AQA A-Level Physics Paper 1, June 2025: Question 31

1 mark · Medium difficulty · Multiple Choice

Calculate the mass of an object attached to a vibrating spring-mass system at resonance given the spring constant and driving frequency.

Practise this question

Question

A multiple-choice question showing a horizontal bar that can oscillate vertically as indicated by a double-ended arrow. Hanging from the center of the bar is a vertical helical spring attached to a block of mass m. The text states that the spring has a constant of 50 N m⁻¹ and undergoes resonance at a driving frequency of 2.6 Hz with damping ignored. Four options for mass m are given: A: 190 g, B: 370 g, C: 490 g, and D: 590 g.
Question text

31 A spring with a spring constant of 50 N m−1 is attached to a rigid horizontal bar.

An object of mass m is attached to the bottom of the spring.

When the bar is made to move vertically up and down with a frequency of 2.6 Hz, the

mass–spring system undergoes resonance.

Ignore the effects of damping.

What is m?

[1 mark]

A 190 g

B 370 g

C 490 g

D 590 g

Mark scheme

Show the mark scheme Mark scheme entry for question 31 showing the correct answer is A (190 g) with assessment objective AO2.

31 A 190 g AO2

How to answer it

Resonance of a Vertical Mass–Spring System

What this question tests

This question assesses your understanding of forced vibrations and resonance in simple harmonic motion (SHM). Specifically, it requires you to recognise the resonance condition (driving frequency = natural frequency), apply the formula for the time period of a mass-spring system, rearrange it to solve for unknown mass m, and correctly handle unit conversions from kilograms to grams.

Question 31 • Multiple Choice [1 Mark]

Determining Unknown Mass at Resonance

AQA A-Level Physics • Further Mechanics & Thermal Physics (Periodic Motion)

✅ Correct Answer

A — 190 g

Awarded for correctly calculating m ≈ 0.187 kg and rounding to 2 significant figures (190 g).

💡 Key Knowledge

  • Condition for Resonance: Occurs when the periodic driving frequency ( f = 2.6 Hz ) equals the natural frequency ( f₀ ) of the oscillating system.
  • Time Period Formula: T = 2π√(m / k)
  • Frequency Relation: Since f = 1 / T : f = (1 / 2π) × √(k / m)

📐 Step-by-Step Calculation

  1. Identify given values:
    Spring constant, k = 50 N m⁻¹
    Resonant frequency, f = 2.6 Hz
  2. Calculate the angular frequency (ω):
    ω = 2πf = 2 × π × 2.6 = 16.336 rad s⁻¹
  3. Rearrange the natural frequency equation for mass (m):
    Since ω = √(k / m) , squaring both sides yields ω² = k / m
    m = k / ω² = k / (2πf)²
  4. Substitute the numbers:
    m = 50 / (16.336)² = 50 / 266.87 = 0.18736 kg
  5. Convert kilograms to grams:
    m = 0.18736 × 1000 g = 187.4 g ≈ 190 g (2 s.f.)
    → Matches Option A.

🧠 Exam Technique

  • Spot the trigger word: The word "resonance" is your clue that the driving frequency equals the system's natural frequency ( fdriving = f0 ).
  • Use the Data Sheet efficiently: The formula is given as T = 2π√(m/k) . You can either find T = 1 / 2.6 = 0.3846 s first, or jump directly using ω = 2πf .
  • Check units immediately: The answers are listed in grams ( g ), but SI standard equations give mass in kilograms ( kg ). Always multiply by 1000 before picking an option.

❌ Common Errors & Distractors

  • Forgetting to square (2π): Entering 50 / 2π(2.6)² on a calculator instead of 50 / (2π × 2.6)² leads to an incorrect denominator of 42.47 , yielding m ≈ 1.18 kg .
  • Inverting the ratio: Confusing m/k with k/m gives m ≈ 5.3 kg .
  • Using the pendulum formula: Accidental retrieval of T = 2π√(l/g) instead of the mass-spring equation.
  • Distractor values: Option B ( 370 g ) is roughly double the correct answer, commonly obtained if the student forgets the factor of 2 or squares π incorrectly.
Assessment Objective breakdown: AO2 (Application of knowledge and understanding of scientific concepts to solve problems involving periodic motion and mechanical resonance). 1 mark total.

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 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.