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

Simple harmonic motion (SHM) is a fundamental topic in A-level Physics that describes oscillations where the restoring force is proportional to displacement. It underpins many physical systems, from pendulums and springs to molecular vibrations and electromagnetic waves. Understanding SHM is crucial for analysing energy transfers in oscillating systems and for later topics such as resonance and damping.

In the AQA A-Level specification, SHM is a key part of the 'Further Mechanics and Thermal Physics' module. You will learn the defining equation a = -ω²x, derive expressions for velocity and displacement, and study the interchange between kinetic and potential energy. This topic also connects to practical work on investigating oscillations and to real-world applications like car suspensions and building stability.

• →Definition of SHM: acceleration is directly proportional to displacement from equilibrium and always directed towards equilibrium, expressed as a = -ω²x.
• →Key equations: x = A cos(ωt) or x = A sin(ωt), v = ±ω√(A² - x²), v_max = ωA, a_max = ω²A, and T = 2π/ω.
• →Energy in SHM: total energy is constant and equals ½mω²A²; kinetic energy is maximum at equilibrium and zero at maximum displacement, while potential energy is the opposite.
• →Phase relationships: velocity leads displacement by π/2 radians, and acceleration is π radians out of phase with displacement.
• →Examples of SHM: mass-spring system (T = 2π√(m/k)) and simple pendulum (T = 2π√(L/g)) for small angles.

• States the defining condition a = −ω²x, with acceleration proportional to displacement and directed towards equilibrium.
• Identifies amplitude, period, frequency and angular frequency, and links ω = 2πf = 2π/T.
• Describes the phase relationship between displacement, velocity and acceleration over one cycle.
• Explains that velocity is maximum at equilibrium and zero at maximum displacement, while acceleration is maximum at maximum displacement and zero at equilibrium.
• Interprets or sketches graphs of x, v and a against time, recognising the same period and the phase differences.
• Uses x = A cos(ωt) or x = A sin(ωt) and differentiates to obtain velocity and acceleration expressions.
• State that the gradient of a displacement–time graph at a given instant gives the instantaneous velocity.
• State that the gradient of a velocity–time graph at a given instant gives the instantaneous acceleration.
• Apply the chain of gradients to SHM: where x is maximum, gradient is zero so v = 0; where x crosses zero, gradient is steepest so v is maximum in magnitude.
• Relate the sign of the v–t gradient to the direction of acceleration, showing that a is directed towards the equilibrium position and is opposite in sign to x.
• Use the standard SHM expressions x = A cos(ωt), v = −Aω sin(ωt) and a = −Aω² cos(ωt) to confirm the graphical relationships and the result a = −ω²x.

• 💡Start by quoting a = −ω²x and then link each characteristic to this defining relationship.
• 💡When describing graphs, state the phase difference in fractions of a cycle or in radians rather than vaguely saying they are shifted.
• 💡Check whether the question gives period or frequency before calculating ω, and show the conversion clearly.
• 💡Sketch x–t, v–t and a–t on the same time axis so that maxima, minima and zero crossings line up; this makes the gradient relationships visible.
• 💡When asked to compare graphs, quote the gradient relationship explicitly, for example state that the value of v at a time equals the gradient of x–t at that time.
• 💡Always define SHM in terms of acceleration, not force, unless the question specifically asks about force. The AQA mark scheme expects 'acceleration is proportional to displacement and directed towards equilibrium'.
• 💡When using the equation v = ±ω√(A² - x²), remember that the ± indicates direction. In calculations, you can ignore the sign if only the magnitude is required, but state the direction if asked.
• 💡For graph questions, label axes with quantities and units, and ensure the curves have the correct phase relationships. Use radians for phase differences if required.

• Omitting the minus sign in a = −ω²x; the correction is to include it to show acceleration is directed towards equilibrium.
• Stating that velocity and acceleration are in phase; the correction is to recognise that velocity leads displacement by a quarter cycle and acceleration is in antiphase with displacement.
• Confusing period with frequency when finding ω; the correction is to use ω = 2π/T for period and ω = 2πf for frequency, keeping the two distinct.
• Assuming acceleration always opposes velocity. Correction: in SHM, acceleration points towards equilibrium, so it opposes velocity when moving away from equilibrium, but acts in the same direction when moving towards it.
• Confusing the value of the graph with its gradient: for example, reading the height of the x–t graph as velocity. Correction: velocity is the gradient of the x–t graph, not its displacement value.
• Treating the gradient as constant over a curved section. Correction: for SHM the gradient changes continuously, so the instantaneous gradient must be found at a point, for example by drawing a tangent.
• Students often think that acceleration is maximum at maximum displacement but in the same direction as displacement. In fact, acceleration is maximum at maximum displacement but directed towards equilibrium (opposite to displacement).
• Many believe that the period of a pendulum depends on its mass or amplitude. For small angles, the period is independent of mass and amplitude, depending only on length and g.
• Some confuse the terms 'frequency' and 'angular frequency'. Angular frequency ω is 2π times the frequency f, and it appears in the SHM equations.

1. 1Day 1-2: Learn the definition of SHM and the equation a = -ω²x. Practice identifying SHM from given equations or graphs.
2. 2Day 3-4: Derive and memorise the equations for displacement, velocity, and acceleration as functions of time. Solve problems using these equations.
3. 3Day 5-6: Study energy in SHM. Learn the expressions for kinetic and potential energy and how they vary with displacement. Practice sketching energy-displacement graphs.
4. 4Day 7-8: Investigate the mass-spring and simple pendulum systems. Derive their periods and solve numerical problems.
5. 5Day 9-10: Review past exam questions on SHM, focusing on graph interpretation and multi-step calculations. Use mark schemes to refine answers.

• 📋Definition and explanation questions: e.g., 'State what is meant by simple harmonic motion.' These require precise wording: acceleration proportional to displacement, directed towards equilibrium.
• 📋Calculation questions: e.g., 'Calculate the maximum speed of an oscillator given its amplitude and period.' Use v_max = ωA or v_max = 2πfA.
• 📋Graph sketching and interpretation: e.g., 'Sketch graphs of displacement, velocity, and acceleration against time for an object in SHM.' Ensure correct phase relationships and labels.
• 📋Practical and data analysis questions: e.g., 'Describe an experiment to determine the spring constant using SHM.' Focus on measurements, variables, and graph plotting (T² against m).