A-level · Forces & Newton's laws

Connected particles on slopes and pulleysExam-style questions

Two particles joined by a light inextensible string over a smooth pulley — one on a slope, one hanging — share the same acceleration and string tension; write F = ma along each particle's line of motion (including the slope component and friction) and solve simultaneously.

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18 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
Figure 1 shows a particle $A$ of mass $3$ kg on a plane inclined at $30^\circ$ to the horizontal. $A$ is connected by a light inextensible string, passing over a smooth pulley at the top of the plane, to a particle $B$ of mass $5$ kg which hangs freely.Figure 1 (not accurately drawn)

The system is released from rest with the string taut.
(a) Given that the plane is smooth, find the acceleration of the system and the tension in the string, each to $3$ significant figures. (6 marks)
(b) The plane is now rough, with coefficient of friction $0.2$ between $A$ and the plane. Find the new acceleration of the system, giving your answer to $3$ significant figures. (4 marks)
Level 5 · 10 marksTry it
2
Two particles $P$ and $Q$ , of masses $3$ kg and $5$ kg respectively, are connected by a light inextensible string. Particle $P$ lies on a rough plane inclined at an angle $\alpha$ to the horizontal, where $\tan\alpha = \dfrac{3}{4}$ . The string passes over a small smooth pulley fixed at the top edge of the plane, and $Q$ hangs freely below the pulley. The part of the string from $P$ to the pulley lies along a line of greatest slope of the plane. The coefficient of friction between $P$ and the plane is $\dfrac{1}{2}$ .

Initially $P$ is $2.5$ m from the pulley and $Q$ is $1.6$ m above horizontal ground. The system is released from rest with the string taut. Take $g = 9.8$ m s⁻².
(a) Show that the acceleration of $P$ is $2.45$ m s⁻² and find the tension in the string. (6 marks)
(b) Find the time taken, from release, for $Q$ to reach the ground, and the speed of $Q$ at that instant. (3 marks)
(c) $Q$ does not rebound when it hits the ground. Find the distance $P$ moves up the plane after $Q$ hits the ground before $P$ first comes to instantaneous rest. (3 marks)
(d) Explain why $P$ does not remain at rest at this position. (1 mark)
Level 5 · 13 marksTry it
3
A package, $P$ , of mass $30$ kilograms, rests near the foot of a rough slope that makes an angle of $45^\circ$ with the horizontal.

A rope tied to $P$ runs up the slope along a line of greatest slope and passes over a smooth pulley fixed at the top of the slope. A second package, $Q$ , of mass $60$ kilograms, is tied to the other end of the rope and hangs vertically below the pulley, as shown in the diagram below.The system is released from rest. $P$ then slides up the slope and $Q$ falls vertically. Both packages have acceleration of magnitude $a\ \text{m s}^{-2}$ , and the tension in the rope is $T$ newtons.
(a) By considering the forces acting on $Q$ , explain why

$60g - T = 60a$

(1 mark)
(b) Show that $R = 15\sqrt{2}\,g$ , where $R$ newtons is the normal reaction force acting on $P$

(1 mark)
(c) It is known that $\mu$ , the coefficient of friction between $P$ and the slope, satisfies $0 \le \mu \le 1$

Show that

$a \ge \frac{\left(2 - \sqrt{2}\right)g}{3}$

(6 marks)
(d) State one assumption that has been made in modelling the motion of the packages.

(1 mark)
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4
Particle $A$ , of mass $1.5$ kg, rests on a rough plane inclined at $35°$ to the horizontal. A light inextensible string is attached to $A$ and runs up the plane, parallel to a line of greatest slope, to a small smooth pulley $P$ fixed at the top of the plane. After passing over $P$ , the string supports particle $B$ , of mass $2.5$ kg, which hangs freely $0.9$ m above horizontal ground (see diagram).The coefficient of friction between $A$ and the plane is $\mu$ . The particles are released from rest with the string taut, and $B$ strikes the ground $0.75$ seconds later. $A$ does not reach $P$ at any stage of the motion.
(a) Show that the acceleration of $B$ as it falls is $3.2$ m s $^{-2}$ .

(1 mark)
(b) Show that the tension in the string during this stage of the motion is $16.5$ N.

(3 marks)
(c) Determine the value of $\mu$ .

(5 marks)

$B$ does not rebound from the ground, and $A$ continues to slide up the plane until it comes to instantaneous rest.
(d) Determine the distance that $A$ slides up the plane after $B$ strikes the ground.

(4 marks)
Level 5 · 13 marksTry it
5
The two upper faces of a fixed wedge are rough planes which meet at a horizontal ridge. The first face is inclined at $\alpha$ to the horizontal and the second face is inclined at $\beta$ to the horizontal, where $\tan\alpha = \frac{4}{3}$ and $\tan\beta = \frac{3}{4}$ . A small smooth pulley is fixed at the ridge.

Particle $A$ , of mass $5$ kg, is on the first face and particle $B$ , of mass $2$ kg, is on the second face. The particles are attached to the ends of a light inextensible string which passes over the pulley. The string is taut, and each part of it lies along a line of greatest slope of the face on which its particle rests (see diagram).The coefficient of friction between $A$ and the first face is $4\mu$ and the coefficient of friction between $B$ and the second face is $3\mu$ , where $\mu$ is a constant.

The particles are released from rest and $A$ starts to slide down the first face. Neither particle reaches the pulley or the foot of its face during the motion considered in this question.
(a) Determine the tension in the string, giving your answer in terms of $g$ .

(7 marks)

The string breaks at the instant when $A$ is moving with speed $1.5$ m s $^{-1}$ . The speed of $A$ is $4.5$ m s $^{-1}$ at the instant $0.5$ seconds after the string breaks.
(b) Determine the deceleration of $B$ while it continues to move up the second face. Give your answer correct to 3 significant figures.

(5 marks)
Level 5 · 12 marksTry it