A-Level Physics: SHM Graphs, Damping and Resonance
Simple harmonic motion questions at A level rarely stop at recalling a formula. They ask you to read and sketch graphs, to link displacement, velocity, acceleration and energy at a given moment, and to explain what happens when an oscillator loses energy or is driven by a periodic force. This year-13 material concentrates on exactly those skills.
The quiz (12 questions) starts with the state of an oscillator described by x = A cos ωt at the instant it is released, then moves to calculations: the speed at a given displacement using v = ±ω√(A² − x²), the kinetic energy of a mass on a spring part-way through its swing, the displacement at a given time (with the reminder that the angle is in radians), and the maximum acceleration of a vibrating loudspeaker cone. Energy questions check that total energy is proportional to amplitude squared and that kinetic and potential energy vary at twice the oscillation frequency. The second half covers damping and forced oscillations: what makes damping critical, the difference between free and forced oscillations, the phase of an oscillator driven well above its natural frequency, the effect of damping on the shape of a resonance curve, Barton's pendulums, and the driving frequency at which a mass on a spring resonates.
The flashcards (12 cards) summarise the defining equation a = −ω²x, the phase relationships between displacement, velocity and acceleration, the velocity–displacement equation, total energy, maximum acceleration, critical damping, resonance, the phase at resonance, the effect of damping on the resonance peak and Barton's pendulums.
The written work (8 questions) asks you to describe the three motion graphs and their phase relationships, carry out a full set of mass–spring calculations, explain energy–displacement graphs and show that three-quarters of the energy is kinetic at half the amplitude, compare light, heavy and critical damping with car suspension as an example, plan an experiment to obtain a resonance curve, explain Barton's pendulums including the phases, discuss useful and harmful resonance and how engineers reduce it, and work out the energy lost by a lightly damped pendulum from its falling amplitude.
The content matches the oscillations section common to A level Physics specifications in England (for example AQA 3.6.1.2 to 3.6.1.4, simple harmonic motion, simple harmonic systems and forced vibrations and resonance). It builds on, and does not repeat, the introductory circular motion and SHM material in this category.
- Relate displacement, velocity, acceleration and energy at any point of an SHM cycle and interpret the graphs
- Use x = A cos ωt, v = ±ω√(A² − x²) and a_max = ω²A in calculations
- Describe how kinetic, potential and total energy vary with displacement and amplitude
- Distinguish light, heavy and critical damping and give practical examples
- Explain forced oscillations, resonance, resonance curves with damping and the phase of a driven oscillator
Practice material written by Zestly, based on the DfE GCE AS and A level subject content for physics (DFE-00356-2014): oscillations, simple harmonic motion, damping, forced vibrations and resonance.
Sample question
An oscillator's displacement is given by $x = A\cos(\omega t)$. Which statements are correct at the instant $t = 0$?
- The acceleration has its maximum magnitude, $\omega^2 A$, and is directed towards the equilibrium position
- The acceleration is zero
- The velocity is zero
- The kinetic energy is zero and the potential energy is at its maximum
- The velocity has its maximum magnitude
See the answer
The acceleration has its maximum magnitude, $\omega^2 A$, and is directed towards the equilibrium position, The velocity is zero, The kinetic energy is zero and the potential energy is at its maximum
At $t = 0$, $x = A$: the oscillator is at maximum displacement. There it is momentarily at rest ($v = -A\omega\sin 0 = 0$), so its kinetic energy is zero and all the energy is potential. From $a = -\omega^2 x$ the acceleration is $-\omega^2 A$: maximum magnitude, pointing back towards equilibrium. On the graphs, the $v$–$t$ curve is a negative sine and the $a$–$t$ curve is a negative cosine.