Browse Forces & Newton's laws

182 questions at your level

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Newton's laws and F = ma

43 questions

LessonNot started

1200

kg travelling along a straight road inclined at an angle

\theta

to the horizontal, where

\sin\theta = \dfrac{1}{14}

.Figure 3 (not accurately drawn)

The car experiences a constant resistance to motion of

400

N. Take

g = 9.8

m s⁻².
(a) The car accelerates up the slope at

0.8

m s⁻². Find the magnitude of the driving force. (5 marks)
(b) When the car is travelling up the slope at

20

m s⁻¹ the engine is switched off. The resistance remains

400

N.

Find the distance the car travels up the slope before coming to instantaneous rest, giving your answer to

3

significant figures. (5 marks)
(c) Find the time taken for the car to come to instantaneous rest after the engine is switched off, giving your answer to

3

significant figures. (2 marks)

●●●●●Level 512 marksStart

Figure 1 shows a rough plane inclined to the horizontal at an angle

\alpha

, where

\tan\alpha = \dfrac{8}{15}

A small block

B

of mass

8

kg is held in equilibrium on the plane by a horizontal force of magnitude

X

newtons, as shown in Figure 1.

The block

B

is modelled as a particle.

The force acts in a vertical plane which contains a line of greatest slope of the inclined plane.

The magnitude of the normal reaction of the plane on

B

96

N.Figure 1 — not accurately drawn

Using the model,
(a) find the magnitude of the frictional force acting on

B

,

(3 marks)
(b) state the direction of the frictional force acting on

B

.

(1 mark)

The horizontal force of magnitude

X

newtons is now removed and

B

moves down the plane.

Given that the coefficient of friction between

B

and the plane is

0.28

(c) find the acceleration of

B

down the plane.

(6 marks)

●●●●●Level 510 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

A child's wooden toy comprises a tractor and a cart connected to each other by a light, inextensible rod.

The tractor has a mass of

0.6

kilograms. The cart has a mass of

0.4

kilograms.

A string inclined at an angle of

30^\circ

above the horizontal is attached to the front of the tractor. The tension in the string is

5

newtons.

As a result the toy moves forward, from rest, in a straight line along a horizontal floor with acceleration

1.5\ \text{m s}^{-2}

as shown in the diagram below.As it moves the tractor experiences a total resistance force of

1.2

N
(a) Show that the total resistance force experienced by the cart is approximately

1.6

N

(4 marks)
(b) At the instant that the toy reaches a speed of

2.4\ \text{m s}^{-1}

the string breaks.

The tractor and cart then slow down with constant deceleration, travelling a further

h

metres before coming to rest.

You may assume that the resistance forces on the tractor and on the cart do not change.
(b) (i) Find the tension in the rod after the string has broken.

(ii) Find

h

(7 marks)
(c) State one modelling assumption that you have used about the rod when answering part (b)(i).

(1 mark)

●●●●●Level 512 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

A rough plank is

5

metres long and is inclined at an angle of

20^\circ

above the horizontal. The bottom of the plank is at the point

O

.

A sack of gravel of mass

30

kg is at rest at the point

A

on the plank. The sack is pulled up the plank using a rope attached to the sack.

Once in motion, the rope remains taut and parallel to the line of greatest slope of the plank.
(a) The tension in the rope is

200

N. The sack accelerates up the plank at

0.8\ \text{m s}^{-2}

Find the coefficient of friction between the sack and the plank.

(7 marks)
(b) (i) The sack takes

3

seconds to reach the top of the plank.

Find the distance

OA

.

(ii) Other than air resistance, state one assumption you have made about the sack in answering part (b)(i).

(4 marks)

●●●●●Level 511 marksStart

A sack of logs,

M

, and a bucket of stones,

N

, are connected securely by a length of rope. The mass of

M

20

kilograms. The mass of

N

30

kilograms.

M

is placed near the bottom of a rough slope. The slope is inclined at

60^\circ

above the horizontal.

The rope is passed over a smooth pulley at the top end of the slope so that

N

hangs with the rope vertical.

M

and

N

are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below.When

M

and

N

are released,

M

moves up the slope as

N

descends, with acceleration

a\ \text{m s}^{-2}

The tension in the rope is

T

newtons.
(a) Explain why the equation of motion for

N

30g - T = 30a

(1 mark)
(b) Show that the normal reaction force between

M

and the slope is

10g

newtons.

(1 mark)
(c) The coefficient of friction,

\mu

, between the slope and

M

is such that

0 \leq \mu \leq 1

Show that

a \geq \left(0.4 - 0.2\sqrt{3}\right)g

(6 marks)
(d) State one modelling assumption you have made throughout this question.

(1 mark)

●●●●●Level 59 marksStart

Figure 1 (not accurately drawn)

A sledge

P

of mass

m

is pulled up a rough slope by a rope. The slope is modelled as a rough plane inclined to the horizontal at an angle

\alpha

, where

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

The sledge is modelled as a particle and the rope is modelled as a light inextensible string.

The rope is inclined to the slope at an angle

\beta

, where

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

, and the tension in the rope is

\frac{4}{5}mg

, as shown in Figure 1. The rope lies in the vertical plane containing a line of greatest slope of the plane.

The coefficient of friction between the sledge and the slope is

\frac{1}{4}

The sledge starts from rest at the point

A

on the slope and moves up the line of greatest slope through

A

.
(a) Show that the acceleration of the sledge is

\frac{6}{25}g

(8 marks)

The sledge passes through the point

B

, where

AB = 3

m.
(b) Find the speed of the sledge at

B

.

(2 marks)

When the sledge reaches

B

, the rope is released. The sledge continues to move up the slope and comes to instantaneous rest at the point

C

.
(c) Find the distance

BC

.

(3 marks)
(d) Determine whether the sledge remains at rest at

C

. You must justify your reasoning.

(2 marks)

●●●●●Level 515 marksStart

In this question the unit vectors

\mathbf{i}

and

\mathbf{j}

are in the directions east and north respectively.

A particle

P

of mass

1.2

kg is moving on a smooth horizontal surface under the action of a single horizontal force

\mathbf{F}

N. At time

t

seconds, where

t \geq 0

, the velocity

\mathbf{v}\ \text{m s}^{-1}

P

, relative to a fixed origin

O

, is given by

\mathbf{v} = (t^2 - 8t + 20)\mathbf{i} + (t - 6)\mathbf{j}.

(a) Show that

P

is never stationary.

(2 marks)
(b) Find, in terms of

\mathbf{i}

and

\mathbf{j}

, the acceleration of

P

at time

t

.

(1 mark)
(c) Determine the magnitude of

\mathbf{F}

when

P

is moving in the direction of the vector

5\mathbf{i} - \mathbf{j}

. Give your answer correct to 3 significant figures.

(5 marks)

When

t = 3

P

is at the point with position vector

(-9\mathbf{i} + 1.5\mathbf{j})

m.
(d) Determine the bearing of

P

from

O

at time

t = 6

.

(5 marks)

●●●●●Level 513 marksStart

Figure 3 shows a block of mass

15

kg resting on a rough horizontal floor. A force of magnitude

P

N is applied to the block at an angle of

20^\circ

above the horizontal.Figure 3 (not accurately drawn)

The coefficient of friction between the block and the floor is

0.35

. Take

g = 9.8

m s⁻².
(a) Find the value of

P

for which the block is on the point of moving, giving your answer to

3

significant figures. (6 marks)
(b) Find the acceleration of the block when

P = 70

, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 510 marksStart

Figure 1Diagram not accurately drawn.

A particle

A

of mass

3m

is held at rest on a rough plane which is inclined to horizontal ground at an angle

\alpha

, where

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

One end of a light inextensible string is attached to

A

. The string passes over a small smooth pulley which is fixed at the top of the plane, and the other end of the string is attached to a particle

B

of mass

4m

. Particle

B

hangs vertically below the pulley, with the string taut, at a height

h

above the ground, as shown in Figure 1.

The part of the string between

A

and the pulley lies along a line of greatest slope of the plane, and

A

B

and the pulley all lie in the same vertical plane.

The coefficient of friction between

A

and the plane is

\frac{1}{4}

Particle

A

is released from rest and begins to move up the plane.
(a) Show that, as

A

moves up the plane, the frictional force acting on

A

has magnitude

\frac{9}{13}mg

(3 marks)
(b) Write down an equation of motion for

B

.

(2 marks)
(c) Show that the acceleration of

A

immediately after its release is

\frac{4}{13}g

(4 marks)

Particle

B

hits the ground and does not rebound. In the subsequent motion,

A

comes to rest before it reaches the pulley.
(d) Find, in terms of

h

, the total distance travelled by

A

from the instant it is released to the instant it first comes to rest.

(6 marks)

●●●●●Level 515 marksStart

Figure 1Diagram not accurately drawn.

A particle

A

of mass

4m

is held at rest on a rough horizontal table. One end of a light inextensible string is attached to

A

. The string passes over a small smooth pulley which is fixed at the edge of the table, and the other end of the string is attached to a particle

B

of mass

3m

. Particle

B

hangs vertically below the pulley at a height

h

above the horizontal floor, as shown in Figure 1.

The string between

A

and the pulley is horizontal and perpendicular to the edge of the table. The particles, the string and the pulley all lie in the same vertical plane.

The coefficient of friction between

A

and the table is

\mu

.

Particle

A

is released from rest with the string taut. Before

B

reaches the floor, the tension in the string is

\frac{15}{7}mg

(a) Write down an equation of motion for

B

.

(2 marks)
(b) Find the value of

\mu

.

(4 marks)
(c) Find, in terms of

m

and

g

, the magnitude of the force exerted on the pulley by the string before

B

reaches the floor.

(3 marks)

Particle

B

hits the floor and does not rebound. In the subsequent motion,

A

does not reach the pulley.
(d) Find, in terms of

h

and

g

, the time from the instant

A

is released to the instant

A

comes to rest.

(5 marks)

●●●●●Level 514 marksStart

Figure 2Diagram not accurately drawn.

Two small blocks,

A

and

B

, are at rest on a rough horizontal floor. Block

A

has mass

0.6

kg and block

B

has mass

0.9

kg. The distance between

A

and

B

1.25

m, as shown in Figure 2. The blocks are modelled as particles.

The coefficient of friction between each block and the floor is

\frac{2}{7}

Block

A

is hit so that it receives an impulse of magnitude

2.4

N s in the direction

AB

.
(a) Find the speed of

A

immediately after the impulse is applied.

(2 marks)
(b) Show that the speed of

A

immediately before it collides with

B

3\text{ m s}^{-1}

(5 marks)

In the collision,

A

and

B

join together and move as a single particle.
(c) Find the speed of the combined particle immediately after the collision.

(2 marks)
(d) Find the magnitude of the impulse exerted on

A

B

in the collision, stating the units of your answer.

(2 marks)
(e) Find the distance travelled by the combined particle after the collision until it comes to rest.

(3 marks)

●●●●●Level 514 marksStart

In this question the unit vectors

\mathbf{i}

and

\mathbf{j}

are in the directions east and north respectively.

A particle

P

of mass

2.5

kg moves on a smooth horizontal surface. At time

t

seconds, where

t \geq 0

, the only horizontal forces acting on

P

are a constant force

(-9\mathbf{i} - 12.5\mathbf{j})

N and a force

\left(3t\mathbf{i} + (1.5t^2 - 1)\mathbf{j}\right)

N which varies with time.
(a) Determine the value of

t

when the forces acting on

P

are in equilibrium.

(2 marks)

It is given that

P

is at rest when

t = 0

.
(b) Determine the speed of

P

at the instant when

P

is moving due north.

(6 marks)
(c) Determine the distance between the positions of

P

when

t = 0

and

t = 5

.

(5 marks)

●●●●●Level 513 marksStart

Figure 1Diagram not accurately drawn.

A small block

A

has mass

2M

and a small ball

B

has mass

kM

, where

k > 2

. The block and the ball are attached to the ends of a light inextensible string that passes over a small smooth fixed pulley.

The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The system is released from rest and, in the subsequent motion,

A

moves upwards with an acceleration of magnitude

\frac{3}{7}g

The block and the ball are modelled as particles.
(a) Write down an equation of motion for

A

.

(2 marks)
(b) Find the value of

k

.

(3 marks)

Given that

M = 0.4

kg,
(c) find the magnitude of the force exerted on the pulley by the string while

B

is moving downwards.

(3 marks)

At the instant when the system is released,

B

1.4

m above horizontal ground and

A

is more than

1.4

m below the pulley. After hitting the ground,

B

rebounds with speed

0.7\ \text{m s}^{-1}

(d) Find the magnitude of the impulse received by

B

when it hits the ground.

(5 marks)

In the subsequent motion,

A

does not reach the pulley.
(e) Find the total time from the instant the system is released until

A

first comes to rest.

(4 marks)

●●●●●Level 517 marksStart

Connected particles

19 questions

LessonNot started

Figure 3 shows a block

A

of mass

4

kg lying on a rough horizontal table. The block is attached to one end of a light inextensible string which passes over a smooth pulley fixed at the edge of the table. The other end of the string is attached to a block

B

of mass

3

kg, which hangs freely.Figure 3 (not accurately drawn)

The coefficient of friction between

A

and the table is

0.25

. The system is released from rest with the string taut.
(a) Find the acceleration of the system and the tension in the string. (7 marks)
(b)

B

hits the ground after falling

1.5

m. Find the speed of

B

as it hits the ground, giving your answer to

3

significant figures. (3 marks)
(c) After

B

hits the ground the string becomes slack and

A

continues to move. Find the further distance

A

travels before coming to rest, giving your answer to

3

significant figures. (2 marks)

●●●●●Level 512 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

A child's wooden toy comprises a tractor and a cart connected to each other by a light, inextensible rod.

The tractor has a mass of

0.6

kilograms. The cart has a mass of

0.4

kilograms.

A string inclined at an angle of

30^\circ

above the horizontal is attached to the front of the tractor. The tension in the string is

5

newtons.

As a result the toy moves forward, from rest, in a straight line along a horizontal floor with acceleration

1.5\ \text{m s}^{-2}

as shown in the diagram below.As it moves the tractor experiences a total resistance force of

1.2

N
(a) Show that the total resistance force experienced by the cart is approximately

1.6

N

(4 marks)
(b) At the instant that the toy reaches a speed of

2.4\ \text{m s}^{-1}

the string breaks.

The tractor and cart then slow down with constant deceleration, travelling a further

h

metres before coming to rest.

You may assume that the resistance forces on the tractor and on the cart do not change.
(b) (i) Find the tension in the rod after the string has broken.

(ii) Find

h

(7 marks)
(c) State one modelling assumption that you have used about the rod when answering part (b)(i).

(1 mark)

●●●●●Level 512 marksStart

A van of mass

1500

kg is moving along a straight horizontal road. The engine of the van is working at a constant rate of

24

kW. The resistance to the motion of the van is modelled as a constant force of magnitude

R

newtons.

At the instant when the speed of the van is

20\ \text{m s}^{-1}

, the acceleration of the van is

0.4\ \text{m s}^{-2}

(a) Show that

R = 600

(4 marks)

Later on, the van is towing a horse trailer of mass

800

kg up a straight road inclined at an angle

\alpha

to the horizontal, where

\sin\alpha = \frac{1}{30}

, as shown in Figure 1.

Figure 1Diagram not accurately drawn.

The trailer is attached to the van by a towbar which is parallel to the direction of motion of the van and the trailer. The towbar is modelled as a light rod.

The resistance to the motion from non-gravitational forces is modelled as

a constant force of magnitude $600$ N on the van
a constant force of magnitude $300$ N on the trailer

The engine of the van is working at a constant rate of

24

kW.
(b) Find the acceleration of the van at the instant when the van and the trailer are moving with speed

10\ \text{m s}^{-1}

(4 marks)
(c) Find the tension in the towbar at this instant.

(3 marks)

At the instant when the van and the trailer are moving up the road at

11\ \text{m s}^{-1}

, the towbar breaks. The trailer continues to move in a straight line up the road until it comes to instantaneous rest. The resistance to the motion of the trailer is unchanged.
(d) Use the work–energy principle to find the distance moved by the trailer from the instant the towbar breaks to the instant it comes to rest.

(4 marks)

●●●●●Level 515 marksStart

Figure 1Diagram not accurately drawn.

A particle

A

of mass

4m

is held at rest on a rough horizontal table. One end of a light inextensible string is attached to

A

. The string passes over a small smooth pulley which is fixed at the edge of the table, and the other end of the string is attached to a particle

B

of mass

3m

. Particle

B

hangs vertically below the pulley at a height

h

above the horizontal floor, as shown in Figure 1.

The string between

A

and the pulley is horizontal and perpendicular to the edge of the table. The particles, the string and the pulley all lie in the same vertical plane.

The coefficient of friction between

A

and the table is

\mu

.

Particle

A

is released from rest with the string taut. Before

B

reaches the floor, the tension in the string is

\frac{15}{7}mg

(a) Write down an equation of motion for

B

.

(2 marks)
(b) Find the value of

\mu

.

(4 marks)
(c) Find, in terms of

m

and

g

, the magnitude of the force exerted on the pulley by the string before

B

reaches the floor.

(3 marks)

Particle

B

hits the floor and does not rebound. In the subsequent motion,

A

does not reach the pulley.
(d) Find, in terms of

h

and

g

, the time from the instant

A

is released to the instant

A

comes to rest.

(5 marks)

●●●●●Level 514 marksStart

Figure 1Diagram not accurately drawn.

A small block

A

has mass

2M

and a small ball

B

has mass

kM

, where

k > 2

A

moves upwards with an acceleration of magnitude

\frac{3}{7}g

The block and the ball are modelled as particles.
(a) Write down an equation of motion for

A

.

(2 marks)
(b) Find the value of

k

.

(3 marks)

Given that

M = 0.4

kg,
(c) find the magnitude of the force exerted on the pulley by the string while

B

is moving downwards.

(3 marks)

At the instant when the system is released,

B

1.4

m above horizontal ground and

A

is more than

1.4

m below the pulley. After hitting the ground,

B

rebounds with speed

0.7\ \text{m s}^{-1}

(d) Find the magnitude of the impulse received by

B

when it hits the ground.

(5 marks)

In the subsequent motion,

A

does not reach the pulley.
(e) Find the total time from the instant the system is released until

A

first comes to rest.

(4 marks)

●●●●●Level 517 marksStart

Pulleys

22 questions

LessonNot started

Figure 1 shows a particle

A

of mass

3

kg on a plane inclined at

30^\circ

to the horizontal.

A

is connected by a light inextensible string, passing over a smooth pulley at the top of the plane, to a particle

B

of mass

5

kg which hangs freely.Figure 1 (not accurately drawn)

The system is released from rest with the string taut.
(a) Given that the plane is smooth, find the acceleration of the system and the tension in the string, each to

3

significant figures. (6 marks)
(b) The plane is now rough, with coefficient of friction

0.2

between

A

and the plane. Find the new acceleration of the system, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 510 marksStart

Two particles

P

and

Q

, of masses

3

kg and

5

kg respectively, are connected by a light inextensible string. Particle

P

lies on a rough plane inclined at an angle

\alpha

to the horizontal, where

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

. The string passes over a small smooth pulley fixed at the top edge of the plane, and

Q

hangs freely below the pulley. The part of the string from

P

to the pulley lies along a line of greatest slope of the plane. The coefficient of friction between

P

and the plane is

\dfrac{1}{2}

.

Initially

P

2.5

m from the pulley and

Q

1.6

m above horizontal ground. The system is released from rest with the string taut. Take

g = 9.8

m s⁻².
(a) Show that the acceleration of

P

2.45

m s⁻² and find the tension in the string. (6 marks)
(b) Find the time taken, from release, for

Q

to reach the ground, and the speed of

Q

at that instant. (3 marks)
(c)

Q

does not rebound when it hits the ground. Find the distance

P

moves up the plane after

Q

hits the ground before

P

first comes to instantaneous rest. (3 marks)
(d) Explain why

P

does not remain at rest at this position. (1 mark)

●●●●●Level 513 marksStart

Figure 3 shows a block

A

of mass

4

B

of mass

3

kg, which hangs freely.Figure 3 (not accurately drawn)

The coefficient of friction between

A

and the table is

0.25

. The system is released from rest with the string taut.
(a) Find the acceleration of the system and the tension in the string. (7 marks)
(b)

B

hits the ground after falling

1.5

m. Find the speed of

B

as it hits the ground, giving your answer to

3

significant figures. (3 marks)
(c) After

B

hits the ground the string becomes slack and

A

continues to move. Find the further distance

A

travels before coming to rest, giving your answer to

3

significant figures. (2 marks)

●●●●●Level 512 marksStart

A sack of logs,

M

, and a bucket of stones,

N

, are connected securely by a length of rope. The mass of

M

20

kilograms. The mass of

N

30

kilograms.

M

is placed near the bottom of a rough slope. The slope is inclined at

60^\circ

above the horizontal.

The rope is passed over a smooth pulley at the top end of the slope so that

N

hangs with the rope vertical.

M

and

N

are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below.When

M

and

N

are released,

M

moves up the slope as

N

descends, with acceleration

a\ \text{m s}^{-2}

The tension in the rope is

T

newtons.
(a) Explain why the equation of motion for

N

30g - T = 30a

(1 mark)
(b) Show that the normal reaction force between

M

and the slope is

10g

newtons.

(1 mark)
(c) The coefficient of friction,

\mu

, between the slope and

M

is such that

0 \leq \mu \leq 1

Show that

a \geq \left(0.4 - 0.2\sqrt{3}\right)g

(6 marks)
(d) State one modelling assumption you have made throughout this question.

(1 mark)

●●●●●Level 59 marksStart

Particles

A

and

B

, of masses

3

kg and

4

kg respectively, are attached to the ends of a light inextensible string.

A

is in contact with a rough plane inclined at

\theta

to the horizontal, where

\sin\theta = \frac{7}{25}

. The string is taut and passes over a small smooth pulley

P

at the top of the plane, with the part of the string from

A

P

parallel to a line of greatest slope of the plane.

B

hangs freely below

P

1.25

m above horizontal ground, as shown in the diagram.The coefficient of friction between

A

and the plane is

\mu

. The system is released from rest and in the subsequent motion

B

hits the ground before

A

reaches

P

. The speed of

B

at the instant that it hits the ground is

2.5

m s

^{-1}

.
(a) For the motion before

B

hits the ground, show that the acceleration of

B

2.5

m s

^{-2}

.

(1 mark)
(b) For the motion before

B

hits the ground, show that the tension in the string is

29.2

N.

(3 marks)
(c) Determine the value of

\mu

.

(5 marks)

After

B

hits the ground,

A

continues to move up the plane and comes to instantaneous rest before it reaches

P

.
(d) Determine the distance that

A

moves up the plane from the instant that

B

hits the ground until

A

comes to instantaneous rest.

(4 marks)

●●●●●Level 513 marksStart

Figure 1Diagram not accurately drawn.

A particle

A

of mass

3m

is held at rest on a rough plane which is inclined to horizontal ground at an angle

\alpha

, where

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

One end of a light inextensible string is attached to

A

. The string passes over a small smooth pulley which is fixed at the top of the plane, and the other end of the string is attached to a particle

B

of mass

4m

. Particle

B

hangs vertically below the pulley, with the string taut, at a height

h

above the ground, as shown in Figure 1.

The part of the string between

A

and the pulley lies along a line of greatest slope of the plane, and

A

B

and the pulley all lie in the same vertical plane.

The coefficient of friction between

A

and the plane is

\frac{1}{4}

Particle

A

is released from rest and begins to move up the plane.
(a) Show that, as

A

moves up the plane, the frictional force acting on

A

has magnitude

\frac{9}{13}mg

(3 marks)
(b) Write down an equation of motion for

B

.

(2 marks)
(c) Show that the acceleration of

A

immediately after its release is

\frac{4}{13}g

(4 marks)

Particle

B

hits the ground and does not rebound. In the subsequent motion,

A

comes to rest before it reaches the pulley.
(d) Find, in terms of

h

, the total distance travelled by

A

from the instant it is released to the instant it first comes to rest.

(6 marks)

●●●●●Level 515 marksStart

Figure 1Diagram not accurately drawn.

A particle

A

of mass

4m

is held at rest on a rough horizontal table. One end of a light inextensible string is attached to

A

. The string passes over a small smooth pulley which is fixed at the edge of the table, and the other end of the string is attached to a particle

B

of mass

3m

. Particle

B

hangs vertically below the pulley at a height

h

above the horizontal floor, as shown in Figure 1.

The string between

A

and the pulley is horizontal and perpendicular to the edge of the table. The particles, the string and the pulley all lie in the same vertical plane.

The coefficient of friction between

A

and the table is

\mu

.

Particle

A

is released from rest with the string taut. Before

B

reaches the floor, the tension in the string is

\frac{15}{7}mg

(a) Write down an equation of motion for

B

.

(2 marks)
(b) Find the value of

\mu

.

(4 marks)
(c) Find, in terms of

m

and

g

, the magnitude of the force exerted on the pulley by the string before

B

reaches the floor.

(3 marks)

Particle

B

hits the floor and does not rebound. In the subsequent motion,

A

does not reach the pulley.
(d) Find, in terms of

h

and

g

, the time from the instant

A

is released to the instant

A

comes to rest.

(5 marks)

●●●●●Level 514 marksStart

A fixed wedge has two plane faces which meet along a horizontal ridge. One face is inclined at

\alpha

to the horizontal and the other at

\beta

to the horizontal, where

\sin\alpha = \frac{12}{13}

and

\sin\beta = \frac{5}{13}

. Both faces are rough.

A particle

A

of mass

6

kg on the first face and a particle

B

of mass

3

kg on the second face are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the ridge. The parts of the string from

A

to the pulley and from

B

to the pulley are parallel to lines of greatest slope of the corresponding faces (see diagram).The coefficient of friction between

A

and the first face is

12\mu

and the coefficient of friction between

B

and the second face is

5\mu

, where

\mu

is a constant.

The system is released from rest, and in the subsequent motion

A

moves down the first face and

B

does not reach the pulley.
(a) Determine, in terms of

g

, the tension in the string.

(7 marks)

When

A

is moving at

1.5

m s

^{-1}

the string breaks. In the

0.4

seconds after the string breaks

A

moves

1.15

m down the first face.
(b) Determine the deceleration of

B

after the string breaks. Give your answer correct to 3 significant figures.

(5 marks)

●●●●●Level 512 marksStart

Figure 1Diagram not accurately drawn.

A small block

A

has mass

2M

and a small ball

B

has mass

kM

, where

k > 2

A

moves upwards with an acceleration of magnitude

\frac{3}{7}g

The block and the ball are modelled as particles.
(a) Write down an equation of motion for

A

.

(2 marks)
(b) Find the value of

k

.

(3 marks)

Given that

M = 0.4

kg,
(c) find the magnitude of the force exerted on the pulley by the string while

B

is moving downwards.

(3 marks)

At the instant when the system is released,

B

1.4

m above horizontal ground and

A

is more than

1.4

m below the pulley. After hitting the ground,

B

rebounds with speed

0.7\ \text{m s}^{-1}

(d) Find the magnitude of the impulse received by

B

when it hits the ground.

(5 marks)

In the subsequent motion,

A

does not reach the pulley.
(e) Find the total time from the instant the system is released until

A

first comes to rest.

(4 marks)

●●●●●Level 517 marksStart

Resolving forces

42 questions

LessonNot started

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

A box of mass

8

kg rests on a rough horizontal floor. The coefficient of friction between the box and the floor is

0.25

. A rope pulls the box with a force of

40

N at an angle of

30^\circ

above the horizontal. Take

g = 9.8

m s⁻².(a) Show that the normal reaction of the floor on the box is

58.4

N. (3 marks)
(b) Explain why the normal reaction is less than the weight of the box, and find the acceleration of the box. (6 marks)
(c) When the box is moving at

6

m s⁻¹, the rope is released. Find the distance the box travels before coming to rest. (4 marks)

●●●●●Level 513 marksStart

Figure 3 shows a car of mass

1200

kg travelling along a straight road inclined at an angle

\theta

to the horizontal, where

\sin\theta = \dfrac{1}{14}

.Figure 3 (not accurately drawn)

The car experiences a constant resistance to motion of

400

N. Take

g = 9.8

m s⁻².
(a) The car accelerates up the slope at

0.8

m s⁻². Find the magnitude of the driving force. (5 marks)
(b) When the car is travelling up the slope at

20

m s⁻¹ the engine is switched off. The resistance remains

400

N.

Find the distance the car travels up the slope before coming to instantaneous rest, giving your answer to

3

significant figures. (5 marks)
(c) Find the time taken for the car to come to instantaneous rest after the engine is switched off, giving your answer to

3

significant figures. (2 marks)

●●●●●Level 512 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 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

A parcel of mass

11

kg is at rest on a rough plane inclined at an angle

\alpha

to the horizontal, where

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

. The coefficient of friction between the parcel and the plane is

\dfrac{1}{2}

.

A horizontal force of magnitude

P

newtons acts on the parcel, in the vertical plane containing a line of greatest slope of the inclined plane, in the direction that tends to push the parcel up the slope. Take

g = 9.8

m s⁻².
(a) Given that the parcel is on the point of sliding up the plane, show that

P = 215.6

. (5 marks)
(b) Find the range of values of

P

for which the parcel remains at rest. (4 marks)
(c) The force is removed. Show that the parcel begins to slide down the plane, and find its speed after it has travelled

2

m. (4 marks)

●●●●●Level 513 marksStart

Figure 1 shows a rough plane inclined to the horizontal at an angle

\alpha

, where

\tan\alpha = \dfrac{8}{15}

A small block

B

of mass

8

kg is held in equilibrium on the plane by a horizontal force of magnitude

X

newtons, as shown in Figure 1.

The block

B

is modelled as a particle.

The force acts in a vertical plane which contains a line of greatest slope of the inclined plane.

The magnitude of the normal reaction of the plane on

B

96

N.Figure 1 — not accurately drawn

Using the model,
(a) find the magnitude of the frictional force acting on

B

,

(3 marks)
(b) state the direction of the frictional force acting on

B

.

(1 mark)

The horizontal force of magnitude

X

newtons is now removed and

B

moves down the plane.

Given that the coefficient of friction between

B

and the plane is

0.28

(c) find the acceleration of

B

down the plane.

(6 marks)

●●●●●Level 510 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

A child's wooden toy comprises a tractor and a cart connected to each other by a light, inextensible rod.

The tractor has a mass of

0.6

kilograms. The cart has a mass of

0.4

kilograms.

A string inclined at an angle of

30^\circ

above the horizontal is attached to the front of the tractor. The tension in the string is

5

newtons.

As a result the toy moves forward, from rest, in a straight line along a horizontal floor with acceleration

1.5\ \text{m s}^{-2}

as shown in the diagram below.As it moves the tractor experiences a total resistance force of

1.2

N
(a) Show that the total resistance force experienced by the cart is approximately

1.6

N

(4 marks)
(b) At the instant that the toy reaches a speed of

2.4\ \text{m s}^{-1}

the string breaks.

The tractor and cart then slow down with constant deceleration, travelling a further

h

metres before coming to rest.

You may assume that the resistance forces on the tractor and on the cart do not change.
(b) (i) Find the tension in the rod after the string has broken.

(ii) Find

h

(7 marks)
(c) State one modelling assumption that you have used about the rod when answering part (b)(i).

(1 mark)

●●●●●Level 512 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

Figure 3 shows a block of mass

15

kg resting on a rough horizontal floor. A force of magnitude

P

N is applied to the block at an angle of

20^\circ

above the horizontal.Figure 3 (not accurately drawn)

The coefficient of friction between the block and the floor is

0.35

. Take

g = 9.8

m s⁻².
(a) Find the value of

P

for which the block is on the point of moving, giving your answer to

3

significant figures. (6 marks)
(b) Find the acceleration of the block when

P = 70

, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 510 marksStart

In this question the unit vectors

\mathbf{i}

and

\mathbf{j}

are in the directions east and north respectively.

A particle

P

of mass

2.5

kg moves on a smooth horizontal surface. At time

t

seconds, where

t \geq 0

, the only horizontal forces acting on

P

are a constant force

(-9\mathbf{i} - 12.5\mathbf{j})

N and a force

\left(3t\mathbf{i} + (1.5t^2 - 1)\mathbf{j}\right)

N which varies with time.
(a) Determine the value of

t

when the forces acting on

P

are in equilibrium.

(2 marks)

It is given that

P

is at rest when

t = 0

.
(b) Determine the speed of

P

at the instant when

P

is moving due north.

(6 marks)
(c) Determine the distance between the positions of

P

when

t = 0

and

t = 5

.

(5 marks)

●●●●●Level 513 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

Forces on an inclined plane

29 questions

LessonNot started

A parcel of mass

11

kg is at rest on a rough plane inclined at an angle

\alpha

to the horizontal, where

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

. The coefficient of friction between the parcel and the plane is

\dfrac{1}{2}

.

A horizontal force of magnitude

P

newtons acts on the parcel, in the vertical plane containing a line of greatest slope of the inclined plane, in the direction that tends to push the parcel up the slope. Take

g = 9.8

m s⁻².
(a) Given that the parcel is on the point of sliding up the plane, show that

P = 215.6

. (5 marks)
(b) Find the range of values of

P

for which the parcel remains at rest. (4 marks)
(c) The force is removed. Show that the parcel begins to slide down the plane, and find its speed after it has travelled

2

m. (4 marks)

●●●●●Level 513 marksStart

Figure 1 shows a rough plane inclined to the horizontal at an angle

\alpha

, where

\tan\alpha = \dfrac{8}{15}

A small block

B

of mass

8

kg is held in equilibrium on the plane by a horizontal force of magnitude

X

newtons, as shown in Figure 1.

The block

B

is modelled as a particle.

The force acts in a vertical plane which contains a line of greatest slope of the inclined plane.

The magnitude of the normal reaction of the plane on

B

96

N.Figure 1 — not accurately drawn

Using the model,
(a) find the magnitude of the frictional force acting on

B

,

(3 marks)
(b) state the direction of the frictional force acting on

B

.

(1 mark)

The horizontal force of magnitude

X

newtons is now removed and

B

moves down the plane.

Given that the coefficient of friction between

B

and the plane is

0.28

(c) find the acceleration of

B

down the plane.

(6 marks)

●●●●●Level 510 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

A rough plank is

5

metres long and is inclined at an angle of

20^\circ

above the horizontal. The bottom of the plank is at the point

O

.

A sack of gravel of mass

30

kg is at rest at the point

A

200

N. The sack accelerates up the plank at

0.8\ \text{m s}^{-2}

Find the coefficient of friction between the sack and the plank.

(7 marks)
(b) (i) The sack takes

3

seconds to reach the top of the plank.

Find the distance

OA

.

(ii) Other than air resistance, state one assumption you have made about the sack in answering part (b)(i).

(4 marks)

●●●●●Level 511 marksStart

A box of mass

6

kg is pulled at constant speed up a rough plane inclined at

20^\circ

to the horizontal, by a force of magnitude

P

newtons acting parallel to the plane.

The coefficient of friction between the box and the plane is

0.3

. (

g = 9.8\ \text{m s}^{-2}

.)

Find the value of

P

, giving your answer correct to 3 significant figures. (6 marks)

●●●●●Level 56 marksStart

Figure 1 (not accurately drawn)

A sledge

P

of mass

m

is pulled up a rough slope by a rope. The slope is modelled as a rough plane inclined to the horizontal at an angle

\alpha

, where

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

The sledge is modelled as a particle and the rope is modelled as a light inextensible string.

The rope is inclined to the slope at an angle

\beta

, where

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

, and the tension in the rope is

\frac{4}{5}mg

, as shown in Figure 1. The rope lies in the vertical plane containing a line of greatest slope of the plane.

The coefficient of friction between the sledge and the slope is

\frac{1}{4}

The sledge starts from rest at the point

A

on the slope and moves up the line of greatest slope through

A

.
(a) Show that the acceleration of the sledge is

\frac{6}{25}g

(8 marks)

The sledge passes through the point

B

, where

AB = 3

m.
(b) Find the speed of the sledge at

B

.

(2 marks)

When the sledge reaches

B

, the rope is released. The sledge continues to move up the slope and comes to instantaneous rest at the point

C

.
(c) Find the distance

BC

.

(3 marks)
(d) Determine whether the sledge remains at rest at

C

. You must justify your reasoning.

(2 marks)

●●●●●Level 515 marksStart

The points

A

and

B

are the lower and upper ends, respectively, of a line of greatest slope on a plane inclined at an angle

\theta

to the horizontal, where

\sin\theta = \frac{8}{17}

and

AB = 2.55

m (see diagram).A particle

P

is projected up the plane with speed

7

m s

^{-1}

from

A

towards

B

.

The plane at

A

is fixed to horizontal ground.

The surface of the plane is rough and the coefficient of friction between

P

and the plane is

0.3

.
(a) Show that the speed of

P

B

3.5

m s

^{-1}

.

(6 marks)

The particle leaves the plane at

B

and moves freely under gravity. It first lands at a point

C

on the horizontal ground. The time taken for

P

to travel from

A

C

T

seconds.
(b) Determine the value of

T

.

(6 marks)

●●●●●Level 512 marksStart

A cyclist and her bicycle have a combined mass of

72

kg. The cyclist is riding along a straight horizontal road and is working at a constant rate of

300

W. The total resistance to her motion is modelled as a constant force of magnitude

R

newtons.

At the instant when her speed is

5\ \text{m s}^{-1}

, her acceleration is

0.5\ \text{m s}^{-2}

(a) Using the model, show that

R = 24

(4 marks)

On another day, the cyclist rides along a straight track from a point

A

to a point

B

. The distance travelled along the track from

A

B

350

m and the point

B

12

m vertically below the point

A

. Her speed at

A

3\ \text{m s}^{-1}

and her speed at

B

8\ \text{m s}^{-1}

The total resistance to her motion from non-gravitational forces is modelled as a constant force of magnitude

24

N.
(b) Using this model, find the total work done by the cyclist in the journey from

A

B

.

(5 marks)

Later, the cyclist rides up a straight road which is inclined at an angle

\alpha

to the horizontal, where

\sin\alpha = \frac{1}{24}

, at a constant speed of

4.5\ \text{m s}^{-1}

The total resistance to her motion from non-gravitational forces is again modelled as a constant force of magnitude

24

N.
(c) Find the rate at which the cyclist is working.

(4 marks)

●●●●●Level 513 marksStart

Friction

68 questions

LessonNot started

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

A box of mass

8

kg rests on a rough horizontal floor. The coefficient of friction between the box and the floor is

0.25

. A rope pulls the box with a force of

40

N at an angle of

30^\circ

above the horizontal. Take

g = 9.8

m s⁻².(a) Show that the normal reaction of the floor on the box is

58.4

N. (3 marks)
(b) Explain why the normal reaction is less than the weight of the box, and find the acceleration of the box. (6 marks)
(c) When the box is moving at

6

m s⁻¹, the rope is released. Find the distance the box travels before coming to rest. (4 marks)

●●●●●Level 513 marksStart

Figure 1 shows a particle

A

of mass

3

kg on a plane inclined at

30^\circ

to the horizontal.

A

is connected by a light inextensible string, passing over a smooth pulley at the top of the plane, to a particle

B

of mass

5

3

significant figures. (6 marks)
(b) The plane is now rough, with coefficient of friction

0.2

between

A

and the plane. Find the new acceleration of the system, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 510 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

\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 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

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

Two particles

P

and

Q

, of masses

3

kg and

5

kg respectively, are connected by a light inextensible string. Particle

P

lies on a rough plane inclined at an angle

\alpha

to the horizontal, where

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

. The string passes over a small smooth pulley fixed at the top edge of the plane, and

Q

hangs freely below the pulley. The part of the string from

P

to the pulley lies along a line of greatest slope of the plane. The coefficient of friction between

P

and the plane is

\dfrac{1}{2}

.

Initially

P

2.5

m from the pulley and

Q

1.6

m above horizontal ground. The system is released from rest with the string taut. Take

g = 9.8

m s⁻².
(a) Show that the acceleration of

P

2.45

m s⁻² and find the tension in the string. (6 marks)
(b) Find the time taken, from release, for

Q

to reach the ground, and the speed of

Q

at that instant. (3 marks)
(c)

Q

does not rebound when it hits the ground. Find the distance

P

moves up the plane after

Q

hits the ground before

P

first comes to instantaneous rest. (3 marks)
(d) Explain why

P

does not remain at rest at this position. (1 mark)

●●●●●Level 513 marksStart

A parcel of mass

11

kg is at rest on a rough plane inclined at an angle

\alpha

to the horizontal, where

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

. The coefficient of friction between the parcel and the plane is

\dfrac{1}{2}

.

A horizontal force of magnitude

P

newtons acts on the parcel, in the vertical plane containing a line of greatest slope of the inclined plane, in the direction that tends to push the parcel up the slope. Take

g = 9.8

m s⁻².
(a) Given that the parcel is on the point of sliding up the plane, show that

P = 215.6

. (5 marks)
(b) Find the range of values of

P

for which the parcel remains at rest. (4 marks)
(c) The force is removed. Show that the parcel begins to slide down the plane, and find its speed after it has travelled

2

m. (4 marks)

●●●●●Level 513 marksStart

Figure 1 shows a rough plane inclined to the horizontal at an angle

\alpha

, where

\tan\alpha = \dfrac{8}{15}

A small block

B

of mass

8

kg is held in equilibrium on the plane by a horizontal force of magnitude

X

newtons, as shown in Figure 1.

The block

B

is modelled as a particle.

The force acts in a vertical plane which contains a line of greatest slope of the inclined plane.

The magnitude of the normal reaction of the plane on

B

96

N.Figure 1 — not accurately drawn

Using the model,
(a) find the magnitude of the frictional force acting on

B

,

(3 marks)
(b) state the direction of the frictional force acting on

B

.

(1 mark)

The horizontal force of magnitude

X

newtons is now removed and

B

moves down the plane.

Given that the coefficient of friction between

B

and the plane is

0.28

(c) find the acceleration of

B

down the plane.

(6 marks)

●●●●●Level 510 marksStart

Figure 3 shows a block

A

of mass

4

B

of mass

3

kg, which hangs freely.Figure 3 (not accurately drawn)

The coefficient of friction between

A

and the table is

0.25

. The system is released from rest with the string taut.
(a) Find the acceleration of the system and the tension in the string. (7 marks)
(b)

B

hits the ground after falling

1.5

m. Find the speed of

B

as it hits the ground, giving your answer to

3

significant figures. (3 marks)
(c) After

B

hits the ground the string becomes slack and

A

continues to move. Find the further distance

A

travels before coming to rest, giving your answer to

3

significant figures. (2 marks)

●●●●●Level 512 marksStart

In this question use

g = 9.8\ \text{m s}^{-2}

A rough plank is

5

metres long and is inclined at an angle of

20^\circ

above the horizontal. The bottom of the plank is at the point

O

.

A sack of gravel of mass

30

kg is at rest at the point

A

200

N. The sack accelerates up the plank at

0.8\ \text{m s}^{-2}

Find the coefficient of friction between the sack and the plank.

(7 marks)
(b) (i) The sack takes

3

seconds to reach the top of the plank.

Find the distance

OA

.

(ii) Other than air resistance, state one assumption you have made about the sack in answering part (b)(i).

(4 marks)

●●●●●Level 511 marksStart

A sack of logs,

M

, and a bucket of stones,

N

, are connected securely by a length of rope. The mass of

M

20

kilograms. The mass of

N

30

kilograms.

M

is placed near the bottom of a rough slope. The slope is inclined at

60^\circ

above the horizontal.

The rope is passed over a smooth pulley at the top end of the slope so that

N

hangs with the rope vertical.

M

and

N

are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below.When

M

and

N

are released,

M

moves up the slope as

N

descends, with acceleration

a\ \text{m s}^{-2}

The tension in the rope is

T

newtons.
(a) Explain why the equation of motion for

N

30g - T = 30a

(1 mark)
(b) Show that the normal reaction force between

M

and the slope is

10g

newtons.

(1 mark)
(c) The coefficient of friction,

\mu

, between the slope and

M

is such that

0 \leq \mu \leq 1

Show that

a \geq \left(0.4 - 0.2\sqrt{3}\right)g

(6 marks)
(d) State one modelling assumption you have made throughout this question.

(1 mark)

●●●●●Level 59 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

Particles

A

and

B

, of masses

3

kg and

4

kg respectively, are attached to the ends of a light inextensible string.

A

is in contact with a rough plane inclined at

\theta

to the horizontal, where

\sin\theta = \frac{7}{25}

. The string is taut and passes over a small smooth pulley

P

at the top of the plane, with the part of the string from

A

P

parallel to a line of greatest slope of the plane.

B

hangs freely below

P

1.25

m above horizontal ground, as shown in the diagram.The coefficient of friction between

A

and the plane is

\mu

. The system is released from rest and in the subsequent motion

B

hits the ground before

A

reaches

P

. The speed of

B

at the instant that it hits the ground is

2.5

m s

^{-1}

.
(a) For the motion before

B

hits the ground, show that the acceleration of

B

2.5

m s

^{-2}

.

(1 mark)
(b) For the motion before

B

hits the ground, show that the tension in the string is

29.2

N.

(3 marks)
(c) Determine the value of

\mu

.

(5 marks)

After

B

hits the ground,

A

continues to move up the plane and comes to instantaneous rest before it reaches

P

.
(d) Determine the distance that

A

moves up the plane from the instant that

B

hits the ground until

A

comes to instantaneous rest.

(4 marks)

●●●●●Level 513 marksStart

A particle of mass

5

kg lies on a rough slope inclined at

30^\circ

, with coefficient of friction

0.25

. It is held in limiting equilibrium by a force

P

acting up the slope. Find the two possible values of

P

, to 3 s.f. (

g = 9.8

●●●●●Level 53 marksStart

A particle of mass

2

kg is projected up a rough slope inclined at

30^{\circ}

with speed

6

m s

^{-1}

. The coefficient of friction is

0.4

.
(a) Find the distance the particle travels up the slope before coming instantaneously to rest. (4 marks)
(b) Determine whether the particle then slides back down. (2 marks)

●●●●●Level 56 marksStart

Figure 1 (not accurately drawn)

A sledge

P

of mass

m

is pulled up a rough slope by a rope. The slope is modelled as a rough plane inclined to the horizontal at an angle

\alpha

, where

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

The sledge is modelled as a particle and the rope is modelled as a light inextensible string.

The rope is inclined to the slope at an angle

\beta

, where

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

, and the tension in the rope is

\frac{4}{5}mg

, as shown in Figure 1. The rope lies in the vertical plane containing a line of greatest slope of the plane.

The coefficient of friction between the sledge and the slope is

\frac{1}{4}

The sledge starts from rest at the point

A

on the slope and moves up the line of greatest slope through

A

.
(a) Show that the acceleration of the sledge is

\frac{6}{25}g

(8 marks)

The sledge passes through the point

B

, where

AB = 3

m.
(b) Find the speed of the sledge at

B

.

(2 marks)

When the sledge reaches

B

, the rope is released. The sledge continues to move up the slope and comes to instantaneous rest at the point

C

.
(c) Find the distance

BC

.

(3 marks)
(d) Determine whether the sledge remains at rest at

C

. You must justify your reasoning.

(2 marks)

●●●●●Level 515 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

The points

A

and

B

are the lower and upper ends, respectively, of a line of greatest slope on a plane inclined at an angle

\theta

to the horizontal, where

\sin\theta = \frac{8}{17}

and

AB = 2.55

m (see diagram).A particle

P

is projected up the plane with speed

7

m s

^{-1}

from

A

towards

B

.

The plane at

A

is fixed to horizontal ground.

The surface of the plane is rough and the coefficient of friction between

P

and the plane is

0.3

.
(a) Show that the speed of

P

B

3.5

m s

^{-1}

.

(6 marks)

The particle leaves the plane at

B

and moves freely under gravity. It first lands at a point

C

on the horizontal ground. The time taken for

P

to travel from

A

C

T

seconds.
(b) Determine the value of

T

.

(6 marks)

●●●●●Level 512 marksStart

Figure 3 shows a block of mass

15

kg resting on a rough horizontal floor. A force of magnitude

P

N is applied to the block at an angle of

20^\circ

above the horizontal.Figure 3 (not accurately drawn)

The coefficient of friction between the block and the floor is

0.35

. Take

g = 9.8

m s⁻².
(a) Find the value of

P

for which the block is on the point of moving, giving your answer to

3

significant figures. (6 marks)
(b) Find the acceleration of the block when

P = 70

, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 510 marksStart

Figure 1Diagram not accurately drawn.

A particle

A

of mass

3m

is held at rest on a rough plane which is inclined to horizontal ground at an angle

\alpha

, where

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

One end of a light inextensible string is attached to

A

. The string passes over a small smooth pulley which is fixed at the top of the plane, and the other end of the string is attached to a particle

B

of mass

4m

. Particle

B

hangs vertically below the pulley, with the string taut, at a height

h

above the ground, as shown in Figure 1.

The part of the string between

A

and the pulley lies along a line of greatest slope of the plane, and

A

B

and the pulley all lie in the same vertical plane.

The coefficient of friction between

A

and the plane is

\frac{1}{4}

Particle

A

is released from rest and begins to move up the plane.
(a) Show that, as

A

moves up the plane, the frictional force acting on

A

has magnitude

\frac{9}{13}mg

(3 marks)
(b) Write down an equation of motion for

B

.

(2 marks)
(c) Show that the acceleration of

A

immediately after its release is

\frac{4}{13}g

(4 marks)

Particle

B

hits the ground and does not rebound. In the subsequent motion,

A

comes to rest before it reaches the pulley.
(d) Find, in terms of

h

, the total distance travelled by

A

from the instant it is released to the instant it first comes to rest.

(6 marks)

●●●●●Level 515 marksStart

Figure 1Diagram not accurately drawn.

A particle

A

of mass

4m

is held at rest on a rough horizontal table. One end of a light inextensible string is attached to

A

. The string passes over a small smooth pulley which is fixed at the edge of the table, and the other end of the string is attached to a particle

B

of mass

3m

. Particle

B

hangs vertically below the pulley at a height

h

above the horizontal floor, as shown in Figure 1.

The string between

A

and the pulley is horizontal and perpendicular to the edge of the table. The particles, the string and the pulley all lie in the same vertical plane.

The coefficient of friction between

A

and the table is

\mu

.

Particle

A

is released from rest with the string taut. Before

B

reaches the floor, the tension in the string is

\frac{15}{7}mg

(a) Write down an equation of motion for

B

.

(2 marks)
(b) Find the value of

\mu

.

(4 marks)
(c) Find, in terms of

m

and

g

, the magnitude of the force exerted on the pulley by the string before

B

reaches the floor.

(3 marks)

Particle

B

hits the floor and does not rebound. In the subsequent motion,

A

does not reach the pulley.
(d) Find, in terms of

h

and

g

, the time from the instant

A

is released to the instant

A

comes to rest.

(5 marks)

●●●●●Level 514 marksStart

Figure 2Diagram not accurately drawn.

Two small blocks,

A

and

B

, are at rest on a rough horizontal floor. Block

A

has mass

0.6

kg and block

B

has mass

0.9

kg. The distance between

A

and

B

1.25

m, as shown in Figure 2. The blocks are modelled as particles.

The coefficient of friction between each block and the floor is

\frac{2}{7}

Block

A

is hit so that it receives an impulse of magnitude

2.4

N s in the direction

AB

.
(a) Find the speed of

A

immediately after the impulse is applied.

(2 marks)
(b) Show that the speed of

A

immediately before it collides with

B

3\text{ m s}^{-1}

(5 marks)

In the collision,

A

and

B

join together and move as a single particle.
(c) Find the speed of the combined particle immediately after the collision.

(2 marks)
(d) Find the magnitude of the impulse exerted on

A

B

in the collision, stating the units of your answer.

(2 marks)
(e) Find the distance travelled by the combined particle after the collision until it comes to rest.

(3 marks)

●●●●●Level 514 marksStart

A fixed wedge has two plane faces which meet along a horizontal ridge. One face is inclined at

\alpha

to the horizontal and the other at

\beta

to the horizontal, where

\sin\alpha = \frac{12}{13}

and

\sin\beta = \frac{5}{13}

. Both faces are rough.

A particle

A

of mass

6

kg on the first face and a particle

B

of mass

3

kg on the second face are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the ridge. The parts of the string from

A

to the pulley and from

B

to the pulley are parallel to lines of greatest slope of the corresponding faces (see diagram).The coefficient of friction between

A

and the first face is

12\mu

and the coefficient of friction between

B

and the second face is

5\mu

, where

\mu

is a constant.

The system is released from rest, and in the subsequent motion

A

moves down the first face and

B

does not reach the pulley.
(a) Determine, in terms of

g

, the tension in the string.

(7 marks)

When

A

is moving at

1.5

m s

^{-1}

the string breaks. In the

0.4

seconds after the string breaks

A

moves

1.15

m down the first face.
(b) Determine the deceleration of

B

after the string breaks. Give your answer correct to 3 significant figures.

(5 marks)

●●●●●Level 512 marksStart

A particle of mass

m

kg sits on a rough slope inclined at angle

\alpha

, where

\tan\alpha = \tfrac34

. The coefficient of friction is

0.1

. A force of magnitude

F

N acts up the slope.
(a) Given that the particle accelerates up the slope, show that

F > 0.68\,mg

.
(b) Given further that

F = 30

and the particle accelerates up the slope at

1.1\ \text{m s}^{-2}

, find

m

, to 3 s.f. (

g = 9.8

●●●●●Level 54 marksStart

A sledge of mass

8

kg is pulled across rough horizontal ground (

\mu = 0.35

) at constant velocity by a rope inclined at

25^{\circ}

above the horizontal.

Find the tension in the rope, giving your answer to 3 significant figures. (5 marks)

●●●●●Level 55 marksStart

Statics — equilibrium under several forces

31 questions

LessonNot started

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

A parcel of mass

11

kg is at rest on a rough plane inclined at an angle

\alpha

to the horizontal, where

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

. The coefficient of friction between the parcel and the plane is

\dfrac{1}{2}

.

A horizontal force of magnitude

P

newtons acts on the parcel, in the vertical plane containing a line of greatest slope of the inclined plane, in the direction that tends to push the parcel up the slope. Take

g = 9.8

m s⁻².
(a) Given that the parcel is on the point of sliding up the plane, show that

P = 215.6

. (5 marks)
(b) Find the range of values of

P

for which the parcel remains at rest. (4 marks)
(c) The force is removed. Show that the parcel begins to slide down the plane, and find its speed after it has travelled

2

m. (4 marks)

●●●●●Level 513 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 particle of mass

5

kg rests on a rough slope of

30^\circ

with coefficient of friction

0.4

. A force

P

acts up the slope.
(a) Find the least value of

P

that keeps the particle from sliding down.
(b) Find the greatest value of

P

before it slides up. Give answers to 3 s.f. (

g = 9.8

●●●●●Level 54 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

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 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

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

A particle of mass

4

kg rests on a rough plane inclined at

30^{\circ}

to the horizontal, with coefficient of friction

0.25

. A force

P

acts up the slope, and the particle is on the point of sliding down the slope.

Find

P

, giving your answer to 3 significant figures. (5 marks)

●●●●●Level 55 marksStart

Connected particles on slopes and pulleys

19 questions

LessonNot started

Figure 1 shows a particle

A

of mass

3

kg on a plane inclined at

30^\circ

to the horizontal.

A

is connected by a light inextensible string, passing over a smooth pulley at the top of the plane, to a particle

B

of mass

5

3

significant figures. (6 marks)
(b) The plane is now rough, with coefficient of friction

0.2

between

A

and the plane. Find the new acceleration of the system, giving your answer to

3

significant figures. (4 marks)

●●●●●Level 510 marksStart

Two particles

P

and

Q

, of masses

3

kg and

5

kg respectively, are connected by a light inextensible string. Particle

P

lies on a rough plane inclined at an angle

\alpha

to the horizontal, where

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

. The string passes over a small smooth pulley fixed at the top edge of the plane, and

Q

hangs freely below the pulley. The part of the string from

P

to the pulley lies along a line of greatest slope of the plane. The coefficient of friction between

P

and the plane is

\dfrac{1}{2}

.

Initially

P

2.5

m from the pulley and

Q

1.6

m above horizontal ground. The system is released from rest with the string taut. Take

g = 9.8

m s⁻².
(a) Show that the acceleration of

P

2.45

m s⁻² and find the tension in the string. (6 marks)
(b) Find the time taken, from release, for

Q

to reach the ground, and the speed of

Q

at that instant. (3 marks)
(c)

Q

does not rebound when it hits the ground. Find the distance

P

moves up the plane after

Q

hits the ground before

P

first comes to instantaneous rest. (3 marks)
(d) Explain why

P

does not remain at rest at this position. (1 mark)

●●●●●Level 513 marksStart

A sack of logs,

M

, and a bucket of stones,

N

, are connected securely by a length of rope. The mass of

M

20

kilograms. The mass of

N

30

kilograms.

M

is placed near the bottom of a rough slope. The slope is inclined at

60^\circ

above the horizontal.

The rope is passed over a smooth pulley at the top end of the slope so that

N

hangs with the rope vertical.

M

and

N

are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below.When

M

and

N

are released,

M

moves up the slope as

N

descends, with acceleration

a\ \text{m s}^{-2}

The tension in the rope is

T

newtons.
(a) Explain why the equation of motion for

N

30g - T = 30a

(1 mark)
(b) Show that the normal reaction force between

M

and the slope is

10g

newtons.

(1 mark)
(c) The coefficient of friction,

\mu

, between the slope and

M

is such that

0 \leq \mu \leq 1

Show that

a \geq \left(0.4 - 0.2\sqrt{3}\right)g

(6 marks)
(d) State one modelling assumption you have made throughout this question.

(1 mark)

●●●●●Level 59 marksStart

Particles

A

and

B

, of masses

3

kg and

4

kg respectively, are attached to the ends of a light inextensible string.

A

is in contact with a rough plane inclined at

\theta

to the horizontal, where

\sin\theta = \frac{7}{25}

. The string is taut and passes over a small smooth pulley

P

at the top of the plane, with the part of the string from

A

P

parallel to a line of greatest slope of the plane.

B

hangs freely below

P

1.25

m above horizontal ground, as shown in the diagram.The coefficient of friction between

A

and the plane is

\mu

. The system is released from rest and in the subsequent motion

B

hits the ground before

A

reaches

P

. The speed of

B

at the instant that it hits the ground is

2.5

m s

^{-1}

.
(a) For the motion before

B

hits the ground, show that the acceleration of

B

2.5

m s

^{-2}

.

(1 mark)
(b) For the motion before

B

hits the ground, show that the tension in the string is

29.2

N.

(3 marks)
(c) Determine the value of

\mu

.

(5 marks)

After

B

hits the ground,

A

continues to move up the plane and comes to instantaneous rest before it reaches

P

.
(d) Determine the distance that

A

moves up the plane from the instant that

B

hits the ground until

A

comes to instantaneous rest.

(4 marks)

●●●●●Level 513 marksStart

Figure 1Diagram not accurately drawn.

A particle

A

of mass

3m

is held at rest on a rough plane which is inclined to horizontal ground at an angle

\alpha

, where

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

One end of a light inextensible string is attached to

A

. The string passes over a small smooth pulley which is fixed at the top of the plane, and the other end of the string is attached to a particle

B

of mass

4m

. Particle

B

hangs vertically below the pulley, with the string taut, at a height

h

above the ground, as shown in Figure 1.

The part of the string between

A

and the pulley lies along a line of greatest slope of the plane, and

A

B

and the pulley all lie in the same vertical plane.

The coefficient of friction between

A

and the plane is

\frac{1}{4}

Particle

A

is released from rest and begins to move up the plane.
(a) Show that, as

A

moves up the plane, the frictional force acting on

A

has magnitude

\frac{9}{13}mg

(3 marks)
(b) Write down an equation of motion for

B

.

(2 marks)
(c) Show that the acceleration of

A

immediately after its release is

\frac{4}{13}g

(4 marks)

Particle

B

hits the ground and does not rebound. In the subsequent motion,

A

comes to rest before it reaches the pulley.
(d) Find, in terms of

h

, the total distance travelled by

A

from the instant it is released to the instant it first comes to rest.

(6 marks)

●●●●●Level 515 marksStart

A fixed wedge has two plane faces which meet along a horizontal ridge. One face is inclined at

\alpha

to the horizontal and the other at

\beta

to the horizontal, where

\sin\alpha = \frac{12}{13}

and

\sin\beta = \frac{5}{13}

. Both faces are rough.

A particle

A

of mass

6

kg on the first face and a particle

B

of mass

3

kg on the second face are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the ridge. The parts of the string from

A

to the pulley and from

B

to the pulley are parallel to lines of greatest slope of the corresponding faces (see diagram).The coefficient of friction between

A

and the first face is

12\mu

and the coefficient of friction between

B

and the second face is

5\mu

, where

\mu

is a constant.

The system is released from rest, and in the subsequent motion

A

moves down the first face and

B

does not reach the pulley.
(a) Determine, in terms of

g

, the tension in the string.

(7 marks)

When

A

is moving at

1.5

m s

^{-1}

the string breaks. In the

0.4

seconds after the string breaks

A

moves

1.15

m down the first face.
(b) Determine the deceleration of

B

after the string breaks. Give your answer correct to 3 significant figures.

(5 marks)

●●●●●Level 512 marksStart