A-level · Forces & Newton's laws

PulleysExam-style questions

Why a light inextensible string over a smooth pulley gives the two masses the same tension and the same size of acceleration in opposite directions, the two-equations-add method for finding a and T, and the classic twist — when one mass lands the string goes slack and the other flies on freely under gravity.

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10 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 1Diagram not accurately drawn.

A rough plane is fixed with its surface inclined to the horizontal at an angle $\alpha$ , where $\tan\alpha = \frac{4}{3}$ . A small smooth pulley is fixed at the top of the plane.

Particle $A$ , of mass $3m$ , is held at rest on the plane. It is connected to particle $B$ , of mass $m$ , by a light inextensible string which passes over the pulley, and $B$ hangs freely below the pulley, as shown in Figure 1. The string is taut and the section of string from $A$ to the pulley is parallel to a line of greatest slope of the plane. The whole system lies in one vertical plane.

The coefficient of friction between $A$ and the plane is $\frac{1}{3}$

Particle $A$ is released. It slides down the plane and $B$ moves upwards.
(a) Show that, as $A$ slides down the plane, the frictional force acting on $A$ has magnitude $\frac{3}{5}mg$

(3 marks)
(b) Write down an equation of motion for $B$ .

(2 marks)
(c) Show that the acceleration of $A$ is $\frac{1}{5}g$

(4 marks)

When $A$ has moved a distance $d$ down the plane, the string breaks. In the subsequent motion, $B$ does not reach the pulley.
(d) Find, in terms of $d$ , the greatest height of $B$ above its initial position.

(6 marks)
Level 5 · 15 marksTry it
2
Figure 1Diagram not accurately drawn.

Particle $A$ has mass $3M$ and particle $B$ has mass $kM$ , where $k$ is a constant and $k > 3$ . The particles hang on either side of a small smooth fixed pulley, attached to the ends of a light inextensible string which passes over the pulley.

Particle $B$ is $1.8$ m above horizontal ground and the string is taut, as shown in Figure 1. When the particles are released from rest, $B$ descends with acceleration $\frac{1}{4}g$
(a) Write down an equation of motion for $B$ .

(2 marks)
(b) Find the value of $k$ .

(3 marks)

Given that $M = 0.6$ kg,
(c) find the magnitude of the force exerted on the pulley by the string before $B$ hits the ground.

(3 marks)

When $B$ hits the ground, it receives an impulse of magnitude $10.5$ N s from the ground and rebounds vertically upwards.
(d) Find the speed with which $B$ rebounds from the ground.

(5 marks)

In the subsequent motion, $A$ does not reach the pulley.
(e) Find the time that elapses between the release of the particles and the instant at which $A$ first comes to rest.

(4 marks)
Level 5 · 17 marksTry it
3
Figure 1Diagram not accurately drawn.

A particle $A$ of mass $4m$ is held at rest on a rough horizontal table. One end of a light inextensible string is attached to $A$ . The string passes over a small smooth pulley which is fixed at the edge of the table, and the other end of the string is attached to a particle $B$ of mass $3m$ . Particle $B$ hangs vertically below the pulley at a height $h$ above the horizontal floor, as shown in Figure 1.

The string between $A$ and the pulley is horizontal and perpendicular to the edge of the table. The particles, the string and the pulley all lie in the same vertical plane.

The coefficient of friction between $A$ and the table is $\mu$ .

Particle $A$ is released from rest with the string taut. Before $B$ reaches the floor, the tension in the string is $\frac{15}{7}mg$
(a) Write down an equation of motion for $B$ .

(2 marks)
(b) Find the value of $\mu$ .

(4 marks)
(c) Find, in terms of $m$ and $g$ , the magnitude of the force exerted on the pulley by the string before $B$ reaches the floor.

(3 marks)

Particle $B$ hits the floor and does not rebound. In the subsequent motion, $A$ does not reach the pulley.
(d) Find, in terms of $h$ and $g$ , the time from the instant $A$ is released to the instant $A$ comes to rest.

(5 marks)
Level 5 · 14 marksTry it