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LearnCambridge97022025–2027Paper 4 · A Level Structured QuestionsOscillations

Paper 4 · Module

Oscillations

3 subtopics≈ 105 min in total

Start the module: Simple harmonic motion Open graphing calculator

Simple harmonic motion
An oscillation is simple harmonic when the acceleration is proportional to the displacement from a fixed point and always directed towards that point: $a=-\omega^2x$ . The angular frequency is $\omega=2\pi f=\dfrac{2\pi}T$ . The solution is sinusoidal: $x=x_0\sin\omega t$ (starting from equilibrium). The period doesn't depend on the amplitude. A mass on a spring and a small-angle pendulum are both approximately simple harmonic.
≈ 35 min
Velocity and energy in SHM
In SHM the speed is greatest at equilibrium, $v_0=\omega x_0$ , and zero at the extremes. At displacement $x$ , $v=\pm\omega\sqrt{x_0^2-x^2}$ . Energy continually swaps between kinetic and potential, but the total stays constant (with no damping): $E=\tfrac12m\omega^2x_0^2$ . Kinetic energy is $\tfrac12m\omega^2(x_0^2-x^2)$ and potential energy is $\tfrac12m\omega^2x^2$ .
≈ 35 min
Damping and resonance
A resistive force (friction, air or fluid resistance) removes energy from an oscillator, so its amplitude decays: damping. Light damping gives slowly decaying oscillations; critical damping returns the system to equilibrium in the shortest time without oscillating; heavy damping returns it slowly without oscillating. When a periodic driving force matches the natural frequency, the amplitude becomes a maximum: resonance. More damping lowers and broadens the resonance peak.
≈ 35 min