).
3. How does heavy damping affect oscillation? - No oscillation; system returns slowly to equilibrium.
4. At resonance, how do driving frequency and natural frequency compare? - They are equal (or very close with damping), and the oscillator lags the driver by about 90° so the driving force is in quadrature with displacement.
5. What happens to phase difference between displacement and driving force far above resonance? - It approaches 180° (force opposite to displacement).

Revision workflow

1. Re-derive the mass-spring and pendulum SHM equations without notes twice a week.
2. Solve one energy-based SHM problem and one resonance graph question per revision session.
3. Prepare comparison tables for under/critical/over damping and their response curves.
4. Practise drawing displacement, velocity and acceleration graphs for different starting conditions within five minutes.

Practice Quiz

Test yourself on the key concepts from this guide.

10 Further reading

• SEAB H2 Physics 9478 syllabus (2026)
• Lumen - Simple Harmonic Motion
• OpenStax - Damped Harmonic Motion
• OpenStax - Forced Oscillations & Resonance

11 Call-to-action

Parents: schedule a 60-min SHM clinic two weeks before WA 2 to pre-empt resonance graph slips. Students: screenshot the damping table and test yourself - can you state the response curve shape from memory?

Last updated 14 Jul 2025. Next review when SEAB issues the 2027 draft syllabus.