A-level · Forces & Newton's laws

Resolving forcesExam-style questions

Splitting a force at an angle into two perpendicular components, F cosθ and F sinθ — to add forces in any directions and to solve equilibrium by resolving in two perpendicular directions.

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

DifficultyAll Level 2 Level 3 Level 4

1
[In this question $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors.]

A particle $Q$ of mass $4$ kg is at rest at the point $O$ on a smooth horizontal plane.

From time $t = 0$ , two constant horizontal forces act on $Q$ . These forces are $(p\mathbf{i} - 2\mathbf{j})$ N and $(3\mathbf{i} + q\mathbf{j})$ N, where $p$ and $q$ are constants, and they are the only horizontal forces acting on $Q$ .

The particle $Q$ moves away from $O$ along a straight line, in the direction of the vector $(24\mathbf{i} + 7\mathbf{j})$ .
(a) Show that $7p - 24q + 69 = 0$

(4 marks)

Given also that $q = 9$
(b) find the speed of $Q$ at the instant when $t = 2$ seconds.

(5 marks)
Level 4 · 9 marksTry it
2
Figure 1 (not accurately drawn)

Two horizontal forces, $\mathbf{P}$ $P$ and $\mathbf{Q}$ $Q$ , act on a particle at the point $O$ $O$ .
- The force $\mathbf{P}$ has magnitude $X$ newtons and acts due north
- The force $\mathbf{Q}$ has magnitude $40$ N and acts on a bearing of $060°$
The forces are shown in Figure 1. The magnitude of the resultant of $\mathbf{P}$ $P$ and $\mathbf{Q}$ $Q$ is $65$ $65$ N.

Find

(i) the value of $X$ $X$

(ii) the angle between the direction of $\mathbf{P}$ $P$ and the direction of the resultant, giving your answer to the nearest degree.

(8 marks)
Level 4 · 8 marksTry it
3
A spotlight, of mass $m$ kilograms, hangs below a horizontal lighting bar in a theatre.

The spotlight is held by two light, inextensible cables, $AP$ and $BP$ , which are fixed to the bar at $A$ and $B$

The spotlight is at $P$ , which is $1.2$ metres below the bar. $P$ is $0.5$ metres horizontally from $A$ and $0.9$ metres horizontally from $B$ , as shown in the diagram below.The spotlight hangs in equilibrium and is modelled as a particle.
(a) Show that the tension in cable $AP$ is more than one and a half times the tension in cable $BP$

(4 marks)
(b) It is given that the tension in cable $BP$ is $5g$ newtons.

Find the value of $m$

(4 marks)
Level 4 · 8 marksTry it
4
A block of mass $m$ kg rests on smooth horizontal ground. Two light inextensible strings are fixed to the top of the block. The first string passes over a small smooth pulley $P$ and supports a particle of mass $4$ kg, which hangs vertically below $P$ . The second string passes over a small smooth pulley $Q$ , on the other side of the block, and supports a particle of mass $M$ kg, which hangs vertically below $Q$ . The section of the first string between the block and $P$ is inclined at $25°$ to the horizontal, and the section of the second string between the block and $Q$ is inclined at $60°$ to the horizontal. The block, the pulleys and the strings all lie in one vertical plane (see diagram).

The system is in equilibrium.
(a) Determine the value of $M$ .

(3 marks)

The magnitude of the force exerted on the block by the ground is $120$ N.
(b) Determine the value of $m$ .

(3 marks)
Level 4 · 6 marksTry it
5
A small ring is held at rest at a point $O$ on a smooth horizontal table by three light horizontal strings. One string pulls on the ring with a force of $14$ N, directed on a bearing of $035°$ ; a second string pulls on it with a force of $9$ N, directed on a bearing of $110°$ (see diagram).(a) Determine the magnitude and bearing of the resultant of these two forces.

(6 marks)

The third string pulls on the ring with a force of magnitude $F$ N.
(b) Write down the value of $F$ , and the bearing on which the third string pulls the ring.

(2 marks)
Level 4 · 8 marksTry it