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

The question

A mass of 0.5 kg is attached to a spring with a spring constant of 20 N/m. Calculate the period of oscillation. [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

  1. Identify the given values: m = 0.5 kg , k = 20 N/m .

• S2

  2. Use the formula for the period of a mass-spring system: T = 2π√(m/k) .

• S3

  3. Substitute the values into the formula: T = 2π√(0.5/20) = 2π√(0.025) ≈ 2π × 0.158 ≈ 0.994 s

• S4

  4. Round the answer to an appropriate number of significant figures: T ≈ 0.99 s .

Model answer

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

1. S1

   1. Identify the given values: m = 0.5 kg , k = 20 N/m .

2. S2

   2. Use the formula for the period of a mass-spring system: T = 2π√(m/k) .

3. S3

   3. Substitute the values into the formula: T = 2π√(0.5/20) = 2π√(0.025) ≈ 2π × 0.158 ≈ 0.994 s

4. S4

   4. Round the answer to an appropriate number of significant figures: T ≈ 0.99 s .

5. Final answer: T ≈ 0.99 s

Common mistakes

• Forgetting to use the small-angle approximation in the derivation of the period for a simple pendulum. — Always remember that the small-angle approximation ( sin(θ) ≈ θ ) is necessary for deriving the period of a simple pendulum. This approximation is valid only for small angles (typically less than 10°).
• Confusing the formulas for the periods of mass-spring systems and simple pendulums. — Memorize the specific formulas for each system: T = 2π√(m/k) for mass-spring systems and T = 2π√(l/g) for simple pendulums. Practice using these formulas in different contexts to reinforce their differences.
• Failing to recognize the conservation of total energy in SHM. — Understand that in SHM, the total energy E total is conserved and given by E total = 0.5kA 2 . Practice problems involving the variation of kinetic and potential energies with displacement and time to reinforce this concept.
• Incorrectly identifying the points where kinetic and potential energies are maximum in SHM. — Remember that kinetic energy is maximum at the equilibrium position (where displacement is zero) and potential energy is maximum at the maximum displacement. Practice drawing energy graphs and solving problems involving these points to solidify your understanding.
• Misunderstanding the effects of different types of damping on oscillations. — Understand that light damping results in a slow decrease in amplitude with nearly constant period, critical damping returns the system to equilibrium as quickly as possible without oscillating, and heavy (over) damping returns the system to equilibrium slowly without oscillating, taking longer than critical damping. Practice problems involving damped oscillators to reinforce these concepts.
• Forgetting to use the correct units when calculating period or frequency. — Always check that all values are in consistent units before substituting them into formulas. For example, ensure that mass is in kilograms, spring constant is in newtons per meter, and length is in meters. Practice unit conversions if necessary.

Where the marks go

• Full worked solution (all marking points)4 marks

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