Browse Moments

86 questions at your level

DifficultyAll Level 2 Level 3 Level 4 Level 5

Moments of forces

59 questions

LessonNot started

Figure 1

A uniform ladder

AB

5

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

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

3

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

●●●●●Level 35 marksStart

Figure 1

A uniform rod

AB

2.4

50

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

4

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

●●●●●Level 36 marksStart

Figure 1

A uniform rod

AB

5

80

N, rests with its end

A

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

C

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

●●●●●Level 37 marksStart

Figure 1

A uniform ladder

AB

W

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

●●●●●Level 37 marksStart

A uniform shelf

PQ

2L

metres and mass

m

kilograms.

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

C

D

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

L - x = k

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

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

20

N acts at the end of a rod

3

30^\circ

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

●●●●●Level 32 marksStart

A rod is pivoted at a point

O

10

N force acts downward

2

m to the left of

O

5

N force acts downward

3

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

O

●●●●●Level 32 marksStart

40

N acts at the end of a rod

2

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

AB

A

30

N acts downward at the point

2

A

, and a force of

20

B

4

A

40^\circ

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

A

●●●●●Level 32 marksStart

AB

4

120

N is pivoted at a point

1.5

A

. A vertical force

F

applied downward at

A

keeps the beam horizontal.

Find

F

●●●●●Level 34 marksStart

AB

6

300

N rests horizontally on supports at

A

B

. A child of weight

200

2

A

.

Find the reaction at each support. (5 marks)

●●●●●Level 35 marksStart

A uniform plank of length

4

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

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

5

100

N rests horizontally on supports at

A

B

. The reactions are

R_{A} = 60

R_{B} = 40

N.

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

A

●●●●●Level 34 marksStart

AB

8

400

N rests horizontally on supports at

C

D

AC = 1

AD = 6

m.
(a) Find the reactions at

C

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

M

2a

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

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

S = 2Mg\cot\theta

●●●●●Level 33 marksStart

Equilibrium of rigid bodies

38 questions

LessonNot started

A uniform shelf

PQ

2L

metres and mass

m

kilograms.

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

C

D

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

L - x = k

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

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

5a

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

C

AC = 3a

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

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

a

●●●●●Level 36 marksStart

Figure 1 (not accurately drawn)

A non-uniform pole

AB

5

6

kg. The centre of mass of the pole is

2.2

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

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

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

●●●●●Level 36 marksStart

AB

4

120

N rests on supports at

A

C

AC = 3

m. Find the reactions at

A

C

●●●●●Level 33 marksStart

A uniform plank

AB

5

300

N rests on supports at

A

B

. A child of weight

400

1

A

. Find the reactions at

A

B

●●●●●Level 33 marksStart

AB

8

160

N rests on supports at

A

B'

AB' = 6

m. Find the reactions at

A

B'

●●●●●Level 33 marksStart

AB

6

100

N rests on supports at

A

B

50

80

2

4

A

. Find the reactions at

A

B

●●●●●Level 33 marksStart

A uniform plank

AB

4

200

N rests on supports at

A

D

AD = 3

m. A person of weight

600

1

A

. Find the reactions at

A

D

●●●●●Level 33 marksStart

AB

4

m rests on supports at

A

B

30

50

1

3

A

. Find the reactions at

A

B

●●●●●Level 33 marksStart

AB

8

240

N rests on supports at

C

D

AC = 2

AD = 6

100

A

. Find the reactions at

C

D

●●●●●Level 33 marksStart

AB

8

120

N rests on supports at

C

D

AC = 2

AD = 6

m. A load of weight

W

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

M

2a

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

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

S = 2Mg\cot\theta

●●●●●Level 33 marksStart

Non-uniform rods and tilting

18 questions

LessonNot started

A non-uniform rod

AB

5

60

N balances horizontally on a single support placed

2

A

. State the distance of its centre of mass from

A

●●●●●Level 32 marksStart

AB

6

100

N rests on supports at

C

D

AC = 1

AD = 4

W

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

6

m rests horizontally on supports at

A

B

. The reactions are

R_A = 40

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

4

80

N rests on supports at

1

3

A

. A load is placed at

A

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

3

●●●●●Level 32 marksStart

A non-uniform rod

AB

8

120

N has its centre of mass

3

A

. It rests horizontally on supports at

A

B

. Find the reactions at

A

B

●●●●●Level 33 marksStart

A non-uniform rod

AB

5

100

N has its centre of mass

2

A

. It rests on supports at

C

D

AC = 1

AD = 4

W

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

4

90

N rests on supports at

A

B

. The reaction at

A

is twice the reaction at

B

. Find the distance of the centre of mass from

A

●●●●●Level 33 marksStart

AB

6

300

N rests on supports at

C

D

AC = 1

AD = 5

m. A person of weight

700

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

C

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

4

80

N rests horizontally on supports at

A

B

. The reaction at

A

is three times the reaction at

B

.
(a) Find the reactions at

A

B

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

1

A

●●●●●Level 34 marksStart

A uniform plank

AB

5

200

N rests on supports at

C

B

AC = 2

m. A child of weight

300

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

C

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

5a

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

C

AC = 3a

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

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

a

●●●●●Level 36 marksStart

Figure 1 (not accurately drawn)

A non-uniform pole

AB

5

6

kg. The centre of mass of the pole is

2.2

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

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

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

●●●●●Level 36 marksStart

A non-uniform rod

AB

4

m rests horizontally on supports at

A

B

. The reaction at

A

30

B

50

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

A

●●●●●Level 33 marksStart