Mechanics 2: Statics
1. [WME02/01, session 1401, question 5]
A
2a
2 C
a
3
4
a
3
B
Figure 1
Figure 1 shows a uniform rod AB, of mass m and length 2a, with the end B resting on
rough horizontal ground. The rod is held in equilibrium at an angle θ to the vertical
by a light inextensible string. One end of the string is attached to the rod at the point
C, where AC = 23 a. The other end of the string is attached to the point D, which is
vertically above B, where BD = 2a.
(a) By taking moments about D, show that the magnitude of the frictional force acting
on the rod at B is 12 mg sin θ
(3)
(b) Find the magnitude of the normal reaction on the rod at B.
(5)
The rod is in limiting equilibrium when tan θ = 4
3
(c) Find the coefficient of friction between the rod and the ground.
(3)
(Total 11 marks)
1
2. [WME02/01, session 1406, question 3]
60°
A
Figure 2
A uniform rod AB of weight W is freely hinged at end A to a vertical wall. The rod is
supported in equilibrium at an angle of 60◦ to the wall by a light rigid strut CD. The strut
is freely hinged to the rod at the point D and to the wall at the point C, which is vertically
below A, as shown in Figure 2. The rod and the strut lie in the same vertical plane, which
is perpendicular to the wall. The length of the rod is 4a and AC = AD = 2.5a.
√
(a) Show that the magnitude of the thrust in the strut is 4 3
5
W
(3)
(b) Find the magnitude of the force acting on the rod at A.
(6)
(Total 9 marks)
2
3. [WME02/01, session 1501, question 5]
F B
C
5
a
4
A
Figure 3
A uniform rod AB, of mass m and length 2a, is freely hinged to a fixed point A. A
particle of mass km is fixed to the rod at B. The rod is held in equilibrium, at an angle
θ to the horizontal, by a force of magnitude F acting at the point C on the rod, where
AC = 54 a, as shown in Figure 3. The line of action of the force at C is at right angles to
AB and in the vertical plane containing AB.
Given that tan θ = 3
4
(a) show that F = 16
25
mg(1 + 2k),
(4)
(b) find, in terms of m, g and k,
(i) the horizontal component of the force exerted by the hinge on the rod at A,
(ii) the vertical component of the force exerted by the hinge on the rod at A.
(5)
Given also that the force acting on the rod at A acts at 45 above the horizontal,
◦
(c) find the value of k.
(3)
(Total 12 marks)
3
4. [WME02/01, session 1506, question 6]
4a C
3a
A
Figure 4
A uniform rod AB has length 4a and weight W . A particle of weight kW , k < 1, is
attached to the rod at B. The rod rests in equilibrium against a fixed smooth horizontal
peg. The end A of the rod is on rough horizontal ground, as shown in Figure 4. The rod
rests on the peg at C, where AC = 3a, and makes an angle α with the ground, where
tan α = 13 . The peg is perpendicular to the vertical plane containing AB.
(a) Give a reason why the force acting on the rod at C is perpendicular to the rod.
(1)
(b) Show that the magnitude of the force acting on the rod at C is
√
10
W (1 + 2k)
5
(4)
The coefficient of friction between the rod and the ground is 3
4
.
(c) Show that for the rod to remain in equilibrium k ≤ 2
11
.
(7)
(Total 12 marks)
4
5. [WME02/01, session 1601, question 6]
C
b
2a
Figure 5
A uniform rod AB, of mass 3m and length 2a, is freely hinged at A to a fixed point
on horizontal ground. A particle of mass m is attached to the rod at the end B. The
system is held in equilibrium by a force F acting at the point C, where AC = b. The rod
makes an acute angle θ with the ground, as shown in Figure 5. The line of action of F is
perpendicular to the rod and in the same vertical plane as the rod.
(a) Show that the magnitude of F is 5mga
b
cos θ
(4)
The force exerted on the rod by the hinge at A is R, which acts upwards at an angle φ
above the horizontal, where φ > θ.
(b) Find
(i) the component of R parallel to the rod, in terms of m, g and θ,
(ii) the component of R perpendicular to the rod, in terms of a, b, m, g and θ.
(5)
(c) Hence, or otherwise, find the range of possible values of b, giving your answer in terms
of a.
(2)
(Total 11 marks)
5
6. [WME02/01, session 1606, question 5]
70°
2m
30°
Figure 6
A uniform rod AB has mass 6 kg and length 2 m. The end A of the rod rests against a
rough vertical wall. One end of a light string is attached to the rod at B. The other end
of the string is attached to the wall at C, which is vertically above A. The angle between
the rod and the string is 30◦ and the angle between the rod and the wall is 70◦ , as shown
in Figure 6. The rod is in a vertical plane perpendicular to the wall and rests in limiting
equilibrium.
Find
(a) the tension in the string,
(4)
(b) the coefficient of friction between the rod and the wall,
(5)
(c) the direction of the force exerted on the rod by the wall at A.
(2)
(Total 11 marks)
6
7. [WME02/01, session 1701, question 7]
30°
A
Figure 7
A uniform rod AB has mass m and length 2a. The end A is in contact with rough
horizontal ground and the end B is in contact with a smooth vertical wall. The rod rests
in equilibrium in a vertical plane perpendicular to the wall and makes an angle of 30◦
with the wall, as shown in Figure 7. The coefficient of friction between the rod and the
ground is µ.
(a) Find, in terms of m and g, the magnitude of the force exerted on the rod by the wall.
(4)
√
(b) (b) Show that µ ≥ 6
3
.
(3)
√
A particle of mass km is now attached to the rod at B. Given that µ = 5
3
and that the
rod is now in limiting equilibrium,
(c) find the value of k.
(6)
(Total 13 marks)
7
8. [WME02/01, session 1706, question 4]
3m
0.5 m
A C B
4m
Figure 8
A uniform rod AB has mass 5 kg and length 4 m. The rod is held in a horizontal position
by a light inextensible string. The end A of the rod rests against a rough vertical wall.
One end of the string is attached to the rod at B and the other end is attached to the
wall at a point D. The point D is vertically above A, with AD = 3 m. A particle of
mass 2 kg is attached to the rod at C, where AC = 0.5 m, as shown in Figure 8. The rod
is in equilibrium in a vertical plane perpendicular to the wall. The coefficient of friction
between the rod and the wall is µ.
Find
(a) the tension in the string,
(4)
(b) the magnitude of the force exerted by the wall on the rod at A,
(5)
(c) the range of possible values of µ.
(2)
(Total 11 marks)
8
9. [WME02/01, session 1801, question 5]
16b
5b
A
13b
Figure 9
A uniform rod, of weight W and length 16b, has one end freely hinged to a fixed point A.
The rod rests against a smooth circular cylinder, of radius 5b, fixed with its axis horizontal
and at the same horizontal level as A. The distance of A from the axis of the cylinder is
13b, as shown in Figure 9. The rod rests in a vertical plane which is perpendicular to the
axis of the cylinder.
(a) Find, in terms of W , the magnitude of the reaction on the rod at its point of contact
with the cylinder.
(4)
(b) Show that the resultant force acting on the rod at A is inclined to the vertical at an
angle α where tan α = 40
73
(6)
(Total 10 marks)
9
10. [WME02/01, session 1806, question 2]
TN
40°
B
A 30°
Figure 10
A uniform rod AB, of mass 6 kg and length 1.6 m, rests with its end A on rough horizontal
ground. The rod is held in equilibrium at 30◦ to the horizontal by a light string attached
to the rod at B. The string is at 40◦ to the horizontal and lies in the same vertical plane
as the rod, as shown in Figure 10. The tension in the string is T newtons. The coefficient
of friction between the ground and the rod is µ.
(a) Show that, to 3 significant figures, T = 27.1
(4)
(b) Find the set of values of µ for which equilibrium is possible.
(5)
(Total 9 marks)
10
