AQA A-Level Physics: Simple Harmonic Motion (SHM) Analysis — mark scheme explained

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

What each mark is really for — in plain English — and the wording trap that loses it.

• S1

  Step 1: Determine the angular frequency ω using the formula ω = 2π/T . ω = 2π/2 = π rad/s

• S2

  Step 2: Calculate the maximum speed using the formula v max = ωA . v max = π × 0.1 = 0.314 m/s

• S3

  Step 3: Calculate the maximum acceleration using the formula a max = ω 2 A . a max = (π) 2 × 0.1 = 0.987 m/s 2

Model answer

Worked through, with each step tagged to the mark it earns.

1. S1

   Step 1: Determine the angular frequency ω using the formula ω = 2π/T . ω = 2π/2 = π rad/s

2. S2

   Step 2: Calculate the maximum speed using the formula v max = ωA . v max = π × 0.1 = 0.314 m/s

3. S3

   Step 3: Calculate the maximum acceleration using the formula a max = ω 2 A . a max = (π) 2 × 0.1 = 0.987 m/s 2

4. Final answer: v max = 0.314 m/s, a max = 0.987 m/s 2

Common mistakes

• Confusing the amplitude with the maximum speed. — 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. — 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. — 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 ). — 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. — 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. — 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

• Full worked solution (all marking points)6 marks

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