A box has mass 40 kg and is on a rough slope with coefficient of friction 0 . 3 . It is pulled up the slope by a force of 300 N at 10 ° above the slope and is in limiting equilibrium. Find the angle that the slope makes with the horizontal.

It is given in the question that, Mass of the box, m = 40 kg Coefficient of friction, μ = 0 . 3 Let's draw its FBD, We know that, λ = tan - 1 0 . 3 On applying the Lami's theorem, we can write 40 g sin 80 ° + λ = 300 sin 180 ° - λ - θ sin 180 ° - λ - θ = 300 sin 80 ° + λ 400 = 0 . 745 Or we can write, ⇒ 180 ° - λ - θ = 48 . 148 ° or 131 . 85 ° ⇒ θ = 115 . 2 ° or 31 . 5 ° But, θ < 90 ° Hence, θ = 31 . 5 °

A ring of mass m kg is at rest on a fixed rough horizontal wire with coefficient of friction μ . It is attached to a string that is at an angle of α above the horizontal. Show that when T < m g sin α cos ( α - θ ) and θ = tan - 1 μ the ring will be in equilibrium. Show further that if α + θ ⩽ 90 ° and T > m g sin α cos ( α - θ ) the ring will always move, but if α + θ > 90 ° and T > m g sin θ sin ( α + θ - 90 ) the ring will remain in equilibrium.

It is given in the question, Mass of the ring, m kg and coefficient of friction, μ And angle of friction, θ = tan - 1 μ The ring is held at an angle α above the angle of the horizontal. The angle above the angle of friction = θ - α Resolving the Tension = T and the weight = m g along the direction parallel to the inclined plane and perpendicular to it, we get m g sin α - T cos ( θ - α ) > 0 ⇒ T < m g sin α T cos ( θ - α ) for the equilibrium condition. Here if, T > m g sin θ sin ( α + θ - 90 ) , the Lami's theorem can't hold and equilibrium is not possible. So the ring always moves.

HARD

AS and A Level

IMPORTANT

Earn 100

A box of mass $12 kg$ is at rest on a rough horizontal surface with coefficient of friction $0.6 .$ A force is exerted on it at an angle $\theta$ above the horizontal so that the force required to break equilibrium is minimised. Show that $θ$ is the angle of friction and find the size of the force required to break equilibrium.

Important Questions on Friction

HARD

AS and A Level

IMPORTANT

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

A box has mass

40 kg

and is on a rough slope with coefficient of friction

0.3 .

It is pulled up the slope by a force of

300 N

10 °

above the slope and is in limiting equilibrium. Find the angle that the slope makes with the horizontal.

HARD

AS and A Level

IMPORTANT

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

A ring of mass

m kg

is at rest on a fixed rough horizontal wire with coefficient of friction

μ .

It is attached to a string that is at an angle of

α

above the horizontal. Show that when

T < \frac{m g \sin α}{\cos (α - θ)}

and

θ = \tan^{- 1} μ

the ring will be in equilibrium. Show further that if

α + θ ⩽ 90 °

and

T > \frac{m g \sin α}{\cos (α - θ)}

the ring will always move, but if

α + θ > 90 °

and

T > \frac{m g \sin θ}{\sin (α + θ - 90)}

the ring will remain in equilibrium.

HARD

AS and A Level

IMPORTANT

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

A particle of weight $W$ is at rest on a rough slope, which makes an angle of $α$ to the horizontal. The coefficient of friction between the particle and the slope is $μ .$ Assuming $θ + α < 90 °,$ where $θ = \tan^{- 1} μ,$ show that the minimum force $F$ required to break equilibrium and make the particle slide up the slope is $F = W \sin (θ + α)$ and that $F$ makes an angle $\theta$ to the slope above the particle.

Show further that in the case where $α < θ,$ the minimum force $F$ required to break equilibrium and make the particle slide down the slope is $F = W \sin (θ - α)$ and that $F$ makes an angle $θ$ to the slope below the particle.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>END-OF-CHAPTER REVIEW EXERCISE 4>Q 1

A horizontal force,

T,

acts on a particle of mass

12 kg,

which is on a rough horizontal plane. Given that the particle is on the point of slipping and that the coefficient of friction is

0.35,

find the size of

T .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>END-OF-CHAPTER REVIEW EXERCISE 4>Q 2

A particle of mass

15 kg

is on a slope at an angle

25 °

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

0.3 .

A force,

P,

acts up the slope along the line of greatest slope. Find the set of values for

P

for the particle to be in equilibrium.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>END-OF-CHAPTER REVIEW EXERCISE 4>Q 3

A bowler rolls a ten-pin bowling ball of mass

4 kg

along a horizontal lane. The ball is released with a speed of

9 m s^{- 1} .

The coefficient of friction between the ball and the lane is

0.04 .

The first pin is

18.5 m

away. Find the speed at which the ball hits the pin.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>END-OF-CHAPTER REVIEW EXERCISE 4>Q 4

A brick of mass

4.3 kg

is being pushed up a slope by a force of

40 N

parallel to the slope. The slope is at

13 °

to the horizontal and the coefficient of friction between the brick and the slope is

0.55 .

Find the acceleration of the brick.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>END-OF-CHAPTER REVIEW EXERCISE 4>Q 5

A boat of mass

5

tonnes is being launched from rest into the sea by sliding it down a ramp. The ramp is at

5 °

to the horizontal and is lubricated so the coefficient of friction is only

0.08 .

The ramp is

40 m

long before the boat enters the sea. Find the speed with which the boat enters the sea.

Important Questions on Friction

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