Simple harmonic motion (SHM) is the “all-purpose grammar” behind pendulums, springs and even the car's shock absorbers.
Key points
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.
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1 Where oscillations sit in the syllabus
2 Free oscillations - the “default setting”
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 equationa=−ω2x, practise turning graphs into equations, and treat resonance with the same respect you give gravity - it can launch bridges or break wine glasses.
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.
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 frequencyf0.
3 The mathematics of SHM
3.1 Defining equation
The motion is SHM if and only if
a=−ω2x,
where a is acceleration, x displacement from equilibrium and ω=2πf the angular frequency.
3.2 Solution set
Using the calculus of the syllabus, two complementary solutions arise:
x=x0sin(ωt+ϕ),v=dtdx=ωx0cos(ωt+ϕ),a=−ω2x.
These forms translate directly into exam graph questions.
All three quantities share the same frequency but differ by phase shifts of π/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 π/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=−ω2x, a positive maximum displacement produces a maximum negative acceleration, and vice versa. There is a phase difference of π rad between them.
Acceleration leads velocity by π/2 - this follows from the two statements above and is the relationship most often tested in graph-matching questions.
Quick-reference table
Displacement
Velocity
Acceleration
Maximum positive (+x0)
Zero
Maximum negative (−ω2x0)
Zero (moving in positive direction)
Maximum positive
Zero
Maximum negative (−x0)
Zero
Maximum positive (+ω2x0)
Zero (moving in negative direction)
Maximum negative
Zero
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+ω2x=0 line up to within experimental uncertainty.
5 Energy interchange
Total mechanical energy in ideal SHM is constant:
E=Ek+Ep=21mv2+21kx2=21kx02.
Kinetic energy peaks at equilibrium; potential dominates at the extremes. Graphs of Ek and Ep 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.
6 Damped oscillations
Real systems lose energy to friction or drag. Three regimes matter:
Damping
Behaviour
Exam cue
Light / underdamped
Gradual amplitude decay, oscillations persist
Car shock absorber with fresh oil
Critical
Returns to rest in minimum time, no overshoot
Door closer, seismograph needle
Heavy / overdamped
Slow crawl to equilibrium, no oscillation
High-viscosity dash-pot
The sharper the damping, the wider and lower the resonance peak becomes.
7 Forced oscillations & resonance
Applying a periodic driving force of frequency fd generates a steady-state amplitude that depends on how close fd is to f0. Maximum response occurs at resonance. The amplitude-frequency curve narrows and drops as damping increases.
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=±ωx02−x2,vmax=ωx0.
Expect these in “find speed at x=0.6x0” type MCQs. They drop straight out of the energy equation by equating Ek and Ep.
9 Three WA timing hacks
Sketch the energy bar first. It organises numbers before you reach for a calculator.
State phase differences (x→v→a) instead of redrawing every graph.
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=−ω2x 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=x0sinωt as a solution to the equation a=−ω2x.
(f) recognise and use the equations v=v0cosωt and v=±ωx02−x2
(g) describe, with graphical illustrations, the relationships between displacement, velocity and acceleration during simple harmonic motion.
(h) describe the interchange between kinetic and potential energy during simple harmonic motion.
(i) describe practical examples of damped oscillations, with particular reference to the effects of the degree of damping (light/under, critical, heavy/over), and to the importance of critical damping in applications such as a car suspension system.
(j) describe graphically how the amplitude of a forced oscillation changes with driving frequency, resulting in maximum amplitude at resonance when the driving frequency is close to or at the natural frequency of the system.
(k) show a qualitative understanding of the effects of damping on the frequency response and sharpness of the resonance.
(l) describe practical examples of forced oscillations and resonance, and show an appreciation that there are some circumstances in which resonance is useful, and other circumstances in which resonance should be avoided.
Concept map (in words)
Identify the restoring force and show it is proportional to displacement → SHM. Use x(t)=x0sin(ωt+ϕ) to generate v(t) and a(t). Track energy swapping between KE and PE. Introduce damping (energy loss) and forcing (energy input); resonance occurs when driving frequency matches natural frequency.
Key relations
Quantity
Expression / comment
SHM definition
a=−ω2x
Displacement as function
x=x0sin(ωt+ϕ)
Velocity & acceleration
v=ωx0cos(ωt+ϕ),a=−ω2x
Energy
E=21kx02=21mω2x02
Mass-spring frequency
ω=mk,
Simple pendulum (small angle)
ω=ℓg
Damped amplitude
x=x0exp(−2mbt)cos(ω′t+ϕ)
Derivations & reasoning to master
Mass-spring SHM: apply Hooke's law and Newton's 2nd law to derive x¨+mkx=0.
Pendulum SHM approximation: linearise sinθ≈θ to show ω=ℓg.
Energy relation: start from x(t) and v(t) to derive E=constant and show phase difference between energies.
Resonance curve shape: explain why peak amplitude decreases and broadens with stronger damping.
Worked example 1 - mass-spring system
A 0.30kg mass oscillates on a spring with k=18N⋅m−1. It is pulled 5.0cm from equilibrium and released from rest. Find the period, maximum speed, and displacement after 0.75s.
Solution path: ω=mk,T=ω2π. vmax=ωx0. Use x=x0cos(ωt) since release from rest at maximum displacement.
Taking x0=0.050m,
ω=0.3018=7.75rad⋅s−1,T=ω2π=0.811s.
vmax=ωx0=7.75(0.050)=0.387m⋅s−1.
x(0.75)=0.050cos(7.75×0.75)≈0.0445m(about 4.45 cm from equilibrium).
Worked example 2 - driven oscillation
A lightly damped oscillator (f0=2.5Hz) is driven by a motor whose frequency increases from 1Hz to 5Hz. Sketch the amplitude response and estimate the frequency at which amplitude peaks if light damping lowers the resonance to 2.3Hz. Explain how the phase between driver and oscillator changes across resonance.
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 ω.
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 Hz and 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 f0.
Quick self-check quiz
What condition must a motion satisfy to be classified as SHM? - Acceleration proportional to displacement and directed towards equilibrium.
For SHM, when is potential energy maximum? - At maximum displacement (turning points).
How does heavy damping affect oscillation? - No oscillation; system returns slowly to equilibrium.
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.
What happens to phase difference between displacement and driving force far above resonance? - It approaches 180° (force opposite to displacement).
Revision workflow
Re-derive the mass-spring and pendulum SHM equations without notes twice a week.
Solve one energy-based SHM problem and one resonance graph question per revision session.
Prepare comparison tables for under/critical/over damping and their response curves.
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.
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.
