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
A uniform rod $AB$ of length $3$ m and weight $60$ N is freely hinged at $A$ and held in equilibrium by a string attached at $B$ . The string makes an angle of $30^\circ$ with the rod, which is horizontal. Find the tension in the string.
Level 5 · 2 marksTry it
2
Figure 2 shows a uniform rod $AB$ of mass $8$ kg and length $3$ m, freely hinged to a vertical wall at $A$ . The rod is held horizontal by a light inextensible string joining $B$ to a point $C$ on the wall vertically above $A$ , where angle $ABC = 40^\circ$ .Figure 2 (not accurately drawn)
(a) Find the tension in the string, giving your answer to $3$ significant figures. (5 marks)
(b) Find the magnitude and direction of the force exerted by the hinge on the rod. (6 marks)
Level 5 · 11 marksTry it
3
A uniform ladder $AB$ , of length $8$ m and mass $12$ kg, rests with $A$ on rough horizontal ground and $B$ against a smooth vertical wall. The ladder makes an angle of $60^\circ$ with the ground. A decorator of mass $60$ kg stands at the point $C$ on the ladder, where $AC = 6$ m. The ladder is on the point of slipping. Take $g = 9.8$ m s⁻².(a) Explain what the modelling assumptions "smooth wall" and "uniform ladder" allow you to say about the forces acting. (2 marks)
(b) Find the magnitude of the normal reaction of the ground at $A$ . (2 marks)
(c) By taking moments about $A$ , find the magnitude of the normal reaction of the wall at $B$ , giving your answer to 3 significant figures. (6 marks)
(d) Find the coefficient of friction between the ladder and the ground, giving your answer in exact form and to 3 significant figures. (4 marks)
(e) The decorator claims he could safely stand at the very top of the ladder. Comment on this claim, using your model. (1 mark)
Level 5 · 15 marksTry it
4
A uniform ladder $AB$ , of length $5$ m and mass $12$ kg, rests in equilibrium with the end $A$ on rough horizontal ground and the end $B$ against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall, and $A$ is $3$ m from the foot of the wall. The coefficient of friction between the ladder and the ground is $0.45$ .

A decorator of mass $60$ kg stands on the ladder at the point $C$ , where $AC = x$ metres. The ladder is modelled as a uniform rod and the decorator as a particle. Take $g = 9.8$ m s⁻².
(a) Show that the magnitude of the force exerted by the wall on the ladder is $(44.1 + 88.2x)$ N. (4 marks)
(b) Find the greatest value of $x$ for which the ladder does not slip. (4 marks)
(c) The decorator needs to stand at $B$ . To make this possible, a second person stands on the ladder at $A$ . Find the least mass of the second person for which the ladder remains in equilibrium with the decorator at $B$ . (5 marks)
(d) In reality the wall is not perfectly smooth. State, with a reason, how this would affect your answer to part (b). (2 marks)
Level 5 · 15 marksTry it
5
A builder uses a uniform plank $AB$ , of length $2a$ and mass $m$ , as a ramp up to a skip. The end $A$ of the plank is on horizontal ground, and the plank rests on the top edge of one side of the skip at the point $C$ . This edge is horizontal, at a height $h$ above the ground, and is perpendicular to the vertical plane containing the plank. The plank is inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha = 0.6$ . A man of mass $6m$ stands on the plank at the point $D$ , where $AD = d$ (see diagram).In the model used, the man is treated as a particle, the ground as rough and the edge of the skip as smooth.
(a) Show that the normal contact force between the plank and the skip is $\dfrac{12mg(a + 6d)}{25h}$ .

(4 marks)

The plank is on the point of slipping, and the coefficient of friction between the plank and the ground is $0.3$ .
(b) Show that $h = k(a + 6d)$ , where $k$ is a constant to be determined.

(7 marks)
(c) Hence find, in terms of $a$ , the greatest possible value of $d$ .

(2 marks)
(d) State one improvement that could be made to the model.

(1 mark)
Level 5 · 14 marksTry it

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