System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 60

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Question 1
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In which of the following cases the centre of mass of a rod is certainly not at its geometrical centre?

A
The density continuously decreases from left to right

B
The density continuously increases from left to right

C
The density decreases from left to right upto the centre and then increases

D
Both $$(a)$$ and $$(b)$$ are correct

Solution
Question 2
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Water is drawn from a well in a $$5\ kg$$ drum of capacity $$55\ L$$ by two ropes connected to the top of the drum. The linear mass density of each rope is $$0.5\ kgm^{-1}$$. The work done in lifting water to the ground from the surface of water in the well $$20\ m$$ below is $$[g=10\ ms^{-2}]$$

A
$$1.4\times 10^{4}\ J$$

B
$$1.5\times 10^{4}\ J$$

C
$$9.8\times 10\times 6\ J$$

D
$$18\ J$$

Solution
Question 3
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A lady dancer is dancing on a turn table. During dancing, she stretches her hands. Then:

A
the angular velocity increases

B
the angular velocity decreases

C
the angular velocity first increases then decreases

D
the angular velocity remains constant

Solution
Question 4
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Mark the correct option or options

A
The angular momentum of a rotating body must be parallel to angular velocity

B
The angular momentum may or may not be parallel to angular velocity

C
The kinetic energy of rotational body is half of product of angular momentum and angular velocity

D
Both $$(b)$$ and $$(c)$$ are correct

Solution
Question 5
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Consider the following two statements :
(A) Linear momentum of the system remains constant.
(B) Centre of mass of the system remains at rest.

A
A implies B and B implies A

B
A does not imply B and B does not imply A.

C
A implies B but B does not imply A.

D
B implies A but A does not imply B.

Solution
Question 6
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The graph between kinetic energy and momentum of a particle is plotted as shown in Fig. The mass of the moving particle is?

A
$$1$$ kg

B
$$2$$ kg

C
$$3$$ kg

D
$$4$$ kg

Solution

Kinetic Energy is given as:
$$K=\dfrac{P^2}{2m}$$
From the graph, $$4=\dfrac{4^2}{2m}\Rightarrow m=2 kg$$.
Question 7
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A cracker is thrown into air with a velocity of $$10$$ m/s at an angle of $$45^o$$ with the vertical. When it is at a height of $$0.5$$m from the ground, it explodes into a number of pieces which follow different parabolic paths. What is the velocity of centre of mass, when it is at a height of $$1$$m from the ground? ($$g=10m/s^2$$)

A
$$4\sqrt{5}$$ m/s

B
$$2\sqrt{5}$$ m/s

C
$$5\sqrt{4}$$ m/s

D
$$10$$ m/s

Solution

Motion of centre of mass is exactly similar to the motion of a body had it not exploded.
$$u_x=u\cos\theta =\dfrac{10}{\sqrt{2}}m/s, u_y=u\sin\theta =\dfrac{10}{\sqrt{2}}m/s$$
$$v_x=u_x=\dfrac{10}{\sqrt{2}}m/s$$
(since there is no change in the horizontal velocity)
$$v^2_y-u^2_y=2(-g)(h)$$
$$\Rightarrow v^2_y=\dfrac{100}{2}-2\times 10\times 1=30$$
Therefore, net velocity of $$CM=\sqrt{v^2_x+v^2_y}$$
$$=\sqrt{\dfrac{100}{2}+30}=\sqrt{80}=4\sqrt{5}m/s$$.
Question 8
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A ball falls vertically onto a floor with momentum p, and then bounces repeatedly. If the coefficient of restitution is e, then the total momentum impareted by the ball on the floor till the ball comes to rest is?

A
$$p(1+e)$$

B
$$\dfrac{p}{1-e}$$

C
$$p\left(1+\dfrac{1}{e}\right)$$

D
$$p\left(\dfrac{1+e}{1-e}\right)$$

Solution
Question 9
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A ball kept in a closed box moves in the box making collisions with the walls. The box is kept on a smooth surface. The velocity of the centre of mass.

A
Of the box remains constant

B
Of the (box $$+$$ ball) system remains constant

C
Of the ball remains constant

D
Of the ball relative to the box remains constant

Solution

As no external force acts on the ball $$+$$ box system, hence velocity of the system remains constant.
Question 10
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A train of mass M is moving on a circular track of radius 'R' with constant speed V. The length of the train is half of the perimeter of the track. The linear momentum of the train will be?

A
$$0$$

B
$$\dfrac{2MV}{\pi}$$

C
$$MVR$$

D
$$MV$$

Solution

If we treat the train as a ring of mass 'M', then its CM will be at a distance $$2R/\pi$$ from the centre of the circle. Velocity of the centre of mass is
$$V_{CM}=R_{CM}\omega =\dfrac{2R}{\pi}\dfrac{V}{R}$$
$$\left(\because \omega =\dfrac{V}{R}\right)$$
$$=\dfrac{2V}{\pi}\Rightarrow MV_{CM}=\dfrac{2MV}{\pi}$$
As the linear momentum of any system is $$MV_{CM}$$, the linear momentum of the train is $$2MV/\pi$$.