A-Level · Physics · AQA · Mark scheme decoded
AQA A-Level Physics: Simple Harmonic Motion (SHM) Analysis — mark scheme explained
The short answer
Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction of the displacement.
The question
A mass-spring system oscillates with an amplitude of 0.1 m and a period of 2 seconds. Calculate the maximum speed and maximum acceleration.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For calculation questions, marks are typically awarded for correct substitution into formulas, showing working, and providing the final answer. For sketching graphs, accuracy in shape, amplitude, and phase relationships is crucial.
What the command words demand
- Calculate
- Perform a numerical calculation using the given data and appropriate formulas.
- Derive
- Show how one equation or relationship can be obtained from another, step by step.
- Sketch
- Draw a graph that accurately represents the given function or relationship.
- Explain
- Provide a clear and concise explanation of a concept or phenomenon.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 5-7 minutes per question to ensure you have enough time to carefully read the question, perform calculations, and check your work.
- Determine the angular frequency ω using the formula ω = 2π/T . ω = 2π/2 = π rad/s2 marks
- Calculate the maximum speed using the formula v max = ωA . v max = π × 0.1 = 0.314 m/s2 marks
- Calculate the maximum acceleration using the formula a max = ω 2 A . a max = (π) 2 × 0.1 = 0.987 m/s 22 marks
Final answer: v max = 0.314 m/s, a max = 0.987 m/s 2
Work through every step correctly and you earn all 6 marks.
Another worked example
A pendulum oscillates with an angular frequency of 5 rad/s and an amplitude of 0.05 m. Sketch the x-t, v-t, and a-t graphs for one complete cycle.
- Use the equation x = A cos(ωt) to sketch the x-t graph. x = 0.05 cos(5t)2 marks
- Use the equation v = ±ω√(A 2 − x 2 ) to derive and sketch the v-t graph. v = ±5√(0.05 2 − (0.05 cos(5t)) 2 )2 marks
- Use the equation a = −ω 2 x to derive and sketch the a-t graph. a = −25 × 0.05 cos(5t) = −1.25 cos(5t)2 marks
Final answer: Graphs of x-t, v-t, and a-t for one complete cycle
Work through every step correctly and you earn all 6 marks.
Common mistakes
Confusing the amplitude with the maximum speed.
Why it happens: Students often mix up the definitions of amplitude and maximum speed. Amplitude is the maximum displacement, while maximum speed is given by v max = ωA .
Fix: Review the definitions: amplitude ( A ) is the maximum displacement, and maximum speed ( v max ) is calculated using v max = ωA .
Incorrectly deriving the v-t graph from the x-t graph.
Why it happens: Students may not understand that the velocity is the gradient of the position-time graph. They might incorrectly assume it is a simple transformation or use the wrong formula.
Fix: Practice finding the slope (gradient) of the x-t graph at various points to derive the v-t graph. Use the relationship v = ±ω√(A 2 − x 2 ) for verification.
Forgetting the negative sign in the acceleration equation.
Why it happens: The negative sign in a = −ω 2 x indicates that the acceleration is always directed towards the equilibrium position. Students might overlook this, leading to incorrect signs in their calculations.
Fix: Always include the negative sign when writing the acceleration equation for SHM. Emphasize that it represents the direction of the restoring force.
Using the wrong units for angular frequency ( ω ).
Why it happens: Students might use degrees instead of radians when calculating ω . Angular frequency is always in radians per second.
Fix: Ensure that all calculations involving angular frequency are done using radians. Convert any given angles from degrees to radians if necessary.
Misinterpreting the phase difference between x-t, v-t, and a-t graphs.
Why it happens: Students might not understand that the v-t graph is out of phase with the x-t graph by π/2 radians (90 degrees), and the a-t graph is out of phase with the x-t graph by π radians (180 degrees).
Fix: Practice sketching all three graphs together, emphasizing the phase differences. Use visual aids to show how the graphs shift relative to each other.
Incorrectly applying the maximum speed and acceleration formulas.
Why it happens: Students might use the wrong values for A or ω when calculating maximum speed and acceleration. They might also confuse the formulas for these quantities.
Fix: Double-check the values of amplitude ( A ) and angular frequency ( ω ) before applying the formulas v max = ωA and a max = ω 2 A . Practice with different values to reinforce understanding.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Calculate SHM Quantities | Find the maximum speed and acceleration of an oscillating mass-spring system from its amplitude and period. | 6 |
| Sketch SHM Graphs | Sketch displacement, velocity, and acceleration graphs for one cycle of a pendulum's oscillation. | 6 |
| Calculate SHM Quantities | Find position, velocity, and acceleration of an oscillator at a given time using SHM equations. | 6 |
| SHM Calculations | Find maximum speed and acceleration of a pendulum from its amplitude and period. | 6 |
| Total across these question types | 24 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.