A boy drags a sledge of mass 4 kg from rest down a rough slope at an angle of 18 ° to the horizontal. He pulls it with a force of 8 N for 3 s by a rope that is angled at 10 ° above the parallel down the slope. After 3 s the rope becomes detached from the sledge. The coefficient of friction between the slope and the sledge is 0 . 4 . Find the total distance the sledge has moved down the slope from when the boy started dragging it until it comes to rest.

It is given in the question, Mass of the sledge, m = 4 kg Coefficient of friction, μ = 0 . 4 Let's draw its FBD when sledge is going down, Resolving the forces in the perpendicular direction, R + T sin 10 ° = 4 g cos 18 ° R = 4 g cos 18 ° - T sin 10 ° F = μ R = 0 . 4 4 g cos 18 ° - T sin 10 ° Resolving in the parallel direction, T cos 10 ° + 4 g sin 18 ° - 0 . 4 4 g cos 18 ° - T sin 10 ° = 4 a When the string is attached, it gives T = 8 N a = 1 . 394 m / s 2 When detached, T = 0 N a = 10 sin 18 ° - 4 g cos 18 ° a = - 0 . 714 m / s 2 Now, when the string is attached we have, u = 0 , v = ? , t = 3 s , a = 1 . 394 m / s 2 v = u + a t v = 4 . 1834 ms - 1 And distance, s = u t + 1 2 a t 2 s = 0 + 1 2 a × 9 = 6 . 27 m When string detached, u = 4 . 1834 m / s , a = - 0 . 714 m / s 2 , s = ? , v = 0 m / s v 2 = u 2 + 2 a s s = 0 - 4 . 134 2 2 a = 12 . 25 ms - 1 Total distance will be, 12 . 25 + 6 . 27 = 18 . 52 m

A particle slides up a slope at angle θ to the horizontal with coefficient of friction μ . It passes a point A on the way up the slope at speed u m s - 1 and passes it on the way down the slope with speed v m s - 1 . Prove that: v 2 = u 2 sin θ - μ cos θ sin θ + μ cos θ so v is independent of the mass of the particle and the value of g . Deduce also that the speed on the way down is always smaller than the speed on the way up.

It is given in the question, The angle of the slope θ , and coefficient of friction, μ Let's draw its free body diagram, With the help of FBD, we can write R = m g cos θ When the particle is going up, F 2 = F m a x = μ m g cos θ F 1 = 0 Travelling down, F 2 = 0 F 1 = F m a x = μ m g cos θ Again when the particle is going up, - F 2 - m g sin θ = m a a = - μ g cos θ - g sin θ v 2 = u 2 + 2 a s 0 = u 2 + 2 - μ g cos θ - g sin θ s s = u 2 2 μ g cos θ + g sin θ When it is going down, m g sin θ - F 1 = m a a = g sin θ - μ g cos θ v 2 = u 2 + 2 a s v 2 = 0 + 2 g sin θ - μ cos θ u 2 2 g μ cos θ + sin θ v 2 = u 2 sin θ - μ cos θ μ cos θ + sin θ As θ is acute, so cos θ > 0 ⇒ sin θ - μ cos θ < μ cos θ + sin θ ⇒ sin θ - μ cos θ μ cos θ + sin θ < 1 ⇒ v 2 < u 2 ⇒ v < u Hence proved.

HARD

AS and A Level

IMPORTANT

Earn 100

A wooden block of mass $3.5 kg$ is sliding up a rough slope and passes a point $A$ with speed $20 m s^{- 1} .$ The slope is at $29 °$ to the horizontal. The block comes to rest $25 m$ up the slope. Find its speed as it passes point $A$ on the way down.

Important Questions on Friction

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4C>Q 11

A boy drags a sledge of mass

4 kg

from rest down a rough slope at an angle of

18 °

to the horizontal. He pulls it with a force of

8 N

for

3 s

by a rope that is angled at

10 °

above the parallel down the slope. After

v^{2} = u^{2} + 2 a s v = \sqrt{0 + 2 (25 (20 \sin (29 °) - 8))} v = 9.21 {ms}^{- 1}

the rope becomes detached from the sledge. The coefficient of friction between the slope and the sledge is

0.4 .

Find the total distance the sledge has moved down the slope from when the boy started dragging it until it comes to rest.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4C>Q 12

A particle slides up a slope at angle

34 °

to the horizontal. It passes a point

P

on the way up the slope with speed

3 m s^{- 1}

and passes it on the way down the slope with speed

2 m s^{- 1} .

Find the coefficient of friction between the particle and the slope.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4C>Q 13

A particle slides up a slope at angle $θ$ to the horizontal with coefficient of friction $μ .$ It passes a point $A$ on the way up the slope at speed $u m s^{- 1}$ and passes it on the way down the slope with speed $v m s^{- 1} .$ Prove that:

$v^{2} = u^{2} (\frac{\sin θ - μ \cos θ}{\sin θ + μ \cos θ})$

so $v$ is independent of the mass of the particle and the value of $g .$ Deduce also that the speed on the way down is always smaller than the speed on the way up.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4C>Q 14

A toy car of mass

80 g

rolls from rest

80 cm

down a rough slope at an angle of

16 °

to the horizontal. When it hits a rubber barrier at the bottom of the slope it bounces back up the slope with its speed halved, and reaches a height of

10 cm .

Find the coefficient of friction between the car and the slope.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4D>Q 1

A boy is trying to drag a box along a rough horizontal surface by pulling horizontally. The box has mass

12 kg .

The coefficient of friction between the box and the surface is

0.4 .

The box is on the point of slipping. Find the size of the force exerted by the boy.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4D>Q 2

A girl tries to drag the box of mass

12 kg

along the rough horizontal surface with coefficient of friction

0.4

She exerts a force at an angle of

10 °

above the horizontal and the box is on the point of slipping. Find the size of the force exerted by the girl.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4D>Q 3

A builder tries dragging a sack of sand of mass

25 kg

along a rough horizontal surface with coefficient of friction

0.5 .

He pulls at

12 °

above the horizontal and the sack is on the point of slipping. Find the size of the total contact force.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4D>Q 5

A trailer has mass

120 kg .

A winch pulls the trailer with a force of

500 N

at an angle

θ

above the horizontal. The trailer is in limiting equilibrium on horizontal ground with coefficient of friction

0.45 .

Find

θ .

A wooden block of mass 3.5 kg is sliding up a rough slope and passes a point A with speed 20 ms-1. The slope is at 29° to the horizontal. The block comes to rest 25 m up the slope. Find its speed as it passes point A on the way down.

Important Questions on Friction

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A wooden block of mass $3.5 kg$ is sliding up a rough slope and passes a point $A$ with speed $20 m s^{- 1} .$ The slope is at $29 °$ to the horizontal. The block comes to rest $25 m$ up the slope. Find its speed as it passes point $A$ on the way down.