Further Mechanics
Lecturer : Chang Mun Wai
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The resultant force in any direction is zero (no linear motion)
❑ The resultant moment about any axis is zero (no turning effect)
❑
Uniform lamina
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• Vertical:
• Horizontal:
sum=0
0
• Moment:
sum=0 No turning effect
• Vertical:
• Horizontal:
sum=0
0
No linear motion
• Moment:
sum≠0
turning effect
No linear motion
equilibrium
NOT
equilibrium
❑
❑
No linear motion ➔ zero resultant force in any direction
No turning effect ➔ zero sum of the moment about any point
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The diagram shows a uniform beam of length 3 m and weight 40 N, suspended in
equilibrium in a horizontal position by two vertical ropes, one attached at the end of A
and the other at C, 1 m from the other end B. Find the tension in each rope.
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A uniform plant of weight W and length 6a rests on two supports at points B and C as
shown in the diagram. The plank carries a load 2W at the end A and a load 3W at the
end D. Find, in terms of W, the force exerted by each support.
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A scaffold board of weight 50 N and length 4 m lies partly on a flat roof and projects
2 m over the edge. A load of weight 30 N is carried on the overhanging end B and the
board is prevented from tipping over the edge by a force applied at the other end A. If
the weight of the scaffold board acts through a point 1 m from the end A (not uniform),
what is the value of the least force needed?
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The diagram shows a set of forces in equilibrium. BCDE is a square of side 0.5 m.
Calculate P, Q and d.
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A ladder of length 4 m and weight W N rests in equilibrium with its foot A on
horizontal ground and resting against a vertical wall at the top B. The ladder is
uniform; contact with the wall is smooth but contact with the ground is rough and the
coefficient of friction is 1/3. Find the angle between the ladder and the wall when the
ladder is on the point of slipping.
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A uniform ladder AB, resting against a smooth vertical wall and a rough horizontal
floor. The ladder has a length of 2a and mass m. The ladder makes an angle of θ with
the wall, where tan θ = 0.75.
Find the range of values for the coefficient of friction that can keep the ladder from
slipping.
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The end A of a uniform rod AB of length 2a and weight W is smoothly pivoted to a
fixed point on a wall. The end B carries a load of weight 2W. The rod is held in a
horizontal position by a light string joining the midpoint G of the rod to a point C on
the wall, vertically above A. The string is inclined at 60º to the wall.
Find, in terms of W, the tension in the string and the horizontal and vertical
components of the force exerted by the pivot on the rod.
Hence, calculate the reaction force exerted by the pivot on the rod.
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A uniform rod AB is hinged at A and is held at 30 above the horizontal by a light
inextensible string attached to B and to a point C vertically above A. The rod is in
equilibrium as shown in the diagram.
Given length AB = AC. The weight of the rod is 40 N. Find the tension in the string.
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A uniform wooden plank AB of mass 60 kg and length 4 m rests with A on rough
horizontal ground where the coefficient of friction is 0.5. The plank rests in rough
contact with the top C of a rail of height 1.5 m and is just about to slip.
Given AC is 3 m, find:
a) the normal component of the contact forces at A and at C.
b) the coefficient of friction at C.
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A uniform rod is hinged at one end A to a wall. The other end B is pulled aside by a
horizontal force until the rod is equilibrium at 60º to the wall. Find the direction of the
hinge force.
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One end A of a uniform rod AB of weight 36 N is hinged to a fixed point. The rod is
held in a horizontal position by a string connecting to a point C vertically above the
hinge. The string is inclined at 45 to the rod. Find:
a) the direction of the force exerted on the rod by the hinge
b) the tension in the string
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A framework consists of three light rods, each of length 2a, smoothly jointed to form a
triangle ABC. The framework is smoothly hinged at B to a smooth vertical wall and
carries a weight W at A. It rests in equilibrium with C resting on the wall at a point
vertically below B. Find the reaction at C and the force in each rod.
B
A
C
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T
T
A
A
C
B
W
B
W
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C
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A uniform triangular lamina is freely suspended from A. Find
the angle between the vertical and the side of AB.
A
0.3m
B
0.3m
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A uniform compound lamina is freely suspended from B and hangs in equilibrium.
Find the angle AB makes with the vertical.
B
4
4
2
A
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The uniform right-angled triangular lamina ABC shown in the diagram is freely
suspended from A.
AB = 0.6m. The angle between AB and the vertical is 40. Find the length of the side
BC.
A
B
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C
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A uniform wire AB, in the shape of a semicircular arc of radius 0.6 m, is freely
suspended from A. Find the angle between AB and the vertical.
A
B
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A letter D is made from a uniform wire. The curve part is in the shape of a
semicircular arc of radius a, is freely suspended from A. Find the angle between AB
and the vertical.
A
B
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As the centre of mass is located at the
central cross-section of the body, the
body can be treated in the same way as
a lamina.
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The diagram shows a solid uniform cone joined to a solid uniform hemisphere. The
cone has base radius r and height 3r, and the hemisphere has radius r. The two shapes
are joined by their plane faces, and AB is a diameter on that plane face.
If the density of the cone is 4 times that of the hemisphere, find the position of the
centre of mass relative to line AB.
The shape is then suspended from point A, Find the angle AB makes with the vertical.
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A solid uniform right circular cone, of height 4a and base radius a, is suspended freely
from a point P on the circumference of the base. Find the angle α between PO and the
vertical, where O is the centre of the base.
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A sculpture is in the form of a uniform solid cylinder of radius 2 cm and mass 4M,
with a small lead bead of mass M let in at a point A on the rim of the base. If the
sculpture is suspended from the point B, directly above A on the upper rim, AB is
inclined to the vertical at an angle α whose tangent is 1/3. Find the height, AB, of the
sculpture.
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An L-shaped uniform lamina is formed by joining two rectangles together, as shown in
the diagram. The shape is suspended from the point A.
Find the angle between AB and the vertical.
A
4r
B
2r
2r
3r
C
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Which object is in equilibrium? Which one will topple?
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Show this expression.
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C
B
W
A
W
W
Which object is in equilibrium? Which one will topple?
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A solid uniform hemisphere is placed on a rough slope, with its plane face against the
slope. Assume the friction force is great enough to prevent the hemisphere from
slipping.
If the slope is inclined at an angle of 65, state whether or not the hemisphere topples.
Justify your answer.
If it does not topple, what is the maximum possible angle of inclination of the slope?
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A shape is formed by joining a solid uniform cylinder to a solid uniform cone. The
cylinder has radius r and height r; the cone has base radius r and height 2r. The two
solids are joined by a place face, and the lines of symmetry of the two solids coincide.
This shape is placed on a rough slope, as shown in diagram.
If the slope is sufficiently rough to prevent sliding, find the angle at which the shape is
about to topple.
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A uniform rectangular prism of height 2h m and base 0.2 m is placed on an inclined
ramp. The ramp, which is rough enough to prevent slipping, is inclined at 40° to the
horizontal. Find the maximum value of h to prevent the cuboid toppling.
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Vertical sum = 0 (W = R)
Moment = 0
➔ equilibrium
W
R
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a
P
When the object is in equilibrium,
Horizontal : P = F ➔ sum = 0
❑ Vertical : W = R ➔ sum = 0
❑
h
W
R
F
d
Moment clockwise
= 𝑃ℎ + 𝑅𝑑
Moment anticlockwise
= 𝑊(𝑎/2)
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No linear motion
No Sliding
Take moment about O,
O
𝑃ℎ + 𝑅𝑑 = 𝑊
➔𝑃=
𝑎/2−𝑑
ℎ
𝑎
2
𝑊 (object is in equilibrium)
How does the dimension of the object affect P?
▪ P is inversely proportional to h
▪ P is proportional to a
❑ How would R react when P increases?
❑
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a
P
h
Assuming friction is large enough to prevent sliding,
❑ Take moment about O,
𝑎/2 − 𝑑
𝑃=
𝑊
ℎ
W
R
F
d
Moment clockwise
= 𝑃ℎ + 𝑅𝑑
Moment anticlockwise
= 𝑊(𝑎/2)
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O
For an object, a, h and W are constant.
❑ When P = 0, d = 0.5a.
❑ When P increases, d decreases.
❑
When 𝑃 =
toppling.
❑
When 𝑃 >
𝑎𝑊
,
2ℎ
𝑎𝑊
2ℎ
d = 0, the object is on the point of
, the object topples.
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a
P
h
❑
When P = 0, the object rests in equilibrium.
❑
When P increases, F increases until it reaches limiting
friction (R). Object is still in equilibrium.
W
❑
R
F
d
O
❑
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When P = R, object is at the point of sliding (limiting
equilibrium).
When P > R, object moves.
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a
P
h
P increases slowly, which will occur first?
▪ Toppling ?
▪ Sliding?
W
R
F
d
𝑎
❑
Object topples if 𝑃𝑇 > 2ℎ 𝑊
❑
Object slides if 𝑃𝑆 > 𝜇𝑅 , R = W, 𝑃𝑆 > 𝜇𝑊
O
𝑎
If 2ℎ > 𝜇, sliding will occur first.
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A uniform right-angled triangular prism, of mass m, is resting on a rough horizontal
surface. The triangle has sides a, 2a and 5a and the depth of the prism is a. A force X
is applied to the top edge, as shown.
Determine in each case whether or not the prism breaks equilibrium.
If it does, determine if it slides or topples.
a) X = 1/5 mg ;  = 0.75
b) X = 1/4 mg ;  = 0.2
c) X = 1/3 mg ;  = 0.5
2a
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A uniform solid cylinder, with radius 0.2 m and height 0.6 m, is resting in equilibrium
with one end on a rough plane inclined at an angle 𝛼 to the horizontal. The inclination
of the plane is gradually increased until the cylinder is just on the point of toppling.
a) Find the greatest possible value of 𝛼.
b) Find the least value of the coefficient of friction between the plane and the cylinder.
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A uniform solid hemisphere of weight 4W rests with its curved surface in contact both
with the rough horizontal ground and with a rough vertical wall. Its place surface is
upper most and is inclined at an angle 𝜃 to the horizontal. A particle of weight W is
attached to the end of the radius perpendicular to the plane base. If the hemisphere is
on the point of slipping, prove that
2𝜇(1 + 𝜇)
sin 𝜃 =
1 + 𝜇2
Where 𝜇 is the coefficient of friction both at the ground and at the wall.
Show that the position of limiting equilibrium is only possible if the value of 𝜇 is less
than a certain number, and find the number.
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The diagram shows a uniform square lamina ABCD of side 2a and weight W which
rests in limiting equilibrium in a vertical plane with the vertex A in contact with rough
horizontal ground and the vertex D in contact with a rough vertical wall. The
coefficients of friction at A and D are 𝜇 and 𝜇′ respectively. Show that
1 − 𝜇𝜇′
tan 𝛼 =
1 + 𝜇𝜇′ + 2𝜇
where 𝛼 is the angle between AD and the horizontal.
C
B
D
𝛼
A
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9231/2002/W/2/Q3
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9231/2010/W/2/Q4
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