A-level · Moments

Equilibrium of rigid bodiesExam-style questions

A rigid body in equilibrium needs both zero resultant force and zero resultant moment — using these two conditions to find the reaction forces on a supported beam, and taking moments about a support to eliminate an unknown.

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

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Figure 1 (not accurately drawn)

A footbridge over a stream is made from a plank $AB$ of length $3$ m and mass $20$ kg. The plank is horizontal and is supported at the points $C$ and $D$ , where $AC = 0.3$ m and $DB = 0.6$ m, as shown in Figure 1.

A man of mass $60$ kg stands on the plank at the point $P$ , where $AP = x$ metres and $0.3 < x < 2.4$

The plank is modelled as a uniform rod and the man is modelled as a particle. The plank is in equilibrium.

Using the model,
(a) show that the magnitude of the reaction on the plank at $D$ is $(280x + 28)$ N.

(3 marks)

The man moves to a position on the plank where the magnitude of the reaction at $D$ is three times the magnitude of the reaction at $C$ .

Using the model,
(b) find the value of $x$

(4 marks)

The man then walks off the bridge. A child of mass $M$ kg stands on the plank at $B$ , and the plank is now on the point of tilting about $D$ . The child is modelled as a particle.

Using the model,
(c) find the value of $M$

(3 marks)
Level 4 · 10 marksTry it
2
Figure 1 (not accurately drawn)

A shelf $AB$ has length $1.5$ m and mass $m$ kg. The shelf hangs from two light vertical chains, which are attached to it at the points $C$ and $D$ , where $AC = 0.2$ m and $DB = 0.35$ m, as shown in Figure 1.

A plant pot of mass $pm$ kg is placed on the shelf at $B$ . The shelf is in equilibrium in a horizontal position, and the tension in the chain at $D$ is twice the tension in the chain at $C$ .

The shelf is modelled as a uniform rod and the plant pot is modelled as a particle.
(a) Using the model, find the exact value of $p$ .

(7 marks)

The plant pot is replaced by a heavier pot, of mass $qm$ kg, which is also placed at $B$ . The shelf stays horizontal, and it is now on the point of tilting.
(b) Using the model, find the exact value of $q$ .

(4 marks)
(c) State how you have used the modelling assumption that the shelf is uniform.

(1 mark)
Level 4 · 12 marksTry it
3
A uniform rod $AB$ of length $2$ m and weight $40$ N is freely hinged to a wall at $A$ and held horizontal by a vertical string attached at $B$ .
(a) Find the tension in the string.
(b) A weight of $20$ N is now hung from the midpoint of the rod. Find the new tension in the string.
Level 4 · 4 marksTry it
4
The face of a retaining wall is a smooth plane sloping at $65°$ to the horizontal. A uniform plank $AB$ , of mass $12$ kg and length $4$ m, leans against the wall, with its end $A$ on rough horizontal ground and its other end $B$ touching the face of the wall. The angle between the plank and the face of the wall is $30°$ , and the plank lies in a vertical plane at right angles to the face of the wall (see diagram).(a) Show that the force exerted on the plank by the wall has magnitude $55.6$ N, correct to 3 significant figures.

(3 marks)
(b) Determine the magnitude of the contact force between the plank and the ground. Give your answer correct to 3 significant figures.

(5 marks)
Level 4 · 8 marksTry it
5
A cupboard is tilted back so that it leans against a smooth vertical wall. In a model, the cupboard is a uniform rectangular lamina $ABCD$ , of mass $30$ kg, with $AB = 0.5$ m and $AD = 1.8$ m. The corner $A$ is on rough horizontal ground, the corner $D$ is in contact with the wall, and the lamina lies in a vertical plane perpendicular to the wall. The edge $AD$ makes an angle of $35°$ with the wall (see diagram). The cupboard is on the point of slipping.(a) By taking moments, show that the magnitude of the normal contact force between the cupboard and the wall is $62.1$ N, correct to 3 significant figures.

(4 marks)
(b) Determine the coefficient of friction between the cupboard and the ground.

(3 marks)
Level 4 · 7 marksTry it

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