Simple harmonic systems (A-level only) — AQA A-Level Physics

Simple harmonic systems are a fundamental topic in AQA A-Level Physics that describes oscillatory motion where the restoring force is proportional to displacement. You will study the defining equation a = -omega squared x, the relationships between displacement, velocity and acceleration, and how these vary sinusoidally with time. This topic also covers the energy transfers in SHM and applies the theory to important practical systems such as a mass on a spring and a simple pendulum.

Understanding SHM is crucial because it appears in many areas of physics, from waves and sound to molecular vibrations and electrical circuits. It builds on your knowledge of forces, energy and circular motion, and it develops your ability to use calculus and trigonometric functions in a physical context. Mastery of SHM is essential for tackling more advanced topics like resonance and damping, and it frequently appears in exam questions that test both conceptual understanding and mathematical application.

• →Definition of SHM: acceleration is directly proportional to displacement from equilibrium and is always directed towards that equilibrium point, expressed as a = -omega squared x.
• →The equations for displacement, velocity and acceleration as functions of time: x = A cos(omega t), v = -A omega sin(omega t), a = -A omega squared cos(omega t).
• →The relationship between period T, frequency f and angular frequency omega: omega = 2 pi f = 2 pi / T.
• →Energy in SHM: total energy is constant and equals 1/2 m omega squared A squared; kinetic energy is maximum at equilibrium and zero at maximum displacement, while potential energy is the opposite.
• →Practical SHM systems: a mass on a spring has T = 2 pi times square root of (m / k), and a simple pendulum has T = 2 pi times square root of (L / g) for small angles.

• State that damping is caused by resistive forces doing work on the oscillator, so mechanical energy is dissipated to the surroundings and the amplitude decreases with time.
• Describe light damping as a gradual exponential-like decrease in amplitude with the oscillator still completing many oscillations and the period remaining approximately constant.
• Describe heavy damping as a slow return to equilibrium with no oscillation, and critical damping as the fastest return to equilibrium without overshooting.
• Explain practical consequences, such as critical damping in door closers, car suspension or moving-coil meters, where overshoot or prolonged oscillation is undesirable.
• Compare energy loss: greater damping removes a larger fraction of energy per cycle, so the amplitude falls more rapidly and fewer oscillations are observed.
• Times a large number of oscillations and divides by that number to obtain a mean period, reducing the effect of human reaction time.
• Uses a fiducial marker at the equilibrium position and starts timing as the oscillator passes it, so each timing begins from the same phase.
• For the mass–spring system, plots T² against m and uses gradient = 4π²/k to determine the spring constant k.
• For the pendulum, plots T² against l and uses gradient = 4π²/g to determine g.
• Keeps the amplitude small so the small-angle approximation holds and the motion remains close to simple harmonic.
• Repeats measurements and calculates a mean, and considers uncertainties in length, time and mass.

• 💡Sketch displacement–time curves for light, heavy and critical damping on the same axes, labelling the equilibrium line and showing that only the lightly damped curve crosses it repeatedly.
• 💡Use the phrase 'returns to equilibrium in the shortest time without overshooting' when defining critical damping, and give one named application to show understanding.
• 💡When explaining energy transfer, refer to work done against resistive forces and the resulting decrease in amplitude rather than saying that energy is simply 'lost'.
• 💡State clearly how you reduce timing uncertainty, for example by timing 20 oscillations and dividing by 20.
• 💡When describing a graph, name the axes and explain how the gradient gives the required constant.
• 💡Mention at least one control variable, such as keeping the amplitude small or using the same spring.
• 💡Always start your answer by writing the defining equation a = -omega squared x and explicitly state that the minus sign indicates acceleration is directed towards the equilibrium position. This secures the definition marks.
• 💡When sketching graphs of displacement, velocity and acceleration against time, ensure you label the axes with the correct quantity and unit, and show the phase relationships: velocity leads displacement by pi/2 radians, and acceleration is in antiphase with displacement.
• 💡In calculations, use standard form and give your answer to an appropriate number of significant figures, usually the same as the least precise value in the question. Always include units in your final answer.

• Thinking damping always changes the period dramatically. Correction: for light damping the period is almost unchanged; the main observable effect is a decrease in amplitude.
• Confusing heavy damping with critical damping. Correction: heavy damping returns to equilibrium slowly without oscillating, whereas critical damping returns in the shortest time without overshooting.
• Believing that damping stops the oscillator instantly. Correction: damping gradually removes energy, so the amplitude decays over many cycles in the lightly damped case.
• Timing only one oscillation, which makes reaction time a large fraction of the measured period; correct by timing many oscillations and dividing.
• Measuring the pendulum length to the top of the bob rather than to its centre, which gives a systematic error in l; correct by measuring to the centre of the bob.
• Using a large amplitude for the pendulum, so the motion is not simple harmonic; correct by keeping the amplitude small.
• Students often think that acceleration is maximum at equilibrium because velocity is maximum. In fact, acceleration is zero at equilibrium and maximum at maximum displacement, because acceleration is proportional to displacement.
• Many students believe that the period of a simple pendulum depends on the mass of the bob. The period depends only on the length of the pendulum and the acceleration due to gravity, provided the angle is small.
• Students sometimes confuse the terms 'frequency' and 'angular frequency'. Frequency f is measured in hertz (cycles per second), while angular frequency omega is measured in radians per second and equals 2 pi f.

1. 1Start by learning the definition of SHM and the equation a = -omega squared x. Practice identifying SHM from given equations or graphs, and ensure you can explain the significance of the negative sign.
2. 2Derive and memorise the expressions for displacement, velocity and acceleration as functions of time. Sketch these graphs for at least two full cycles, noting the phase relationships and the points where each quantity is zero or maximum.
3. 3Study the energy changes in SHM. Learn the equations for kinetic energy, potential energy and total energy, and practice sketching energy-displacement graphs. Solve problems involving energy conservation.
4. 4Apply SHM to the mass-spring and simple pendulum systems. Derive their periods and solve numerical problems. Understand the assumptions and limitations of each model.
5. 5Attempt past paper questions on SHM, focusing on both definitions and calculations. Mark your answers using the mark scheme and note where you lose marks. Revise any weak areas.

• 📋Definition and explanation questions: e.g., 'State what is meant by simple harmonic motion.' These require precise wording including the proportionality to displacement and the direction towards equilibrium. Practice writing the definition concisely.
• 📋Graph sketching and interpretation: e.g., 'Sketch a graph of velocity against displacement for an object in SHM.' You must recall the shapes (ellipse for v-x, sine/cosine for x-t) and label key points such as amplitude and maximum velocity.
• 📋Calculation questions: e.g., 'Calculate the maximum acceleration of a mass-spring system given mass, spring constant and amplitude.' These involve using omega = square root of (k/m) and a_max = omega squared A. Show all steps clearly.
• 📋Practical and experimental questions: e.g., 'Describe an experiment to determine the spring constant using SHM.' You need to know how to measure period and use T = 2 pi times square root of (m/k) to find k, including how to reduce uncertainties.