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 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

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

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