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12
MCQ (Multiple Correct Answer)

Mass $$m_1$$ moves on a slope making an angle $$\theta$$ with the horizontal and is attached to mass $$m_2$$ by a string passing over a frictionless pulley as shown in figure. The coefficient of friction between $$m_1$$ and the sloping surface is $$\mu$$. Which of the following statements are true?

A
$$ \text { If } m_2>m_1 \sin \theta \text {, the body will move up the plane } $$
B
$$ \text { If } m_2>m_1(\sin \theta+\mu \cos \theta) \text {, the body will move up the plane } $$
C
$$ \text { If } m_2< m_1(\sin \theta+\mu \cos \theta) \text {, the body will move up the plane } $$
D
$$ \text { If } m_2< m_1(\sin \theta-\mu \cos \theta) \text {, the body will move down the plane } $$
13
MCQ (Multiple Correct Answer)

In figure a body $$A$$ of mass $$m$$ slides on plane inclined at angle $$\theta_1$$ to the horizontal and $$\mu$$ is the coefficient of friction between $$A$$ and the plane. $$A$$ is connected by a light string passing over a frictionless pulley to another body $$B$$, also of mass $$m$$, sliding on a frictionless plane inclined at an angle $$\theta_2$$ to the horizontal. Which of the following statements are true?

A
A will never move up the plane
B
A will just start moving up the plane when $$ \mu=\frac{\sin \theta_2-\sin \theta_1}{\cos \theta_1} $$
C
For A to move up the plane, $$\theta_2$$ must always be greater than $$\theta_1$$
D
B will always slide down with constant speed
14
MCQ (Multiple Correct Answer)

Two billiard balls $$A$$ and $$B$$, each of mass 50 g and moving in opposite directions with speed of $$5 \mathrm{~m} \mathrm{~s}^{-1}$$ each, collide and rebound with the same speed. If the collision lasts for $$10^{-3} \mathrm{~s}$$, which of the following statements are true?

A
The impulse imparted to each ball is $$0.25 \mathrm{~kg}-\mathrm{ms}^{-1}$$ and the force on each ball is 250 N
B
The impulse imparted to each ball is $$0.25 \mathrm{~kg}-\mathrm{m} \mathrm{s}^{-1}$$ and the force exerted on each ball is $$25 \times 10^{-5} \mathrm{~N}$$
C
The impulse imparted to each ball is $$0.5 \mathrm{~N}-\mathrm{s}$$
D
The impulse and the force on each ball are equal in magnitude and opposite in directions
15
MCQ (Multiple Correct Answer)

A body of mass 10 kg is acted upon by two perpendicular forces, 6 N and 8 N. The resultant acceleration of the body is

A
$$1 \mathrm{~m} \mathrm{~s}^{-2}$$ at an angle of $$\tan ^{-1}\left(\frac{4}{3}\right)$$ w.r.t. 6 N force
B
$$0.2 \mathrm{~m} \mathrm{~s}^{-2}$$ at an angle of $$\tan ^{-1}\left(\frac{4}{3}\right)$$ w.r.t. 6 N force
C
$$1 \mathrm{~m} \mathrm{~s}^{-2}$$ at an angle of $$\tan ^{-1}\left(\frac{3}{4}\right)$$ w.r.t. 8 N force
D
$$0.2 \mathrm{~m} \mathrm{~s}^{-2}$$ at an angle of $$\tan ^{-1}\left(\frac{3}{4}\right)$$ w.r.t. 8 N force
16
Subjective

A girl riding a bicycle along a straight road with a speed of $$5 \mathrm{~ms}^{-1}$$ throws a stone of mass 0.5 kg which has a speed of $$15 \mathrm{~ms}^{-1}$$ with respect to the ground along her direction of motion. The mass of the girl and bicycle is 50 kg . Does the speed of the bicycle change after the stone is thrown? What is the change in speed, if so?

Explanation

Given, total mass of girl, bicycle and stone $$=m_1=(50+0.5) \mathrm{kg}=50.5 \mathrm{~kg}$$.

Velocity of bicycle $$u_1=5 \mathrm{~m} / \mathrm{s}$$, Mass of stone $$m_2=0.5 \mathrm{~kg}$$

Velocity of stone $$u_2=15 \mathrm{~m} / \mathrm{s}$$, Mass of girl and bicycle $$m=50 \mathrm{~kg}$$

Yes, the speed of the bicycle changes after the stone is thrown.

Let after throwing the stone the speed of bicycle be $$v \mathrm{~m} / \mathrm{s}$$.

According to law of conservation of linear momentum,

$$\begin{aligned} m_1 u_1 & =m_2 u_2+m v \\ 50.5 \times 5 & =0.5 \times 15+50 \times v \\ 252.5-7.5 & =50 \mathrm{v} \\ \text{or}\quad v & =\frac{245.0}{50} \\ v & =4.9 \mathrm{~m} / \mathrm{s} \\ \text { Change in speed } & =5-4.9=0.1 \mathrm{~m} / \mathrm{s} . \end{aligned}$$

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