AQA A-Level Physics Paper 1, June 2024: Question 31

1 mark · Medium difficulty · Multiple Choice

Determine the new period of oscillation of mass M when suspended from a different spring that extends by x/2 instead of x.

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Question

Multiple choice question numbered 31 about mass suspended from a spring. Text describes mass M extending a spring by x with period T, then suspended from a different spring extending by x/2. Four options are given: A (T/2), B (T/√2), C (T√2), D (2T).
Question text

31 When a mass M is suspended from a spring, the spring extends by a distance x.

M is displaced vertically and, when released, it oscillates with a period T.

x

M is removed and suspended from a different spring. The spring extends by a distance .

M is again displaced vertically and released.

What is the new period of oscillations of M?

[1 mark]

T

A

T

B

C T 2

D 2T

Mark scheme

Show the mark scheme Mark scheme indicating the correct answer is B, with the formula T/√2 and AO2 assessment objective.

T

31 B AO2

How to answer it

SHM and Spring Constant Proportionalities

What this question tests

This question assesses your understanding of Simple Harmonic Motion (SHM), specifically the time period of a mass-spring system, combined with Hooke's Law. It tests your ability to link static equilibrium extension to spring stiffness and scale proportionalities without needing numerical values.

Question 31: Multiple Choice Solution

Evaluating the New Period of Oscillations

✅ Correct Answer: B

The correct option is B ( T / √2 ). When the extension is halved, the spring constant doubles, causing the period to decrease by a factor of the square root of 2.

💡 Key Knowledge

  • Hooke's Law: Mg = kx , meaning spring constant k = Mg / x .
  • SHM Period Formula: For a mass-spring system, T = 2π√(M / k) .
  • Proportionality: Since mass M is constant, T is inversely proportional to √(k) , and directly proportional to √(x) .

🧠 Exam Technique

For 1-mark proportional reasoning questions, avoid substituting numbers. Instead, set up a ratio or substitute expressions directly to see how scaling factors propagate through equations.

❌ Common Errors

Students often mistake direct proportionality for inverse proportionality, or forget that period depends on the square root of the spring constant, incorrectly selecting T / 2 (Option A).

📐 Step-by-Step Derivation

  1. Initial state: Mg = k₁x , so k₁ = Mg / x . Period T = 2π√(M / k₁) .
  2. New state: The new extension is x / 2 . Therefore, the new spring constant k₂ = Mg / (x / 2) = 2(Mg / x) = 2k₁ .
  3. New period: Substitute k₂ into the SHM formula: T₂ = 2π√(M / k₂) = 2π√(M / 2k₁) .
  4. Simplify: T₂ = (2π√(M / k₁)) / √2 = T / √2 .
Mark Scheme Award: 1 mark for selecting option B ( T / √2 ). Tested via Assessment Objective 2 (AO2) - applying physics principles to a familiar scenario in a novel way.

Topics

Physics · 3.6 Further mechanics and thermal physics (A-level only)

Question and mark scheme from the AQA A-Level Physics examination, Paper 1, June 2024. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.