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What defines simple harmonic motion, and how does energy move during it?

The defining condition for simple harmonic motion, the equations for displacement, velocity and acceleration, energy in SHM, and free, damped and forced oscillations with resonance.

A CCEA A-Level Physics answer on the defining condition for simple harmonic motion, the equations for displacement, velocity and acceleration, the exchange of kinetic and potential energy, and free, damped and forced oscillations including resonance.

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  1. What this dot point is asking
  2. The answer
  3. Examples in context
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What this dot point is asking

CCEA wants you to state the defining condition for simple harmonic motion, use the equations for displacement, velocity and acceleration, describe the exchange of kinetic and potential energy, and describe free, damped and forced oscillations and resonance. Graph sketches and a numerical SHM calculation appear in almost every paper.

The answer

The defining condition

This restoring condition arises whenever the net force is proportional to displacement, F=−kxF = -kx, as for a mass on a spring or, for small angles, a simple pendulum.

The equations of SHM

Energy, damping and resonance

In SHM, kinetic energy is maximum at the centre and potential energy is maximum at the extremes, while the total energy stays constant for an undamped oscillation. The total energy is E=12mω2A2E = \tfrac{1}{2}m\omega^2 A^2, so it scales with the square of the amplitude. Damping is the loss of energy to resistive forces, reducing the amplitude over time; light damping decays slowly, critical damping returns the system to rest in the shortest time without overshoot, and heavy (over)damping returns it slowly with no oscillation.

Worked example: a simple pendulum

Examples in context

Example 1. The Millennium Bridge in London. When the footbridge opened, pedestrians unconsciously synchronised their steps with a slight sideways sway, driving it near its natural frequency. This forced oscillation at resonance built up an alarming amplitude, and the bridge was closed and fitted with dampers (energy-absorbing devices) that lowered and broadened the resonance peak so it could not build up again.

Example 2. A quartz watch crystal. A tiny quartz tuning fork is driven electrically at its natural frequency of 32 768 Hz32\,768\ \text{Hz}. Because the SHM period is independent of amplitude and the crystal is very lightly damped, the oscillation is extremely stable, and dividing the frequency down by powers of two gives an accurate one-second tick.

Try this

Q1. A mass on a spring oscillates with amplitude 0.050 m0.050\ \text{m} and frequency 2.0 Hz2.0\ \text{Hz}. Find its maximum speed. [3 marks]

  • Cue. ω=2πf=12.6 rad s−1\omega = 2\pi f = 12.6\ \text{rad s}^{-1}, so vmax⁡=ωA=12.6×0.050≈0.63 m s−1v_{\max} = \omega A = 12.6 \times 0.050 \approx 0.63\ \text{m s}^{-1}.

Q2. State the condition for resonance. [1 mark]

  • Cue. The driving frequency equals the natural frequency of the system.

Q3. A trolley undergoing SHM has displacement x=0.040 mx = 0.040\ \text{m} at an instant when its amplitude is 0.10 m0.10\ \text{m} and ω=6.0 rad s−1\omega = 6.0\ \text{rad s}^{-1}. Find its speed at that instant. [2 marks]

  • Cue. v=ωA2−x2=6.00.102−0.0402=6.0×0.0917=0.55 m s−1v = \omega\sqrt{A^2 - x^2} = 6.0\sqrt{0.10^2 - 0.040^2} = 6.0 \times 0.0917 = 0.55\ \text{m s}^{-1}.

Exam-style practice questions

Practice questions written in the style of CCEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

CCEA 20206 marksA trolley of mass 0.50 kg is attached to springs and oscillates with simple harmonic motion of amplitude 8.0 cm and period 1.2 s. Calculate the angular frequency, the maximum speed and the maximum acceleration of the trolley, and state where each maximum occurs.
Show worked answer →

The angular frequency is

ω=2πT=2π1.2=5.24\omega = \frac{2\pi}{T} = \frac{2\pi}{1.2} = 5.24 rad per second.

The maximum speed occurs at the equilibrium position (the centre of the oscillation):

vmax⁡=ωA=5.24×0.080=0.42v_{\max} = \omega A = 5.24 \times 0.080 = 0.42 m per second.

The maximum acceleration occurs at the extremes of the motion (maximum displacement):

amax⁡=ω2A=5.242×0.080=27.5×0.080=2.2a_{\max} = \omega^2 A = 5.24^2 \times 0.080 = 27.5 \times 0.080 = 2.2 m per second squared.

Markers reward correct ω\omega from the period, both maxima, and stating that speed peaks at the centre while acceleration peaks at the extremes.

CCEA 20185 marksExplain what is meant by resonance. Describe an experiment to demonstrate resonance in a mechanical system, and explain the effect of increasing the damping on the resonance curve of amplitude against driving frequency.
Show worked answer →

Resonance occurs when the frequency of a periodic driving force equals the natural frequency of the oscillating system. At this frequency energy is transferred most efficiently from the driver to the system, so the amplitude of oscillation reaches a maximum.

A simple demonstration is a Barton's pendulum arrangement, or a mass on a spring driven by a vibration generator whose frequency is varied. As the driving frequency is increased through the natural frequency, the amplitude rises to a sharp peak and then falls again.

Increasing the damping lowers the height of the resonance peak (the maximum amplitude is smaller), broadens it, and shifts the peak slightly to a lower frequency than the undamped natural frequency.

Markers reward the equal-frequency definition, a workable experiment with a way to vary frequency, and the lower, broader peak under heavier damping.

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