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.

MEDIUM

AS and A Level

IMPORTANT

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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.

Important Questions on Friction

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 .

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.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 4 - Friction>EXERCISE 4D>Q 9

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.

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.

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.

Important Questions on Friction

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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.