Home · A-Level Maths 9709 · Newton's laws of motion · Connected particles and pulleys · Practice
AS & A Level · Practice questions · Connected particles and pulleys

Each particle, its own equation.

Six original Cambridge-style questions on connected particles, tow-ropes, tow-bars and pulleys, with full worked solutions.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
The method

F = ma for each particle, then add.

01
Recall
[3 marks]

A car of mass 1000 kg tows a trailer of mass 500 kg with a light horizontal rope on a level road. The driving force is 2000 N and there is no resistance. Find the acceleration and the tension in the rope.

  • Whole: \(2000 = 1500a\), so \(a = 1.33\ \text{m s}^{-2}\). ✓
  • Trailer: \(T = 500 \times \tfrac{4}{3} = 667\) N. ✓
02
Calculation
[4 marks]

A car of mass 1500 kg tows a trailer of mass 500 kg. The driving force is 3000 N; the resistances are 400 N on the car and 200 N on the trailer. Find the acceleration and the tension in the tow-rope.

  • Whole: \(3000 - 600 = 2000a\), so \(a = 1.2\ \text{m s}^{-2}\). ✓
  • Trailer: \(T - 200 = 500(1.2)\), so \(T = 800\) N. ✓
03
Calculation
[4 marks]

Particles of mass 5 kg and 3 kg hang on either side of a smooth fixed pulley, connected by a light inextensible string. They are released from rest. Find the acceleration, the tension and the force exerted by the string on the pulley.

  • \(50 - T = 5a\), \(T - 30 = 3a\); adding, \(a = 2.5\ \text{m s}^{-2}\). ✓
  • \(T = 37.5\) N. ✓
  • Force on pulley \(= 2T = 75\) N, vertically downwards. ✓
04
Analysis
[5 marks]

A block A of mass 3 kg lies on a rough horizontal table and is joined by a light inextensible string over a smooth pulley at the edge to a block B of mass 2 kg hanging freely. When released, the system accelerates at \(2.8\ \text{m s}^{-2}\). Find the coefficient of friction and the tension.

  • B: \(20 - T = 2(2.8)\), so \(T = 14.4\) N. ✓
  • A: \(T - \mu(30) = 3(2.8)\), so \(14.4 - 30\mu = 8.4\). ✓
  • \(\mu = 0.2\). ✓
05
Analysis
[6 marks]

A particle A of mass 4 kg lies on a rough plane inclined at \(30^\circ\), \(\mu = 0.25\). A light inextensible string from A passes up the plane, over a smooth pulley at the top, to a particle B of mass 6 kg hanging freely. The system is released from rest. Find the acceleration and the tension.

  • A: \(R = 40\cos 30^\circ = 34.64\) N; friction \(= 8.660\) N down the plane (A moves up). ✓
  • A: \(T - 20 - 8.660 = 4a\). B: \(60 - T = 6a\). ✓
  • Add: \(31.34 = 10a\), so \(a = 3.13\ \text{m s}^{-2}\). ✓
  • \(T = 60 - 6(3.134) = 41.2\) N. ✓
06
Analysis
[6 marks]

Particles P (5 kg) and Q (3 kg) hang on either side of a smooth pulley, with P 1.2 m above the floor. They are released from rest. P hits the floor and does not rebound; Q never reaches the pulley.

(a) Find the time P takes to reach the floor.

(b) Find the greatest height of Q above its starting point.

  • (a) \(a = 2.5\ \text{m s}^{-2}\); \(1.2 = \tfrac{1}{2}(2.5)t^2\), so \(t^2 = 0.96\) and \(t = 0.980\) s. ✓
  • (b) Speed at impact: \(v^2 = 2(2.5)(1.2) = 6\). ✓
  • Q then moves freely under gravity: further rise \(= \dfrac{6}{20} = 0.3\) m. ✓
  • Greatest height \(= 1.2 + 0.3 = 1.5\) m. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.