Browse Moments

86 questions at your level

DifficultyAll Level 2 Level 3 Level 4 Level 5

Moments of forces

59 questions

LessonNot started

AB

, of length

5

m and weight

336

N, rests with its end

A

on rough horizontal ground and its end

B

against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle

\theta

with the ground, where

\tan\theta = \frac{24}{7}

, as shown in Figure 1.

By taking moments about

A

, find the magnitude of the normal reaction of the wall on the ladder.

(3 marks)

●●●●●Level 23 marksStart

Figure 1

A uniform rod

AB

, of length

3

m and weight

80

N, rests with its end

A

on rough horizontal ground and against a small smooth horizontal peg at

C

, where

AC = 2.4

m. The rod lies in a vertical plane perpendicular to the peg and is inclined at

25^\circ

to the horizontal, as shown in Figure 1.

Find, to 3 significant figures, the magnitude of the force exerted on the rod by the peg.

(3 marks)

●●●●●Level 23 marksStart

A force of

25

N acts at a perpendicular distance of

0.4

m from a pivot. Find its moment about the pivot.

●●●●●Level 22 marksStart

A light rod is pivoted at its centre. A

6

N force acts downward

2

m to the left of the pivot, and an

8

N force acts downward

1.5

m to the right. Find the resultant moment about the pivot.

●●●●●Level 22 marksStart

A force of

15

N acts at a perpendicular distance of

3

m from a pivot, turning it clockwise. State the moment of the force.

●●●●●Level 22 marksStart

A light rod is pivoted at a point

O

. A force of

8

N acts vertically downward at a point

1.5

m to the left of

O

, and a force

F

acts vertically downward

2

m to the right of

O

. The rod is in equilibrium.

Find

F

. (3 marks)

●●●●●Level 23 marksStart

Figure 1

A uniform ladder

AB

, of length

5

m and mass

20

kg, rests with its end

A

on rough horizontal ground and its end

B

against a smooth vertical wall. The ladder is inclined at

65^\circ

to the ground and lies in a vertical plane perpendicular to the wall. A woman of mass

60

kg stands on the ladder at the point

C

, where

AC = 3.5

m, as shown in Figure 1.

The ladder is modelled as a uniform rod and the woman as a particle.

Given that the ladder rests in limiting equilibrium, find the coefficient of friction between the ladder and the ground.

(6 marks)

●●●●●Level 36 marksStart

Figure 1

A uniform rod

AB

, of length

3

m and weight

90

N, rests with its end

A

on rough horizontal ground and its end

B

against a smooth vertical wall. The rod lies in a vertical plane perpendicular to the wall and is inclined to the wall at an angle of

20^\circ

, as shown in Figure 1. The rod rests in equilibrium.
(a) Find the magnitude of the normal reaction of the wall on the rod.

(3 marks)

The coefficient of friction between the rod and the ground is

\mu

.
(b) Find the smallest possible value of

\mu

.

(2 marks)

●●●●●Level 35 marksStart

Figure 1

A uniform rod

AB

, of length

2.4

m and weight

50

N, has its end

A

freely hinged to a fixed point on horizontal ground. The rod is held in equilibrium at an angle

\theta

to the ground, where

\tan\theta = \frac{5}{2}

, by a light inextensible string. One end of the string is attached to the rod at

B

and the other end is attached to the point

D

on a vertical wall. The string is horizontal, and the rod and the string lie in a vertical plane perpendicular to the wall, as shown in Figure 1.
(a) Find the tension in the string.

(3 marks)
(b) Find the magnitude of the force exerted on the rod by the hinge.

(2 marks)
(c) Find the angle between this force and the ground.

(2 marks)

●●●●●Level 37 marksStart

Figure 1

A uniform ladder

AB

, of length

4

m and mass

15

kg, rests with its end

A

on rough horizontal ground and its end

B

against a smooth vertical wall. The ladder is inclined at

60^\circ

to the ground and lies in a vertical plane perpendicular to the wall, as shown in Figure 1. The coefficient of friction between the ladder and the ground is

\frac{1}{2}

(a) Find the magnitude of the normal reaction of the wall on the ladder.

(3 marks)

A horizontal force of magnitude

P

newtons is now applied to the ladder at

A

, towards the wall. The line of action of this force is perpendicular to the wall and the ladder remains in equilibrium.
(b) Find the greatest possible value of

P

.

(3 marks)

●●●●●Level 36 marksStart

Figure 1

A uniform rod

AB

, of length

5

m and weight

80

N, rests with its end

A

on rough horizontal ground. The rod rests on a small smooth horizontal peg at

C

, where

AC = 4

m. A particle of weight

20

N is attached to the rod at

B

. The rod lies in a vertical plane perpendicular to the peg and is inclined at an angle

\theta

to the ground, where

\tan\theta = \frac34

, as shown in Figure 1. The rod is in equilibrium.
(a) Show that the magnitude of the force exerted on the rod by the peg is

60

N.

(3 marks)

The coefficient of friction between the rod and the ground is

\mu

.
(b) Find the smallest possible value of

\mu

.

(4 marks)

●●●●●Level 37 marksStart

Figure 1

A uniform ladder

AB

, of weight

W

and length

2a

, rests with its end

A

on rough horizontal ground and its end

B

against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle

\theta

with the ground, as shown in Figure 1. The ladder rests in equilibrium.

The resultant force exerted on the ladder by the ground acts at an angle

\beta

to the horizontal.
(a) Show that

\tan\beta = 2\tan\theta

(5 marks)
(b) Given that

\beta = 80^\circ

, find the value of

\theta

.

(2 marks)

●●●●●Level 37 marksStart

A uniform shelf

PQ

has length

2L

metres and mass

m

kilograms.

The shelf rests horizontally, in equilibrium, on two fixed supports at the points

C

and

D

The support at

C

x

metres from the end

P

, and the support at

D

(x + 0.4)

metres from the end

Q

, as shown in the diagram below.The reaction force on the shelf at

D

is three times the reaction force on the shelf at

C

Show that

L - x = k

where

k

is a constant to be found.

(4 marks)

●●●●●Level 34 marksStart

A games club checks the masses of its gaming chips using a simple balance. Each chip is a thin disc with a diameter of

40

mm and should have a mass of

10

grams.

The balance is made from a uniform rod

AB

of length

300

mm, supported at its midpoint, and a fixed load of mass

m

grams.

A chip is laid flat on the rod near

A

, so that a point on the chip's curved edge is vertically above

A

and the whole chip is on the rod.

The fixed load hangs from

B

on a light inextensible string, as shown in the diagram below.(a) Given that the rod is horizontal and rests in equilibrium, find

m

(3 marks)
(b) State an assumption you have made about the chip to answer part (a).

(1 mark)

●●●●●Level 34 marksStart

A force of

20

N acts at the end of a rod

3

m long, at

30^\circ

to the rod. Find its moment about the other end.

●●●●●Level 32 marksStart

A rod is pivoted at a point

O

. A

10

N force acts downward

2

m to the left of

O

, and a

5

N force acts downward

3

m to the right. Find the magnitude and sense of the resultant moment about

O

●●●●●Level 32 marksStart

A force of

40

N acts at the end of a rod

2

m long, at

60^\circ

to the rod. Find its moment about the other end, to 3 s.f.

●●●●●Level 32 marksStart

A light rod is pivoted at its centre. A

12

N force acts downward

1.5

m from the pivot on one side. Find the force

F

that must act downward

2

m from the pivot on the other side to balance the rod.

●●●●●Level 32 marksStart

A rod

AB

is pivoted at

A

. A force of

30

N acts downward at the point

2

m from

A

, and a force of

20

N acts at

B

(

4

m from

A

) at

40^\circ

to the rod, both turning it clockwise. Find the total clockwise moment about

A

, to 3 s.f.

●●●●●Level 32 marksStart

A uniform beam

AB

of length

4

m and weight

120

N is pivoted at a point

1.5

m from

A

. A vertical force

F

applied downward at

A

keeps the beam horizontal.

Find

F

. (4 marks)

●●●●●Level 34 marksStart

A uniform beam

AB

of length

6

m and weight

300

N rests horizontally on supports at

A

and

B

. A child of weight

200

N stands

2

m from

A

.

Find the reaction at each support. (5 marks)

●●●●●Level 35 marksStart

A uniform plank of length

4

m and weight

80

N lies on a horizontal roof, with

1

m projecting beyond the roof's edge. A person begins to walk out along the overhang.

Find the greatest weight of a person who can stand at the very end of the plank without it tipping. (4 marks)

●●●●●Level 34 marksStart

A force of

50

N acts at the end of a rod of length

2

m, at an angle of

40^{\circ}

to the rod. The other end of the rod is pivoted.

Find the moment of the force about the pivot, giving your answer to 3 significant figures. (3 marks)

●●●●●Level 33 marksStart

A non-uniform rod

AB

of length

5

m and weight

100

N rests horizontally on supports at

A

and

B

. The reactions are

R_{A} = 60

N and

R_{B} = 40

N.

Find the distance of the rod's centre of mass from

A

. (4 marks)

●●●●●Level 34 marksStart

A uniform beam

AB

of length

8

m and weight

400

N rests horizontally on supports at

C

and

D

, where

AC = 1

m and

AD = 6

m.
(a) Find the reactions at

C

and

D

. (4 marks)
(b) A child of weight

100

N stands on the beam so that the two reactions become equal. Find how far from

C

the child stands. (2 marks)

●●●●●Level 36 marksStart

Figure 1 shows a uniform rod

AB

of mass

M

and length

2a

.

The end

A

of the rod rests on rough horizontal ground and the end

B

rests against a smooth vertical wall.

A particle of mass

2M

is attached to the rod at the point

C

, where

AC = 1.5a

The rod rests in equilibrium in a vertical plane perpendicular to the wall, at an angle

\theta

to the horizontal ground, as shown in Figure 1.Diagram not accurately drawn.

The magnitude of the normal reaction of the wall on the rod at

B

S

.

By taking moments about

A

, show that

S = 2Mg\cot\theta

(3 marks)

●●●●●Level 33 marksStart

Figure 1 shows a uniform rod

AB

of mass

M

and length

2a

.

The end

A

of the rod rests on rough horizontal ground.

The rod rests on a small smooth fixed horizontal peg at the point

C

of the rod, where

AC = 1.6a

The rod lies in a vertical plane perpendicular to the peg, at an angle

\theta

to the horizontal ground, as shown in Figure 1.Diagram not accurately drawn.

The magnitude of the normal reaction of the peg on the rod at

C

S

.
(a) By taking moments about

A

, show that

S = \frac{5}{8}Mg\cos\theta

(3 marks)

The coefficient of friction between the rod and the ground is

\mu

Given that

\tan\theta = \dfrac{5}{12}

and that the rod is in limiting equilibrium,
(b) find the value of

\mu

, giving your answer to 2 significant figures.

(6 marks)

●●●●●Level 49 marksStart

Figure 1

Figure 1 shows a uniform plank

AB

, of weight

65

N and length

6.5

m, propped against a low wall. The end

A

of the plank is on rough horizontal ground, and the plank touches the top edge of the wall at

C

, where

AC = 5

m. The plank makes an angle

\theta

with the ground, where

\tan\theta = \frac{5}{12}

In a model, the plank is a uniform rod, the top edge of the wall is a smooth horizontal peg, and the rod lies in a vertical plane perpendicular to the peg. The force exerted on the plank by the wall at

C

has magnitude

P

newtons.
(a) Show that, according to the model,

P = 39

(3 marks)
(b) Find the magnitude of the total force exerted on the plank by the ground at

A

.

(6 marks)

●●●●●Level 49 marksStart

Figure 1

A uniform ladder

AB

has mass

m

and length

2a

. Its foot

A

stands on rough horizontal ground and its top

B

rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall and makes an angle

\alpha

with the wall, as shown in Figure 1. The coefficient of friction between the ladder and the ground is

\frac{2}{5}

, and the ladder is in limiting equilibrium.
(a) Show that

\tan\alpha = \frac{4}{5}

(6 marks)

A worker then pushes the ladder at its midpoint with a horizontal force of magnitude

kmg

. The force is directed towards the wall and acts in the vertical plane containing the ladder. The ladder stays at rest in the same position.
(b) Find the greatest possible value of

k

.

(4 marks)

●●●●●Level 410 marksStart

Figure 1

A uniform pole

AB

has mass

12

kg and length

5

m. The pole leans with its end

B

against a rough vertical wall and its end

A

on rough horizontal ground, making an angle of

60^\circ

with the ground, as shown in Figure 1. The pole lies in a vertical plane perpendicular to the wall and is modelled as a uniform rod.

The coefficient of friction is

\frac{1}{3}

at the wall and

\mu

at the ground. The pole is on the point of slipping.
(a) Show that the normal reaction of the wall on the pole has magnitude

\dfrac{18g}{3\sqrt{3} + 1}

newtons.

(5 marks)
(b) Hence find the value of

\mu

, giving your answer to 2 significant figures.

(6 marks)

●●●●●Level 411 marksStart

Figure 1

Figure 1 shows a uniform rod

AB

, of length

5a

and weight

W

, with its end

A

freely hinged to a fixed point on a vertical wall. A particle of weight

W

is fixed to the rod at

B

. The rod is held in equilibrium by a light inextensible string

CD

of length

4a

. The end

C

of the string is fixed to the rod, with

AC = 3a

, and the end

D

is fixed to the wall vertically above

A

, with

AD = 5a

. The rod and the string lie in a vertical plane perpendicular to the wall. The tension in the string is

T

.
(a) Show that

T = 2W

(4 marks)

The force exerted on the rod by the hinge has magnitude

kW

.
(b) Find the value of

k

.

(6 marks)

●●●●●Level 410 marksStart

Figure 1

A lantern of weight

W

newtons hangs from the end

B

of a horizontal pole

AB

, as shown in Figure 1. The pole has weight

60

N and length

3

m, and its end

A

is freely hinged to a vertical wall. The pole is supported by a strut

CD

, where

C

is the midpoint of the pole and

D

is the point on the wall

2

m vertically below

A

. The pole and the strut lie in a vertical plane perpendicular to the wall.

The pole is modelled as a uniform rod, the strut as a light rod and the lantern as a particle. The thrust in the strut is

150

N.
(a) Show that

W = 30

(4 marks)
(b) Find the magnitude of the force exerted on the pole by the hinge.

(6 marks)

●●●●●Level 410 marksStart

Figure 1

A uniform rod

AB

, of length

4

m and weight

30

N, rests in a horizontal position with its end

A

in contact with a rough vertical wall. A light inextensible string has one end attached to the rod at

B

and the other end attached to the wall at the point

D

, where

D

is vertically above

A

and

AD = 3

m. A particle of weight

20

N is attached to the rod at the point

C

, where

AC = x

metres, as shown in Figure 1. The rod and the string lie in a vertical plane perpendicular to the wall.

The rod is in equilibrium and the tension in the string is

T

newtons.
(a) Show that

T = \frac{25}{3}(3 + x)

(3 marks)

The coefficient of friction between the rod and the wall is

\frac12

Given that the rod is in limiting equilibrium,
(b) find the value of

x

.

(6 marks)

●●●●●Level 49 marksStart

A uniform rod

AB

has length

3.6

m and weight

45

N. The end

A

of the rod rests on rough horizontal ground and the end

B

rests against a smooth sloping wall, which makes an angle of

65°

with the horizontal. The rod is in equilibrium at

25°

to the horizontal, in a vertical plane perpendicular to the wall (see diagram).(a) Show that the magnitude of the force acting on the rod at

B

26.6

N, correct to 3 significant figures.

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

(5 marks)

●●●●●Level 48 marksStart

A uniform rectangular lamina

ABCD

, of mass

1.5

kg, has

AB = 0.8

m and

AD = 2.5

m. It rests in limiting equilibrium in a vertical plane perpendicular to a smooth vertical wall, with the corner

A

on rough horizontal ground and the corner

D

against the wall. The edge

AD

is inclined at

56°

to the horizontal (see diagram).(a) By taking moments, show that the magnitude of the normal contact force between the lamina and the wall is

2.61

N, correct to 3 significant figures.

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

(3 marks)

●●●●●Level 47 marksStart

A uniform rod

AB

, of mass

4

kg and length

2

m, is freely hinged to a vertical wall at

A

. The rod is held in a horizontal position by a chain

BD

, where

D

is a fixed point on the wall vertically above

A

, and the chain makes an angle of

35°

with the rod. A sign of mass

6

kg hangs from the rod at the point

E

, where

AE = 1.5

m (see diagram).The chain is modelled as a light inextensible string and the sign is modelled as a particle.
(a) By taking moments about

A

, determine the tension in the chain.

(4 marks)
(b) Determine the magnitude of the force exerted on the rod at

A

.

(4 marks)

●●●●●Level 48 marksStart

A non-uniform rod

AB

of length

6

m and mass

40

kg hangs horizontally from a ceiling, supported by two light vertical wires attached at

C

and

D

, where

AC = 1

m and

AD = 5

m. The tension in the wire at

D

is twice the tension in the wire at

C

. Take

g = 9.8

m s⁻².(a) Find the tension in each wire. (2 marks)
(b) Determine the distance of the centre of mass of the rod from

A

. (2 marks)

A block of mass

30

kg is now hung from the rod at

B

.
(c) Find the new tension in each wire. (3 marks)
(d) Determine the greatest mass that could instead be hung at

B

without the wire at

C

becoming slack. (2 marks)

●●●●●Level 49 marksStart

A decorator places a uniform plank

PQ

, of length

5

metres, across two trestles. The plank lies horizontally at rest, in equilibrium, on the trestles at the points

X

and

Y

The distance

PX

0.5

metres and the distance

QY

1.5

metres, as shown in the diagram below.(a) The reaction force on the plank at

X

12g

newtons.
(a) (i) Show that the mass of the plank is

36

kilograms.

(ii) Find the reaction force, in terms of

g

, on the plank at

Y

(4 marks)
(b) The trestle at

Y

is moved so that the distance

QY = 1

metre. The plank remains horizontally at rest in equilibrium.

The decorator claims that the reaction force on the plank at

X

increases.

Explain, with a reason, whether this claim is correct.

(2 marks)

●●●●●Level 46 marksStart

A simple bench in a park is made from a uniform plank

AB

of length

4

metres. The plank rests horizontally, in equilibrium, on two fixed supports at the points

C

and

D

The distance

AC

1

metre and the distance

DB

0.5

metres.

A bag of mass

5

kilograms is placed on the bench at

B

, as shown in the diagram below.(a) The reaction force on the plank at

D

22g

newtons.
(a) (i) Show that the mass of the plank is

40

kilograms.

(ii) Find the reaction force, in terms of

g

, on the plank at

C

(4 marks)
(b) The bag is moved from

B

A

. The bench remains horizontally at rest in equilibrium.

It is claimed that the reaction force on the plank at

C

increases by

5g

newtons.

Explain, with a reason, whether this claim is correct.

(2 marks)

●●●●●Level 46 marksStart

Figure 1 (not accurately drawn)

A decorator's plank

AB

has mass

36

kg and length

4.5

m.

The plank rests on two trestles at

C

and

D

, where

AC = 0.5

m and

DB = 1

m, as shown in Figure 1.

A decorator of mass

60

kg stands on the plank between

C

and

D

.

The decorator is modelled as a particle at the point

P

, where

CP = x

metres and

0 < x < 3

The plank is modelled as a uniform rod resting in equilibrium in a horizontal position.

Using the model,
(a) show that the magnitude of the reaction at

C

(735 - 196x)

N.

(3 marks)

The magnitude of the reaction at

D

is twice the magnitude of the reaction at

C

.

Using the model,
(b) find the value of

x

(4 marks)

The decorator steps off the plank. A stack of tiles of mass

M

kg is placed on the plank at

B

. The stack of tiles is modelled as a particle at

B

.

The plank is now on the point of tilting about

D

.

Using the model,
(c) find the value of

M

(3 marks)

●●●●●Level 410 marksStart

Figure 1 (not accurately drawn)

A plank

AB

forms the floor of a window-cleaning platform. The plank has mass

m

kg and length

3

m and it hangs from two light vertical ropes. One rope is attached to the plank at

C

and the other rope is attached at

D

, where

AC = 0.5

m and

DB = 0.8

m, as shown in Figure 1.

A bucket of mass

pm

kg is placed on the plank at

B

and the plank rests in equilibrium in a horizontal position.

The plank is modelled as a uniform rod and the bucket is modelled as a particle.
(a) Given that the tension in the rope attached at

D

is three times the tension in the rope attached at

C

, use the model to find the exact value of

p

.

(7 marks)

The bucket is removed and replaced by a toolbox of mass

qm

kg, placed at

B

. The plank remains in equilibrium in a horizontal position but 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 plank is uniform.

(1 mark)

●●●●●Level 412 marksStart

A uniform lamina is in the shape of the segment of a circle cut off by the chord

AB

. The circle has centre

O

and radius

6a

, and angle

AOB = \frac{2\pi}{3}

radians. Figure 1 shows the lamina, shaded, together with the radii

OA

and

OB

.Figure 1
(a) Show that the distance of the centre of mass of the lamina from

O

\frac{18\sqrt{3}}{4\pi - 3\sqrt{3}}\,a

(5 marks)

The lamina is freely suspended from

A

and hangs in equilibrium with

AB

at an angle of

\theta°

to the downward vertical.
(b) Find the value of

\theta

, giving your answer to the nearest whole number.

(3 marks)

●●●●●Level 48 marksStart

The uniform lamina

ABCD

shown in Figure 1 is in the shape of a trapezium, with

AB

parallel to

DC

and angle

DAB

= angle

ADC = 90°

. The lengths of

AB

DC

and

AD

are

6a

2a

and

ka

respectively, where

k

is a positive constant.Figure 1 (not accurately drawn)
(a) Show that the centre of mass of the lamina is a distance

\frac{5ka}{12}

from

AB

.

(3 marks)

The lamina is freely suspended from

D

and hangs in equilibrium with

DA

at an angle

\alpha

to the downward vertical, where

\tan\alpha = \frac{13}{14}

(b) Find the value of

k

.

(6 marks)

●●●●●Level 49 marksStart

A template is formed by removing a circular disc of radius

a

from a uniform rectangular lamina

ABCD

, where

AB = 10a

and

AD = 6a

. The centre

X

of the disc that is removed is a distance

2a

from

AB

and a distance

2a

from

BC

. The template is shown shaded in Figure 1.Figure 1
(a) Show that the centre of mass of the template is a distance

\frac{2(90 - \pi)}{60 - \pi}\,a

from

AB

.

(5 marks)

The template is freely suspended from

D

and hangs in equilibrium.
(b) Find, to the nearest degree, the angle between

DC

and the downward vertical.

(4 marks)

●●●●●Level 49 marksStart

The uniform rectangular lamina

ABCD

has

AB = 14a

and

BC = 6a

. The point

P

AB

and the point

Q

DC

are such that

PB = QC = 4a

, as shown in Figure 1.Figure 1

The lamina is folded along

PQ

so that the rectangle

PBCQ

lies flat on the rest of the lamina, with

B

AP

and

C

DQ

, forming the folded lamina

L

shown in Figure 2.Figure 2
(a) Show that the centre of mass of

L

is a distance

\frac{41a}{7}

from

AD

.

(5 marks)

The weight of

L

W

. A particle of weight

\frac{1}{2}W

is attached to

L

B

. The loaded lamina is freely suspended from

D

and is held in equilibrium, with

DA

vertical, by a horizontal force of magnitude

F

acting at

A

. The line of action of the force lies in the plane of

L

.
(b) Find

F

in terms of

W

.

(4 marks)

●●●●●Level 49 marksStart

A uniform rectangular lamina

ABCD

has mass

6m

, with

AB = 6a

and

BC = 4a

. Particles of mass

2m

and

m

are attached to the lamina at

C

and

D

respectively, as shown in Figure 1.Figure 1
(a) Show that the centre of mass of the loaded lamina is a distance

\frac{10a}{3}

from

AD

.

(4 marks)

The loaded lamina hangs in equilibrium in a vertical plane, supported by two light vertical strings, one attached at

A

and the other attached at

B

, with

AB

horizontal.
(b) Find, in terms of

m

and

g

, the tension in each string.

(4 marks)

●●●●●Level 48 marksStart

The template shown in Figure 1 is made from two uniform laminas that lie in the same plane:

the rectangle $ABCD$ , with $AB = 5a$ and $BC = 4a$
the quarter-circle $BEC$ , with centre $B$ and radius $4a$ , where $E$ lies on $AB$ produced

The mass per unit area of

BEC

is twice the mass per unit area of

ABCD

.Figure 1
(a) Show that the centre of mass of the template is a distance

\frac{139 + 60\pi}{6(5 + 2\pi)}\,a

from

AD

.

(5 marks)

The template is freely suspended from

D

and hangs in equilibrium.
(b) Find, to the nearest degree, the angle between

DA

and the downward vertical.

(4 marks)

●●●●●Level 49 marksStart

Figure 1 shows a rod

AB

of mass

M

and length

2a

.

The end

A

of the rod rests on rough horizontal ground.

A particle of mass

4M

is attached to the rod at the point

C

, where

AC = 1.25a

The rod is held in equilibrium at an angle

\theta

to the horizontal ground by a light string attached to the rod at

B

. The string is perpendicular to the rod, as shown in Figure 1.

The rod is in limiting equilibrium.Diagram not accurately drawn.
(a) State the direction (left or right on Figure 1 above) of the frictional force acting on the rod at

A

. Give a reason for your answer.

(1 mark)

In an initial model, the rod is modelled as being uniform.

Use this initial model to answer parts (b),
(c) and (d).

The tension in the string is

T

.
(b) By taking moments about

A

, show that

T = 3Mg\cos\theta

(3 marks)

The coefficient of friction between the rod and the ground is

\mu

Given that

\tan\theta = \dfrac{4}{3}

(c) find the value of

\mu

, giving your answer to 2 significant figures.

(5 marks)
(d) Find, in terms of

M

and

g

, the magnitude of the resultant force acting on the rod at

A

.

(3 marks)

In a new model, the rod is modelled as being non-uniform, with its centre of mass closer to

A

than it is to

B

.

A new value for

T

is calculated using this new model, with

\tan\theta = \dfrac{4}{3}

(e) State whether this new value for

T

is larger, smaller or equal to the value of

T

that would be found using the initial model. Give a reason for your answer.

(1 mark)

●●●●●Level 513 marksStart

Figure 1

A uniform plank

AB

has mass

m

and length

2a

. The plank rests with its end

A

on rough horizontal ground and leans against a smooth horizontal pipe, which is fixed parallel to the ground. The plank touches the pipe at the point

C

, where

AC = 1.25a

, and makes an angle

\theta

with the ground, as shown in Figure 1. The plank lies in a vertical plane perpendicular to the pipe.

The plank is modelled as a rod and the pipe as a smooth peg.
(a) Show that the magnitude of the force exerted on the plank by the pipe is

\frac45 mg\cos\theta

(3 marks)

The coefficient of friction between the plank and the ground is

\frac23

Given that the plank is in limiting equilibrium,
(b) show that

5\tan^2\theta - 6\tan\theta + 1 = 0

(7 marks)

The plank is now placed so that

\tan\theta = \frac12

, still touching the pipe at

C

. A particle of mass

km

is attached to the plank at the point

D

, where

AD = \frac12 a

. The plank rests in limiting equilibrium.
(c) Find the value of

k

.

(5 marks)

●●●●●Level 515 marksStart

Figure 1

A uniform rod

AB

has length

8a

and weight

W

. The end

A

of the rod is freely hinged to a fixed point on horizontal ground. A smooth cylinder is fixed to the ground with its axis horizontal. The rod rests in equilibrium on the cylinder, touching it at the point

C

, where

AC = 3a

. The rod lies in a vertical plane perpendicular to the axis of the cylinder and is inclined to the ground at an angle

\theta

, where

\tan\theta = \frac34

, as shown in Figure 1.
(a) Show that the magnitude of the normal reaction between the rod and the cylinder is

\frac{16}{15}W

(3 marks)

The force exerted on the rod by the hinge makes an angle

\alpha

with the ground.
(b) Find

(i) the magnitude of this force, as an exact multiple of

W

,

(ii) the exact value of

\tan\alpha

.

(8 marks)

The cylinder is held in place by a bracket, which gives way if the magnitude of the force exerted on the cylinder by the rod exceeds

2W

. A particle of weight

\lambda W

is now attached to the rod at

B

, and the rod remains in equilibrium in the same position.
(c) Find the greatest possible value of

\lambda

.

(4 marks)

●●●●●Level 515 marksStart

Figure 1

A ladder

AB

, of mass

m

and length

2a

, rests with its end

A

on rough horizontal ground and its end

B

against a rough vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle

\theta

with the ground, where

\tan\theta = 2

The coefficient of friction between the ladder and the ground is

\frac25

and the coefficient of friction between the ladder and the wall is

\frac12

A man of mass

5m

climbs the ladder. When he reaches the point

C

, where

AC = x

, the ladder is about to slip, as shown in Figure 1.

The ladder is modelled as a uniform rod and the man as a particle.
(a) Show that the magnitude of the normal reaction of the ground on the ladder is

5mg

(4 marks)
(b) Find

x

in terms of

a

.

(5 marks)

The man climbs back down to the ground. A block of mass

km

is then fixed to the ladder at

A

, and the man climbs the ladder again. He reaches

B

and the ladder is then about to slip.
(c) Find the value of

k

.

(4 marks)

●●●●●Level 513 marksStart

The diagram shows a ladder

AB

, of length

2a

and mass

m

, resting in equilibrium on the top edge

C

of a vertical wall of height

h

. The end

A

is on horizontal ground and the ladder is inclined at an angle

\alpha

to the horizontal, where

\tan\alpha = \frac{3}{4}

. A bag of tools of mass

3m

is placed on the ladder at the point

D

, where

AD = d

. The ladder rests in a vertical plane perpendicular to the wall.The ladder is modelled as uniform, the bag of tools as a particle, the ground as rough and the top edge of the wall as smooth.
(a) Show that the normal contact force between the ladder and the wall is

\dfrac{12mg(a + 3d)}{25h}

.

(4 marks)

It is given that the equilibrium is limiting and that the coefficient of friction between the ladder and the ground is

\frac{1}{2}

.
(b) Show that

h = k(a + 3d)

, 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 514 marksStart

A basket of flowers, of mass

3.2

kg, hangs from the end

B

of a bracket

AB

. The bracket is a rod of mass

1.8

kg and length

1.6

m, and its end

A

is freely hinged to a vertical wall. The rod is held in equilibrium by a chain

CD

, where

C

is the point on the rod with

AC = 1.2

m and

D

is a fixed point on the wall vertically above

A

. The rod makes an angle of

75°

with the upward vertical and the chain makes an angle of

50°

with the wall (see diagram).The chain is modelled as a light inextensible string, the rod is modelled as uniform and the basket is modelled as a particle.
(a) By taking moments about

A

, determine the tension in the chain.

(4 marks)
(b) (i) Determine the magnitude of the force exerted on the rod at

A

.

(ii) Calculate the direction of the force exerted on the rod at

A

.

(6 marks)
(c) Suggest one improvement that could be made to the model to make it more realistic.

(1 mark)

●●●●●Level 511 marksStart

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 511 marksStart

Figure 4 shows a non-uniform plank

AB

of length

6

m and mass

30

kg resting horizontally on two supports at

C

and

D

, where

AC = 1

m and

DB = 1.5

m. The centre of mass of the plank is at the point

G

.Figure 4 (not accurately drawn)

A child of mass

25

kg stands on the plank at

A

. The plank is then on the point of tilting about

C

.
(a) Find the distance

AG

, giving your answer to

3

significant figures. (5 marks)

The child now stands at

B

instead.
(b) Find the magnitude of the reaction at

C

and the magnitude of the reaction at

D

. (6 marks)
(c) Find the greatest mass of a child who could stand at

B

without the plank tilting, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 515 marksStart

Figure 2 shows a non-uniform rod

AB

of length

4

m and mass

25

kg, hanging horizontally in equilibrium from two vertical strings attached at

A

and at the point

C

, where

AC = 3

m. The centre of mass of the rod is at

G

.Figure 2 (not accurately drawn)

The tension in the string at

A

98

N. Take

g = 9.8

m s⁻².
(a) Find the tension in the string at

C

. (2 marks)
(b) Find the distance

AG

. (3 marks)

A particle of mass

m

kg is now attached to the rod at

B

. The rod is then on the point of tilting about

C

.
(c) Find the value of

m

. (4 marks)

●●●●●Level 59 marksStart

[A uniform semicircular arc of radius

r

has its centre of mass at a distance

\frac{2r}{\pi}

from the centre of the circle.]

A uniform wire is used to make the rigid framework shown in Figure 1. The framework consists of

the semicircular arc $APB$ , with centre $O$ and radius $8a$
the straight wire $AB$ , which is a diameter of the arc
the straight wires $AC$ and $BC$ , each of length $17a$

The point

C

is on the opposite side of

AB

from the arc, and all of the framework lies in one plane.Figure 1
(a) Show that the centre of mass of the framework is a distance

\frac{127a}{50 + 8\pi}

from

AB

.

(5 marks)

The framework is freely suspended from

A

and hangs in equilibrium with

AB

at an angle

\theta

to the downward vertical.
(b) Find the exact value of

\tan\theta

.

(3 marks)

The mass of the framework is

M

. A particle of mass

kM

is now attached to the framework at

C

. The framework is again freely suspended from

A

, and it now hangs in equilibrium with

AB

45°

to the downward vertical.
(c) Find the value of

k

, giving your answer to 3 significant figures.

(5 marks)

●●●●●Level 513 marksStart

The template shown in Figure 1 is made from two uniform laminas that lie in the same plane: the square

ABCD

, with sides of length

6a

, and a semicircle with diameter

BC

, which lies outside the square. The mass per unit area of the semicircle is

\lambda

times the mass per unit area of the square, where

\lambda

is a positive constant.Figure 1
(a) Show that the centre of mass of the template is a distance

\frac{2\big(12 + (3\pi + 2)\lambda\big)}{8 + \pi\lambda}\,a

from

AD

.

(5 marks)

Given that the centre of mass of the template lies inside the square

ABCD

,
(b) find the range of possible values of

\lambda

.

(3 marks)

It is now given that

\lambda = 3

and that the mass of the template is

M

. The template is freely pivoted at

A

and is held in equilibrium, with

AD

vertical, by a horizontal force of magnitude

F

acting at

C

. The line of action of the force lies in the plane of the template.
(c) Find

F

in terms of

M

and

g

.

(4 marks)

●●●●●Level 512 marksStart

A uniform semicircular lamina has diameter

AB

, centre

O

and radius

6a

. The point

C

on its arc is such that

OC

is perpendicular to

AB

. A template is formed by removing the square

OPQR

, with sides of length

4a

, from the lamina, where

P

lies on

OB

and

R

lies on

OC

. The template is shown shaded in Figure 1.Figure 1
(a) Show that the centre of mass of the template is a distance

\frac{56a}{9\pi - 8}

from

AB

.

(5 marks)

The template is freely suspended from

A

and hangs in equilibrium with

AB

at an angle

\theta

to the downward vertical.
(b) Find the exact value of

\tan\theta

.

(4 marks)

The mass of the template is

M

. A particle is to be attached to the template so that the template can hang in equilibrium, freely suspended from

C

, with

AB

horizontal.
(c) Find, in terms of

M

, the least possible mass of the particle.

(4 marks)

●●●●●Level 513 marksStart

Equilibrium of rigid bodies

38 questions

LessonNot started

A uniform beam

AB

of length

6

m and weight

200

N rests on supports at

A

and

B

. Find the reactions at

A

and

B

●●●●●Level 22 marksStart

A uniform shelf

PQ

has length

2L

metres and mass

m

kilograms.

The shelf rests horizontally, in equilibrium, on two fixed supports at the points

C

and

D

The support at

C

x

metres from the end

P

, and the support at

D

(x + 0.4)

metres from the end

Q

, as shown in the diagram below.The reaction force on the shelf at

D

is three times the reaction force on the shelf at

C

Show that

L - x = k

where

k

is a constant to be found.

(4 marks)

●●●●●Level 34 marksStart

A games club checks the masses of its gaming chips using a simple balance. Each chip is a thin disc with a diameter of

40

mm and should have a mass of

10

grams.

The balance is made from a uniform rod

AB

of length

300

mm, supported at its midpoint, and a fixed load of mass

m

grams.

A chip is laid flat on the rod near

A

, so that a point on the chip's curved edge is vertically above

A

and the whole chip is on the rod.

The fixed load hangs from

B

on a light inextensible string, as shown in the diagram below.(a) Given that the rod is horizontal and rests in equilibrium, find

m

(3 marks)
(b) State an assumption you have made about the chip to answer part (a).

(1 mark)

●●●●●Level 34 marksStart

Figure 1 (not accurately drawn)

A non-uniform plank

AB

, of length

5a

, rests in equilibrium on a horizontal loading platform. The plank overhangs the edge of the platform at

C

, where

AC = 3a

and

CB = 2a

, as shown in Figure 1.

The plank is perpendicular to the edge of the platform. The plank has weight

W

and its centre of mass is at a distance

d

from

A

.

When a force of magnitude

F

, acting vertically upwards, is applied to the plank at

B

, the plank is on the point of tilting about

A

.

When this force is replaced by a force of magnitude

2F

, acting vertically downwards at

B

, the plank is on the point of tilting about

C

.

The plank is modelled as a non-uniform rod.

Find

d

in terms of

a

.

(6 marks)

●●●●●Level 36 marksStart

Figure 1 (not accurately drawn)

A non-uniform pole

AB

has length

5

m and mass

6

kg. The centre of mass of the pole is

2.2

m from

A

.

The pole lies on a horizontal flat roof and overhangs the edge of the roof at

C

1.4

m, as shown in Figure 1. The pole is perpendicular to the edge of the roof.

A particle of mass

9

kg is attached to the pole at

B

and a second particle, of mass

M

kg, is attached to the pole at the point

D

, where

AD = 0.6

m.

The pole remains in equilibrium in a horizontal position.
(a) Find the smallest possible value of

M

.

(3 marks)

Both particles are now removed. A new particle is attached to the pole at the point

E

, where

EB = 0.4

m.

The pole remains in equilibrium in a horizontal position but is now on the point of tilting about

C

.
(b) Find the magnitude of the force exerted on the pole by the roof at

C

.

(3 marks)

●●●●●Level 36 marksStart

A uniform beam

AB

of length

4

m and weight

120

N rests on supports at

A

and at

C

, where

AC = 3

m. Find the reactions at

A

and

C

●●●●●Level 33 marksStart

A uniform plank

AB

of length

5

m and weight

300

N rests on supports at

A

and

B

. A child of weight

400

N stands

1

m from

A

. Find the reactions at

A

and

B

●●●●●Level 33 marksStart

A uniform beam

AB

of length

8

m and weight

160

N rests on supports at

A

and at

B'

, where

AB' = 6

m. Find the reactions at

A

and

B'

, to 3 s.f.

●●●●●Level 33 marksStart

A uniform beam

AB

of length

6

m and weight

100

N rests on supports at

A

and

B

. Loads of

50

N and

80

N hang at

2

m and

4

m from

A

. Find the reactions at

A

and

B

●●●●●Level 33 marksStart

A uniform plank

AB

of length

4

m and weight

200

N rests on supports at

A

and at

D

, where

AD = 3

m. A person of weight

600

N stands

1

m from

A

. Find the reactions at

A

and

D

, to 3 s.f.

●●●●●Level 33 marksStart

A light rod

AB

of length

4

m rests on supports at

A

and

B

. Loads of

30

N and

50

N hang at

1

m and

3

m from

A

. Find the reactions at

A

and

B

●●●●●Level 33 marksStart

A uniform beam

AB

of length

8

m and weight

240

N rests on supports at

C

and

D

, where

AC = 2

m and

AD = 6

m. A load of

100

N hangs at

A

. Find the reactions at

C

and

D

●●●●●Level 33 marksStart

A uniform beam

AB

of length

8

m and weight

120

N rests on supports at

C

and

D

, where

AC = 2

m and

AD = 6

m. A load of weight

W

N hangs at

A

.
(a) Show that the reaction at

D

R_D = \left(60 - \tfrac12 W\right)

N.
(b) Hence find the value of

W

for which the beam is on the point of tilting about

C

●●●●●Level 34 marksStart

Figure 1 shows a uniform rod

AB

of mass

M

and length

2a

.

The end

A

of the rod rests on rough horizontal ground and the end

B

rests against a smooth vertical wall.

A particle of mass

2M

is attached to the rod at the point

C

, where

AC = 1.5a

The rod rests in equilibrium in a vertical plane perpendicular to the wall, at an angle

\theta

to the horizontal ground, as shown in Figure 1.Diagram not accurately drawn.

The magnitude of the normal reaction of the wall on the rod at

B

S

.

By taking moments about

A

, show that

S = 2Mg\cot\theta

(3 marks)

●●●●●Level 33 marksStart

Figure 1 shows a uniform rod

AB

of mass

M

and length

2a

.

The end

A

of the rod rests on rough horizontal ground.

The rod rests on a small smooth fixed horizontal peg at the point

C

of the rod, where

AC = 1.6a

The rod lies in a vertical plane perpendicular to the peg, at an angle

\theta

to the horizontal ground, as shown in Figure 1.Diagram not accurately drawn.

The magnitude of the normal reaction of the peg on the rod at

C

S

.
(a) By taking moments about

A

, show that

S = \frac{5}{8}Mg\cos\theta

(3 marks)

The coefficient of friction between the rod and the ground is

\mu

Given that

\tan\theta = \dfrac{5}{12}

and that the rod is in limiting equilibrium,
(b) find the value of

\mu

, giving your answer to 2 significant figures.

(6 marks)

●●●●●Level 49 marksStart

A uniform rod

AB

has length

3.6

m and weight

45

N. The end

A

of the rod rests on rough horizontal ground and the end

B

rests against a smooth sloping wall, which makes an angle of

65°

with the horizontal. The rod is in equilibrium at

25°

to the horizontal, in a vertical plane perpendicular to the wall (see diagram).(a) Show that the magnitude of the force acting on the rod at

B

26.6

N, correct to 3 significant figures.

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

(5 marks)

●●●●●Level 48 marksStart

A uniform rectangular lamina

ABCD

, of mass

1.5

kg, has

AB = 0.8

m and

AD = 2.5

m. It rests in limiting equilibrium in a vertical plane perpendicular to a smooth vertical wall, with the corner

A

on rough horizontal ground and the corner

D

against the wall. The edge

AD

is inclined at

56°

to the horizontal (see diagram).(a) By taking moments, show that the magnitude of the normal contact force between the lamina and the wall is

2.61

N, correct to 3 significant figures.

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

(3 marks)

●●●●●Level 47 marksStart

A uniform rod

AB

, of mass

4

kg and length

2

m, is freely hinged to a vertical wall at

A

. The rod is held in a horizontal position by a chain

BD

, where

D

is a fixed point on the wall vertically above

A

, and the chain makes an angle of

35°

with the rod. A sign of mass

6

kg hangs from the rod at the point

E

, where

AE = 1.5

m (see diagram).The chain is modelled as a light inextensible string and the sign is modelled as a particle.
(a) By taking moments about

A

, determine the tension in the chain.

(4 marks)
(b) Determine the magnitude of the force exerted on the rod at

A

.

(4 marks)

●●●●●Level 48 marksStart

A non-uniform plank

AB

has length

6

m and mass

20

kg. Its centre of mass is at the point

G

, where

AG = 2.5

m. The plank rests horizontally on two supports at

C

and

D

, where

AC = 1

m and

AD = 4.5

m. Take

g = 9.8

m s⁻².(a) Find the magnitudes of the reactions at

C

and

D

. (5 marks)
(b) A child stands at

B

, and the plank is on the point of tilting about

D

. Modelling the child as a particle, find the child's mass. (4 marks)
(c) A different child, of mass

40

kg, walks along the plank from

D

towards

B

. Find how far from

A

they can walk before the plank starts to tilt. (3 marks)

●●●●●Level 412 marksStart

A decorator places a uniform plank

PQ

, of length

5

metres, across two trestles. The plank lies horizontally at rest, in equilibrium, on the trestles at the points

X

and

Y

The distance

PX

0.5

metres and the distance

QY

1.5

metres, as shown in the diagram below.(a) The reaction force on the plank at

X

12g

newtons.
(a) (i) Show that the mass of the plank is

36

kilograms.

(ii) Find the reaction force, in terms of

g

, on the plank at

Y

(4 marks)
(b) The trestle at

Y

is moved so that the distance

QY = 1

metre. The plank remains horizontally at rest in equilibrium.

The decorator claims that the reaction force on the plank at

X

increases.

Explain, with a reason, whether this claim is correct.

(2 marks)

●●●●●Level 46 marksStart

A simple bench in a park is made from a uniform plank

AB

of length

4

metres. The plank rests horizontally, in equilibrium, on two fixed supports at the points

C

and

D

The distance

AC

1

metre and the distance

DB

0.5

metres.

A bag of mass

5

kilograms is placed on the bench at

B

, as shown in the diagram below.(a) The reaction force on the plank at

D

22g

newtons.
(a) (i) Show that the mass of the plank is

40

kilograms.

(ii) Find the reaction force, in terms of

g

, on the plank at

C

(4 marks)
(b) The bag is moved from

B

A

. The bench remains horizontally at rest in equilibrium.

It is claimed that the reaction force on the plank at

C

increases by

5g

newtons.

Explain, with a reason, whether this claim is correct.

(2 marks)

●●●●●Level 46 marksStart

Figure 1 (not accurately drawn)

A decorator's plank

AB

has mass

36

kg and length

4.5

m.

The plank rests on two trestles at

C

and

D

, where

AC = 0.5

m and

DB = 1

m, as shown in Figure 1.

A decorator of mass

60

kg stands on the plank between

C

and

D

.

The decorator is modelled as a particle at the point

P

, where

CP = x

metres and

0 < x < 3

The plank is modelled as a uniform rod resting in equilibrium in a horizontal position.

Using the model,
(a) show that the magnitude of the reaction at

C

(735 - 196x)

N.

(3 marks)

The magnitude of the reaction at

D

is twice the magnitude of the reaction at

C

.

Using the model,
(b) find the value of

x

(4 marks)

The decorator steps off the plank. A stack of tiles of mass

M

kg is placed on the plank at

B

. The stack of tiles is modelled as a particle at

B

.

The plank is now on the point of tilting about

D

.

Using the model,
(c) find the value of

M

(3 marks)

●●●●●Level 410 marksStart

Figure 1 (not accurately drawn)

A plank

AB

forms the floor of a window-cleaning platform. The plank has mass

m

kg and length

3

m and it hangs from two light vertical ropes. One rope is attached to the plank at

C

and the other rope is attached at

D

, where

AC = 0.5

m and

DB = 0.8

m, as shown in Figure 1.

A bucket of mass

pm

kg is placed on the plank at

B

and the plank rests in equilibrium in a horizontal position.

The plank is modelled as a uniform rod and the bucket is modelled as a particle.
(a) Given that the tension in the rope attached at

D

is three times the tension in the rope attached at

C

, use the model to find the exact value of

p

.

(7 marks)

The bucket is removed and replaced by a toolbox of mass

qm

kg, placed at

B

. The plank remains in equilibrium in a horizontal position but 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 plank is uniform.

(1 mark)

●●●●●Level 412 marksStart

The uniform rectangular lamina

ABCD

has

AB = 14a

and

BC = 6a

. The point

P

AB

and the point

Q

DC

are such that

PB = QC = 4a

, as shown in Figure 1.Figure 1

The lamina is folded along

PQ

so that the rectangle

PBCQ

lies flat on the rest of the lamina, with

B

AP

and

C

DQ

, forming the folded lamina

L

shown in Figure 2.Figure 2
(a) Show that the centre of mass of

L

is a distance

\frac{41a}{7}

from

AD

.

(5 marks)

The weight of

L

W

. A particle of weight

\frac{1}{2}W

is attached to

L

B

. The loaded lamina is freely suspended from

D

and is held in equilibrium, with

DA

vertical, by a horizontal force of magnitude

F

acting at

A

. The line of action of the force lies in the plane of

L

.
(b) Find

F

in terms of

W

.

(4 marks)

●●●●●Level 49 marksStart

A uniform rectangular lamina

ABCD

has mass

6m

, with

AB = 6a

and

BC = 4a

. Particles of mass

2m

and

m

are attached to the lamina at

C

and

D

respectively, as shown in Figure 1.Figure 1
(a) Show that the centre of mass of the loaded lamina is a distance

\frac{10a}{3}

from

AD

.

(4 marks)

The loaded lamina hangs in equilibrium in a vertical plane, supported by two light vertical strings, one attached at

A

and the other attached at

B

, with

AB

horizontal.
(b) Find, in terms of

m

and

g

, the tension in each string.

(4 marks)

●●●●●Level 48 marksStart

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 44 marksStart

Figure 1 shows a rod

AB

of mass

M

and length

2a

.

The end

A

of the rod rests on rough horizontal ground.

A particle of mass

4M

is attached to the rod at the point

C

, where

AC = 1.25a

The rod is held in equilibrium at an angle

\theta

to the horizontal ground by a light string attached to the rod at

B

A

. Give a reason for your answer.

(1 mark)

In an initial model, the rod is modelled as being uniform.

Use this initial model to answer parts (b),
(c) and (d).

The tension in the string is

T

.
(b) By taking moments about

A

, show that

T = 3Mg\cos\theta

(3 marks)

The coefficient of friction between the rod and the ground is

\mu

Given that

\tan\theta = \dfrac{4}{3}

(c) find the value of

\mu

, giving your answer to 2 significant figures.

(5 marks)
(d) Find, in terms of

M

and

g

, the magnitude of the resultant force acting on the rod at

A

.

(3 marks)

In a new model, the rod is modelled as being non-uniform, with its centre of mass closer to

A

than it is to

B

.

A new value for

T

is calculated using this new model, with

\tan\theta = \dfrac{4}{3}

(e) State whether this new value for

T

is larger, smaller or equal to the value of

T

that would be found using the initial model. Give a reason for your answer.

(1 mark)

●●●●●Level 513 marksStart

The diagram shows a ladder

AB

, of length

2a

and mass

m

, resting in equilibrium on the top edge

C

of a vertical wall of height

h

. The end

A

is on horizontal ground and the ladder is inclined at an angle

\alpha

to the horizontal, where

\tan\alpha = \frac{3}{4}

. A bag of tools of mass

3m

is placed on the ladder at the point

D

, where

AD = d

\dfrac{12mg(a + 3d)}{25h}

.

(4 marks)

It is given that the equilibrium is limiting and that the coefficient of friction between the ladder and the ground is

\frac{1}{2}

.
(b) Show that

h = k(a + 3d)

, 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 514 marksStart

A basket of flowers, of mass

3.2

kg, hangs from the end

B

of a bracket

AB

. The bracket is a rod of mass

1.8

kg and length

1.6

m, and its end

A

is freely hinged to a vertical wall. The rod is held in equilibrium by a chain

CD

, where

C

is the point on the rod with

AC = 1.2

m and

D

is a fixed point on the wall vertically above

A

. The rod makes an angle of

75°

with the upward vertical and the chain makes an angle of

50°

with the wall (see diagram).The chain is modelled as a light inextensible string, the rod is modelled as uniform and the basket is modelled as a particle.
(a) By taking moments about

A

, determine the tension in the chain.

(4 marks)
(b) (i) Determine the magnitude of the force exerted on the rod at

A

.

(ii) Calculate the direction of the force exerted on the rod at

A

.

(6 marks)
(c) Suggest one improvement that could be made to the model to make it more realistic.

(1 mark)

●●●●●Level 511 marksStart

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 511 marksStart

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 515 marksStart

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

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 515 marksStart

Figure 4 shows a non-uniform plank

AB

of length

6

m and mass

30

kg resting horizontally on two supports at

C

and

D

, where

AC = 1

m and

DB = 1.5

m. The centre of mass of the plank is at the point

G

.Figure 4 (not accurately drawn)

A child of mass

25

kg stands on the plank at

A

. The plank is then on the point of tilting about

C

.
(a) Find the distance

AG

, giving your answer to

3

significant figures. (5 marks)

The child now stands at

B

instead.
(b) Find the magnitude of the reaction at

C

and the magnitude of the reaction at

D

. (6 marks)
(c) Find the greatest mass of a child who could stand at

B

without the plank tilting, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 515 marksStart

Figure 2 shows a non-uniform rod

AB

of length

4

m and mass

25

kg, hanging horizontally in equilibrium from two vertical strings attached at

A

and at the point

C

, where

AC = 3

m. The centre of mass of the rod is at

G

.Figure 2 (not accurately drawn)

The tension in the string at

A

98

N. Take

g = 9.8

m s⁻².
(a) Find the tension in the string at

C

. (2 marks)
(b) Find the distance

AG

. (3 marks)

A particle of mass

m

kg is now attached to the rod at

B

. The rod is then on the point of tilting about

C

.
(c) Find the value of

m

. (4 marks)

●●●●●Level 59 marksStart

[A uniform semicircular arc of radius

r

has its centre of mass at a distance

\frac{2r}{\pi}

from the centre of the circle.]

A uniform wire is used to make the rigid framework shown in Figure 1. The framework consists of

the semicircular arc $APB$ , with centre $O$ and radius $8a$
the straight wire $AB$ , which is a diameter of the arc
the straight wires $AC$ and $BC$ , each of length $17a$

The point

C

is on the opposite side of

AB

from the arc, and all of the framework lies in one plane.Figure 1
(a) Show that the centre of mass of the framework is a distance

\frac{127a}{50 + 8\pi}

from

AB

.

(5 marks)

The framework is freely suspended from

A

and hangs in equilibrium with

AB

at an angle

\theta

to the downward vertical.
(b) Find the exact value of

\tan\theta

.

(3 marks)

The mass of the framework is

M

. A particle of mass

kM

is now attached to the framework at

C

. The framework is again freely suspended from

A

, and it now hangs in equilibrium with

AB

45°

to the downward vertical.
(c) Find the value of

k

, giving your answer to 3 significant figures.

(5 marks)

●●●●●Level 513 marksStart

The template shown in Figure 1 is made from two uniform laminas that lie in the same plane: the square

ABCD

, with sides of length

6a

, and a semicircle with diameter

BC

, which lies outside the square. The mass per unit area of the semicircle is

\lambda

times the mass per unit area of the square, where

\lambda

is a positive constant.Figure 1
(a) Show that the centre of mass of the template is a distance

\frac{2\big(12 + (3\pi + 2)\lambda\big)}{8 + \pi\lambda}\,a

from

AD

.

(5 marks)

Given that the centre of mass of the template lies inside the square

ABCD

,
(b) find the range of possible values of

\lambda

.

(3 marks)

It is now given that

\lambda = 3

and that the mass of the template is

M

. The template is freely pivoted at

A

and is held in equilibrium, with

AD

vertical, by a horizontal force of magnitude

F

acting at

C

. The line of action of the force lies in the plane of the template.
(c) Find

F

in terms of

M

and

g

.

(4 marks)

●●●●●Level 512 marksStart

A uniform semicircular lamina has diameter

AB

, centre

O

and radius

6a

. The point

C

on its arc is such that

OC

is perpendicular to

AB

. A template is formed by removing the square

OPQR

, with sides of length

4a

, from the lamina, where

P

lies on

OB

and

R

lies on

OC

. The template is shown shaded in Figure 1.Figure 1
(a) Show that the centre of mass of the template is a distance

\frac{56a}{9\pi - 8}

from

AB

.

(5 marks)

The template is freely suspended from

A

and hangs in equilibrium with

AB

at an angle

\theta

to the downward vertical.
(b) Find the exact value of

\tan\theta

.

(4 marks)

The mass of the template is

M

. A particle is to be attached to the template so that the template can hang in equilibrium, freely suspended from

C

, with

AB

horizontal.
(c) Find, in terms of

M

, the least possible mass of the particle.

(4 marks)

●●●●●Level 513 marksStart

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 52 marksStart

Non-uniform rods and tilting

18 questions

LessonNot started

A non-uniform rod

AB

of length

5

m and weight

60

N balances horizontally on a single support placed

2

m from

A

. State the distance of its centre of mass from

A

●●●●●Level 32 marksStart

A uniform beam

AB

of length

6

m and weight

100

N rests on supports at

C

and

D

, where

AC = 1

m and

AD = 4

m. A weight

W

hangs at

B

. Find the value of

W

for which the beam is on the point of tilting about

D

●●●●●Level 32 marksStart

A non-uniform rod

AB

of length

6

m rests horizontally on supports at

A

and

B

. The reactions are

R_A = 40

N and

R_B = 20

N. Find the weight of the rod and the distance of its centre of mass from

A

●●●●●Level 33 marksStart

A uniform plank

AB

of length

4

m and weight

80

N rests on supports at

1

m and

3

m from

A

. A load is placed at

A

. Find the load for which the plank is on the point of tilting (lifting at the

3

m support).

●●●●●Level 32 marksStart

A non-uniform rod

AB

of length

8

m and weight

120

N has its centre of mass

3

m from

A

. It rests horizontally on supports at

A

and

B

. Find the reactions at

A

and

B

●●●●●Level 33 marksStart

A non-uniform rod

AB

of length

5

m and weight

100

N has its centre of mass

2

m from

A

. It rests on supports at

C

and

D

, where

AC = 1

m and

AD = 4

m. A weight

W

hangs at

B

. Find the value of

W

for which the rod is on the point of tilting about

D

●●●●●Level 32 marksStart

A non-uniform rod

AB

of length

4

m and weight

90

N rests on supports at

A

and

B

. The reaction at

A

is twice the reaction at

B

. Find the distance of the centre of mass from

A

, to 3 s.f.

●●●●●Level 33 marksStart

A uniform beam

AB

of length

6

m and weight

300

N rests on supports at

C

and

D

, where

AC = 1

m and

AD = 5

m. A person of weight

700

N stands on the beam.
(a) Find the reactions at

C

and

D

when the person stands at the midpoint of the beam.
(b) Find how far from

A

the person can stand before the beam is on the point of tilting about

D

●●●●●Level 34 marksStart

A non-uniform rod

AB

of length

4

m and weight

80

N rests horizontally on supports at

A

and

B

. The reaction at

A

is three times the reaction at

B

.
(a) Find the reactions at

A

and

B

.
(b) Show that the centre of mass of the rod is

1

m from

A

●●●●●Level 34 marksStart

A uniform plank

AB

of length

5

m and weight

200

N rests on supports at

C

and

B

, where

AC = 2

m. A child of weight

300

N stands on the plank.
(a) Find the reactions at

C

and

B

when the child stands at the midpoint.
(b) The child walks towards

A

. Find the distance from

A

at which the plank is on the point of tilting about

C

●●●●●Level 34 marksStart

Figure 1 (not accurately drawn)

A non-uniform plank

AB

, of length

5a

, rests in equilibrium on a horizontal loading platform. The plank overhangs the edge of the platform at

C

, where

AC = 3a

and

CB = 2a

, as shown in Figure 1.

The plank is perpendicular to the edge of the platform. The plank has weight

W

and its centre of mass is at a distance

d

from

A

.

When a force of magnitude

F

, acting vertically upwards, is applied to the plank at

B

, the plank is on the point of tilting about

A

.

When this force is replaced by a force of magnitude

2F

, acting vertically downwards at

B

, the plank is on the point of tilting about

C

.

The plank is modelled as a non-uniform rod.

Find

d

in terms of

a

.

(6 marks)

●●●●●Level 36 marksStart

Figure 1 (not accurately drawn)

A non-uniform pole

AB

has length

5

m and mass

6

kg. The centre of mass of the pole is

2.2

m from

A

.

The pole lies on a horizontal flat roof and overhangs the edge of the roof at

C

1.4

m, as shown in Figure 1. The pole is perpendicular to the edge of the roof.

A particle of mass

9

kg is attached to the pole at

B

and a second particle, of mass

M

kg, is attached to the pole at the point

D

, where

AD = 0.6

m.

The pole remains in equilibrium in a horizontal position.
(a) Find the smallest possible value of

M

.

(3 marks)

Both particles are now removed. A new particle is attached to the pole at the point

E

, where

EB = 0.4

m.

The pole remains in equilibrium in a horizontal position but is now on the point of tilting about

C

.
(b) Find the magnitude of the force exerted on the pole by the roof at

C

.

(3 marks)

●●●●●Level 36 marksStart

A non-uniform rod

AB

of length

4

m rests horizontally on supports at

A

and

B

. The reaction at

A

30

N and at

B

50

N. Find the weight of the rod and the distance of its centre of mass from

A

●●●●●Level 33 marksStart

A non-uniform plank

AB

has length

6

m and mass

20

kg. Its centre of mass is at the point

G

, where

AG = 2.5

m. The plank rests horizontally on two supports at

C

and

D

, where

AC = 1

m and

AD = 4.5

m. Take

g = 9.8

m s⁻².(a) Find the magnitudes of the reactions at

C

and

D

. (5 marks)
(b) A child stands at

B

, and the plank is on the point of tilting about

D

. Modelling the child as a particle, find the child's mass. (4 marks)
(c) A different child, of mass

40

kg, walks along the plank from

D

towards

B

. Find how far from

A

they can walk before the plank starts to tilt. (3 marks)

●●●●●Level 412 marksStart

A non-uniform rod

AB

of length

6

m and mass

40

kg hangs horizontally from a ceiling, supported by two light vertical wires attached at

C

and

D

, where

AC = 1

m and

AD = 5

m. The tension in the wire at

D

is twice the tension in the wire at

C

. Take

g = 9.8

m s⁻².(a) Find the tension in each wire. (2 marks)
(b) Determine the distance of the centre of mass of the rod from

A

. (2 marks)

A block of mass

30

kg is now hung from the rod at

B

.
(c) Find the new tension in each wire. (3 marks)
(d) Determine the greatest mass that could instead be hung at

B

without the wire at

C

becoming slack. (2 marks)

●●●●●Level 49 marksStart

Figure 1 shows a rod

AB

of mass

M

and length

2a

.

The end

A

of the rod rests on rough horizontal ground.

A particle of mass

4M

is attached to the rod at the point

C

, where

AC = 1.25a

The rod is held in equilibrium at an angle

\theta

to the horizontal ground by a light string attached to the rod at

B

A

. Give a reason for your answer.

(1 mark)

In an initial model, the rod is modelled as being uniform.

Use this initial model to answer parts (b),
(c) and (d).

The tension in the string is

T

.
(b) By taking moments about

A

, show that

T = 3Mg\cos\theta

(3 marks)

The coefficient of friction between the rod and the ground is

\mu

Given that

\tan\theta = \dfrac{4}{3}

(c) find the value of

\mu

, giving your answer to 2 significant figures.

(5 marks)
(d) Find, in terms of

M

and

g

, the magnitude of the resultant force acting on the rod at

A

.

(3 marks)

In a new model, the rod is modelled as being non-uniform, with its centre of mass closer to

A

than it is to

B

.

A new value for

T

is calculated using this new model, with

\tan\theta = \dfrac{4}{3}

(e) State whether this new value for

T

is larger, smaller or equal to the value of

T

that would be found using the initial model. Give a reason for your answer.

(1 mark)

●●●●●Level 513 marksStart

Figure 4 shows a non-uniform plank

AB

of length

6

m and mass

30

kg resting horizontally on two supports at

C

and

D

, where

AC = 1

m and

DB = 1.5

m. The centre of mass of the plank is at the point

G

.Figure 4 (not accurately drawn)

A child of mass

25

kg stands on the plank at

A

. The plank is then on the point of tilting about

C

.
(a) Find the distance

AG

, giving your answer to

3

significant figures. (5 marks)

The child now stands at

B

instead.
(b) Find the magnitude of the reaction at

C

and the magnitude of the reaction at

D

. (6 marks)
(c) Find the greatest mass of a child who could stand at

B

without the plank tilting, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 515 marksStart

Figure 2 shows a non-uniform rod

AB

of length

4

m and mass

25

kg, hanging horizontally in equilibrium from two vertical strings attached at

A

and at the point

C

, where

AC = 3

m. The centre of mass of the rod is at

G

.Figure 2 (not accurately drawn)

The tension in the string at

A

98

N. Take

g = 9.8

m s⁻².
(a) Find the tension in the string at

C

. (2 marks)
(b) Find the distance

AG

. (3 marks)

A particle of mass

m

kg is now attached to the rod at

B

. The rod is then on the point of tilting about

C

.
(c) Find the value of

m

. (4 marks)

●●●●●Level 59 marksStart