AQA AS Level Mathematics Paper 1, June 2025: Question 21
7 marks · Medium difficulty · Multi-step Problem
Find the acceleration and tension in terms of g for two connected buckets over a pulley after a mass is added to one, and state a limitation of modelling the buckets as particles.
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Question text
21 Two buckets, A and B, each have mass 0.5 kg
Each bucket is attached to one end of a light inextensible rope.
The rope is hung over a smooth fixed pulley.
The system is in equilibrium with both buckets hanging freely at rest, as shown in
the diagram.
A B
A builder then places a brick of mass 3 kg inside bucket A
Bucket A, with the brick inside, then moves vertically downwards.
During the subsequent motion, the magnitude of the acceleration of each bucket is
a m s–2 and the magnitude of the tension in the rope is T N
Assume the buckets and brick can be modelled as particles.
21 (a) Find a and T, leaving your answers in terms of g
Fully justify your answer.
[6 marks]
… 29
(28)
21 (b) Explain a limitation of modelling the buckets as particles.
[1 mark]
Mark scheme
Show the mark scheme
Q Marking instructions AO Marks Typical solution
21(a) Forms a three-term equation 3.3 M1 For bucket A
modelling the motion of 3.5g – T = 3.5a
bucket A with at least one side
correct For bucket B
Forms a three-term equation 3.3 M1 T – 0.5g = 0.5a
modelling the motion of
bucket B with at least one side Eliminating T
correct 3g = 4a
Forms two fully correct 1.1b A1
equations 3g
Eliminates either a or T to form 1.1a M1 a =
a single equation
Substituting
3g 1.1b A1 g
Obtains a = 7
4 T =
OE in terms of g
7g 1.1b A1
Obtains T =
OE in terms of g
Subtotal 6
21(b) Explains that the model is 3.5b E1 Assumes the buckets have no size
unrealistic as buckets have size but this is unrealistic as buckets
have dimensions
Subtotal 1
Question 21 Total 7
How to answer it
Connected Particles: Pulley System & Modelling Assumptions
This question assesses your ability to apply Newton's Second Law to a connected particle system hanging vertically over a smooth fixed pulley, as well as evaluating mechanics modelling assumptions:
- Setting up individual equations of motion ( F = ma ) for connected bodies moving in opposite directions.
- Correctly accounting for composite masses (bucket + contents).
- Solving simultaneous equations to find acceleration ( a ) and tension ( T ) in exact terms of g .
- Critically evaluating the particle model in a real-world physical scenario.
Question 21 (a)
Find acceleration (a) and tension (T) in terms of g [6 marks]
✅ Correct Answers
Acceleration:
a = ³⁄₄ g m s⁻² (or 0.75g )
Tension:
T = ⁷⁄₈ g N (or 0.875g )
💡 Key Knowledge
- Total Mass: Bucket A contains a brick, so total mass m_A = 0.5 + 3 = 3.5 kg . Bucket B has mass m_B = 0.5 kg .
- Newton's Second Law: Resultant Force = m × a along the line of motion.
- Light inextensible rope: Both buckets experience identical acceleration magnitude a .
- Smooth pulley: Tension T is uniform across both sides of the rope.
📐 Step-by-Step Solution
Step 1: Set up the equation of motion for Bucket A (moving downwards ↓)
Forces acting on A: Weight downwards ( 3.5g ) and tension upwards ( T ).
3.5g - T = 3.5a [Equation 1]
Step 2: Set up the equation of motion for Bucket B (moving upwards ↑)
Forces acting on B: Tension upwards ( T ) and weight downwards ( 0.5g ).
T - 0.5g = 0.5a [Equation 2]
Step 3: Eliminate T to find acceleration a
Add [Equation 1] and [Equation 2] together:
(3.5g - T) + (T - 0.5g) = 3.5a + 0.5a
3g = 4a
a = ³⁄₄ g (or 0.75g m s⁻² )
Step 4: Substitute a back to find tension T
Substitute a = 0.75g into [Equation 2]:
T = 0.5g + 0.5(0.75g) = 0.5g + 0.375g = 0.875g = ⁷⁄₈ g N
🧠 Mark Breakdown & Exam Technique
- M1 (AO3.3): Form a 3-term equation for Bucket A with at least one side fully correct.
- M1 (AO3.3): Form a 3-term equation for Bucket B with at least one side fully correct.
- A1 (AO1.1b): Both equations completely correct.
- M1 (AO1.1a): Valid algebraic method to eliminate either a or T .
- A1 (AO1.1b): Correct expression for a = ³⁄₄ g .
- A1 (AO1.1b): Correct expression for T = ⁷⁄₈ g .
- Tip: Always specify the direction you resolve in (e.g. "For A (↓):") so signs are clear to the examiner.
❌ Common Calculation Traps
- Forgetting bucket mass: Using m_A = 3 kg instead of 3.5 kg . You must add the brick and bucket together!
- Sign reversal: Writing T - 3.5g = 3.5a for bucket A. Since A accelerates downwards, downwards force is greater ( 3.5g - T ).
- Evaluating g numerically: The question explicitly states "leaving your answers in terms of g". Do not substitute 9.8 .
Question 21 (b)
Limitation of modelling the buckets as particles [1 mark]
✅ Correct Answers (Any one of the following)
- In reality, buckets have size / dimensions (or volume), whereas particles have zero dimensions / no size.
- The buckets could swing, tilt, or rotate, which cannot happen for a particle.
- The brick could shift or move relative to the bucket.
🧠 Examiner Commentary & Pitfalls
What top candidates did:
- Directly addressed the specific modelling term requested: "particle". A particle has negligible dimensions (mass acts at a single point).
What lost marks:
- Blaming air resistance, rope friction, or pulley weight. These refer to "smooth" or "light" assumptions, NOT the "particle" assumption.
Topics
Mechanics · R: Forces and Newton’s laws
Question and mark scheme from the AQA AS Level Mathematics examination, Paper 1, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.