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 1Diagram not accurately drawn.

A rough plane, inclined to the horizontal at an angle $\alpha$ where $\tan\alpha = \frac{3}{4}$ , is fixed in position. A small smooth pulley is fixed at the highest point of the plane.

Particles $P$ and $Q$ , of masses $2.5$ kg and $1$ kg respectively, are joined by a light inextensible string which passes over the pulley. Particle $P$ rests on the plane and $Q$ hangs freely below the pulley, as shown in Figure 1. The section of the string between $P$ and the pulley is parallel to a line of greatest slope of the plane.

The coefficient of friction between $P$ and the plane is $\mu$ .

The string is taut and $P$ is on the point of sliding down the plane.
(a) Find the value of $\mu$ .

(8 marks)

The string is then cut and $P$ slides down the plane from rest.
(b) Find the time taken for $P$ to slide $3.2$ m.

(4 marks)
Level 4 · 12 marksTry it
2
A particle $A$ of mass $2$ kg lies on a rough plane inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha = \dfrac35$ . The particle is attached to one end of a light inextensible string. The string lies along a line of greatest slope of the plane and passes over a small smooth pulley $P$ fixed at the top of the plane. The other end of the string is attached to a particle $B$ of mass $3$ kg, which hangs freely with the string vertical. $B$ is $1$ m above horizontal ground, and the plane is long enough that $A$ does not reach $P$ in the motion that follows.

The coefficient of friction between $A$ and the plane is $\dfrac12$ . The system is released from rest with the string taut. Take $g = 9.8$ m s⁻².
(a) Show that the acceleration of $A$ is $1.96$ m s⁻² and find the tension in the string. (5 marks)

When $B$ reaches the ground it does not rebound.
(b) Find the further distance that $A$ travels up the plane after $B$ reaches the ground, before $A$ first comes to instantaneous rest. (3 marks)
(c) Determine whether $A$ remains at rest at this point. (2 marks)
Level 4 · 10 marksTry it
3
A smooth plane slopes downwards at $45°$ to the horizontal from one edge of a rough horizontal table, and a small smooth pulley is fixed at this edge. A small block $B$ , of mass $0.8$ kg, rests on the table, and a particle $P$ , of mass $1.2$ kg, rests on the plane. A light inextensible string joins $B$ to $P$ and passes over the pulley. The string is taut; the part between $B$ and the pulley is horizontal and the part between $P$ and the pulley lies along a line of greatest slope of the plane (see diagram).The coefficient of friction between $B$ and the table is $\mu$ .

The system is released from rest. $P$ reaches a speed of $1.5$ m s $^{-1}$ after travelling $0.45$ m down the plane, and at this instant $B$ is still some distance from the pulley.
(a) Find the tension in the string.

(4 marks)
(b) Determine the value of $\mu$ .

(5 marks)
Level 4 · 9 marksTry it