A-Level Physics: Oscillations Guide
Download printable cheat-sheet (CC-BY 4.0)14 Jul 2025, 00:00 Z
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.
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 iff
\[ a = -\omega^2 x, \]
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 = \tfrac{\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.
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:
\[ E = E_k + E_p = \tfrac12 m v^2 + \tfrac12 k x^2 = \tfrac12 k x_0^2. \]
Kinetic energy peaks at equilibrium; potential dominates at the extremes. Graphs of \(E_k\) and \(E_p\) look like “out-of-phase” sine waves.
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 \(f_d\) generates a steady-state amplitude that depends on how close \(f_d\) is to \(f_0\). 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
\[ v = \pm \omega \sqrt{x_0^2 - x^2}, \qquad v_{\text{max}} = \omega x_0. \]
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
- Sketch the energy bar first. It organises numbers before you reach for a calculator.
- State phase differences (\(x\to v\to a\)) instead of redrawing every graph.
- Quote damping type whenever you see a decaying sinusoid — it is a free descriptor mark.
10 Further reading
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.
