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.

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.

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.

It is given in the question that, Mass of the sack of sand, m = 25 kg Coefficient of friction, μ = 0 . 5 The sack is on the point of slipping. So, on drawing the free body diagram of the sack, Here, λ is the angle of friction. We also know that, tan λ = μ = 0 . 5 Now applying the Lami's theorem, C sin 102 ° = 25 g sin 78 ° + λ On solving, ⇒ C = 25 g sin 102 ° sin 78 ° + λ ⇒ C = 250 sin 102 ° sin 78 ° + tan - 1 0 . 5 ⇒ C = 253 N

HARD

AS and A Level

IMPORTANT

Earn 100

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.

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 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

\Rightarrow \sin θ - μ \cos θ < μ \cos θ + \sin θ \Rightarrow \frac{\sin θ - μ \cos θ}{μ \cos θ + \sin θ} < 1 \Rightarrow v^{2} < u^{2} \Rightarrow v < u

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 .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4D>Q 6

A metal block of mass

20 kg

is on a rough slope at an angle of

12 °

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

0.4 .

A boy is trying to move the block up the slope by pushing parallel to the slope. He increases the force until equilibrium breaks. Find the maximum size of the force the boy pushes with before the block slips.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4D>Q 7

A girl drags a sledge up a rough slope, which has an angle of

10 °

to the horizontal. The sledge has mass

8 kg

and the coefficient of friction between the slope and the sledge is

0.3 .

She pulls the sledge with a rope at an angle of

12 °

to the slope and increases the tension until equilibrium is broken. Find the tension in the rope when this happens.

HARD

AS and A Level

IMPORTANT

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

A car is towed down a rough slope, which is at an angle of

5 °

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

0.35 .

The car is towed using a rope at an angle of

13 °

to the slope. Equilibrium is broken when the tension in the rope is

4000 N .

Find the mass of the car.

Important Questions on Friction

Learn and Prepare for any exam you want !