A-level · Forces & Newton's laws

Connected particlesExam-style questions

Why two bodies joined by a light inextensible string share one acceleration and one tension, the two-step method that runs every problem — whole system for the acceleration, a single body for the internal force — and why the connecting force is internal to the system but external to each body.

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13 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 car of mass $1200$ kg has broken down and is being towed up a straight road by a truck of mass $2800$ kg, as shown in Figure 1. The road is inclined to the horizontal at an angle $\alpha$ , where $\sin\alpha = \frac{1}{40}$

The car is attached to the truck by a rigid tow pole. The tow pole is parallel to the road and is modelled as a light rod. The truck and the car are modelled as particles.

The engine of the truck produces a constant driving force of $3000$ N, and the truck and the car move up the road with acceleration $0.25\ \text{m s}^{-2}$

The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude $R$ newtons. The resistance to the motion of the truck from non-gravitational forces is modelled as a constant force of magnitude $2R$ newtons.

Using the model,
(a) show that $R = 340$

(4 marks)
(b) find the tension in the tow pole.

(3 marks)

At the instant when the truck and the car are moving up the road with speed $10\ \text{m s}^{-1}$ , the tow pole becomes detached from the truck. The resistance to the motion of the car is unchanged.
(c) Using the model, find the distance that the car moves up the road from the instant the tow pole becomes detached to the instant the car first comes to rest.

(4 marks)
Level 4 · 11 marksTry it
2
A polar explorer hauls two sledges, $P$ and $Q$ , across a horizontal sheet of ice. The sledges are joined by a light rigid pole, and $Q$ travels directly behind $P$ .

Sledge $P$ has a mass of $60$ kilograms and sledge $Q$ has a mass of $40$ kilograms.

The explorer pulls sledge $P$ with a rope attached to its front. The rope makes an angle of $22^\circ$ with the horizontal and has a constant tension of $110$ newtons.

From rest, the sledges accelerate at $0.3\ \text{m s}^{-2}$ in a straight line, as shown in the diagram below.A resistance force of $40$ N acts on sledge $P$ while it is moving.
(a) Show that a resistance force of approximately $32$ N acts on sledge $Q$

(4 marks)
(b) When the sledges are moving at $1.2\ \text{m s}^{-1}$ , the explorer unclips the rope from her harness. The sledges then slow down at a constant rate until they stop.

The resistance forces on the sledges do not change.

Find the distance that the sledges travel after the rope is unclipped.

(3 marks)
(c) State one modelling assumption about the pole that your answer to part (a) depends on.

(1 mark)
Level 4 · 8 marksTry it
3
A block is fixed to horizontal ground. The upper face of the block is horizontal and rough. One side face of the block is a smooth plane inclined at $\alpha$ to the horizontal, where $\tan\alpha = \frac{3}{4}$ , and the opposite side face is vertical. A small smooth pulley is fixed at each end of the upper face.

Particles $P$ , $Q$ and $S$ have masses $5$ kg, $4$ kg and $1$ kg respectively. $Q$ lies on the upper face, $P$ lies on the inclined face and $S$ hangs freely beside the vertical face. $Q$ is joined to $P$ by one light inextensible string and to $S$ by another, and each string passes over one of the pulleys. Both strings are taut, and the part of the string between $P$ and its pulley is parallel to a line of greatest slope of the inclined face (see diagram).The coefficient of friction between $Q$ and the upper face is $\mu$ . The particles are released from rest. In the motion that follows, $P$ slides down the inclined face and each particle has an acceleration of magnitude $a$ m s $^{-2}$ .
(a) Find an equation connecting $\mu$ , $a$ and $g$ , and hence show that $a \le \frac{1}{5}g$ .

(6 marks)

It is now given that $a = \frac{1}{10}g$ .
(b) Determine the magnitude of the contact force exerted on $Q$ by the upper face of the block. Give your answer correct to 3 significant figures.

(3 marks)
Level 4 · 9 marksTry it
4
A plank $D$ lies at rest on a rough horizontal table. A small box $P$ , of mass $2$ kg, rests on top of the plank. A light inextensible string is fixed to $P$ and runs horizontally from $P$ , parallel to the plank, to a small smooth pulley which is fixed to the table at its edge, beyond one end of the plank. The string passes over the pulley, and a particle $Q$ , of mass $1.5$ kg, hangs freely from its other end. The pulley is not attached to $D$ (see diagram).The box is modelled as a particle. $P$ and $Q$ are released from rest with the string taut, and $P$ slides along the plank while the plank itself remains at rest. At the instant $0.5$ seconds after release, $Q$ is moving with speed $1.12$ m s $^{-1}$ .
(a) Find the acceleration of $Q$ during this motion.

(2 marks)
(b) Determine the coefficient of friction between the box and the plank.

(3 marks)

The coefficient of friction between the plank and the table is $0.1$ .
(c) Determine the least possible mass of the plank.

(3 marks)
Level 4 · 8 marksTry it