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 bin of mass $10 kg$ is at rest on a rough slope at an angle of $32 °$ to the horizontal. It is held on the point of moving up the slope by a force of $90 N$ parallel to the slope. Show that when the force is removed the bin would slide down the slope, and find its acceleration.

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 7

A trolley of mass

5 kg

is rolling up a rough slope, which is at an angle of

25 °

to the horizontal. The coefficient of friction between the trolley and the slope is

0.4 .

It passes a point

A

with speed

12 m s^{- 1} .

Find its speed when it passes

A

on its way back down the slope.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4C>Q 8

A ball of mass

1.5 kg

is sliding up a slope, which is at

30 °

to the horizontal. The coefficient of friction between the ball and the slope is

0.45 .

It passes a point

A

10 m s^{- 1} .

By modelling the ball as a particle, find the time taken to return to

A .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4C>Q 10

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.

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

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.

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.

A bin of mass 10 kg is at rest on a rough slope at an angle of 32° to the horizontal. It is held on the point of moving up the slope by a force of 90 N parallel to the slope. Show that when the force is removed the bin would slide down the slope, and find its acceleration.

Important Questions on Friction

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A bin of mass $10 kg$ is at rest on a rough slope at an angle of $32 °$ to the horizontal. It is held on the point of moving up the slope by a force of $90 N$ parallel to the slope. Show that when the force is removed the bin would slide down the slope, and find its acceleration.