System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 81

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Question 1
1 / -0

Two pendulums each of mass $$m _ { 1 }$$ and $$m _ { 2 }$$ and length l are initially at rest as shown in figure. The ball of $$m _ { 1 } ,$$ is released and it strikes the ball of $$m _ { 2 } ,$$ if collision is perfectly inelastic, find the rise of centre of mass of the system after collision:

A
$$H = h \left( \frac { m _ { 2 } } { m _ { 1 } + m _ { 2 } } \right) ^ { 2 }$$

B
$$H = h \left( \frac { m _ { 1 } } { m _ { 1 } + m _ { 2 } } \right) ^ { 2 }$$

C
$$H = h \left( \frac { m _ { 1 } + m _ { 2 } } { m _ { 1 } } \right) ^ { 2 }$$

D
$$H = h \left( \frac { m _ { 1 } + m _ { 2 } } { m _ { 1 } m _ { 2 } } \right) ^ { 2 }$$

Solution
Question 2
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A ladder PQ of length 5m inclined to a vertical wall is slipping over a horizontal surface with a velocity of $$2 ms^{-1}$$, when Q is at a distance 3 M from the ground. Calculate the velocity of centre of mass of the rod at this moment.

A
$$2.5ms^{-1}$$

B
$$1.25ms^{-1}$$

C
$$1.75ms^{-1}$$

D
$$2.75ms^{-1}$$

Solution
Question 3
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A piece of paper is in the form of square . Two corners of these square are folded to make it appear like figure $$2$$ . Both the corners are put together at center of square $$O$$ . If $$O$$ is taken to be $$(0,0)$$ , the center of mass of new will be :

A
$$\left( - \frac { a } { 8 } , 0 \right)$$

B
$$\left( - \frac { a } { 6 } , 0 \right)$$

C
$$\left( - \frac { a } { 12 } , 0 \right)$$

D
$$\left( \frac { a } { 12 } , 0 \right)$$

Solution
Question 4
1 / -0

Distance of the center of mass of a solid uniform cone from its verted is zo. If the radius of its base isand its height is h, then zo is equal to

A
$$\frac { h ^ { 2 } } { 4 R }$$

B
$$\frac { 3 h } { 4 }$$

C
$$\frac { 5 h } { 8 }$$

D
$$\frac { 3 h ^ { 2 } } { 8 R }$$

Solution
Question 5
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linear mass density of a rod AB ( of length l) waries with distance X from end A of the rod as given by $$\lambda =\lambda _{0}x^{4}$$ Here $$\lambda _{0}$$ is constant. Distance of centre of mass of the rod from end B is

A
$$\frac{l}{6}$$

B
$$\frac{5l}{6}$$

C
$$\frac{3l}{4}$$

D
$$\frac{l}{4}$$

Solution
Question 6
1 / -0

Two hail stones with radii in the ratio of 1 : 2 fall from a great height through the atomosphere. Then the ratio of their momenta after they have a attained terminal velocity is.

A
1 : 1

B
1 : 4

C
1 : 16

D
1 : 32

Solution
Question 7
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Two pith balls carrying equal charges are suspended from a common point by strings of equal length, the equilibrium separation between them is $$r$$ . Now the strings are rigidly clamped at half the height. The equilibrium separation between the balls now become

A
$$\left( \frac { r } { \sqrt [ 3 ] { 2 } } \right)$$

B
$$\left( \frac { 2 r } { \sqrt { 3 } } \right)$$

C
$$\left( \frac { 2 r } { 3 } \right)$$

D
$$\left( \frac { 1 } { \sqrt { 2 } } \right) ^ { 2 }$$
Question 8
1 / -0

Two bodies of masses 2 kg and 4 kg are moving with velocities 2 m/s and 1 m/s towards each other due to mutual gravitational attraction. What is the velocity of their centre of mass ?

A
5.3 m/s

B
6 m/s

C
Zero

D
8.1 m/s

Solution
Question 9
1 / -0

A thin rod of length 4l mass 4 m is bent at the points as shown in the fig. What is the moment of inertia of the rod about the axis passing point O& perpendiular to the plane of the paper

A
$$\dfrac { { m\ell }^{ 2 } }{ 3 } $$

B
$$\dfrac { 10{ m\ell }^{ 2 } }{ 3 } $$

C
$$\dfrac { { m\ell }^{ 2 } }{ 12 } $$

D
$$\dfrac { { m\ell }^{ 2 } }{ 24 } $$

Solution

$$\begin{array}{l}\text { we know moment of Inertia of a Rod about } \\\text { an axis passing perpendicular through its end- } \\\Rightarrow I_{0 B}=I_{0 C}=\frac{m L^{2}}{3}\end{array}$$
$$\begin{array}{l}\text { Now moment of Inertia of Part } C D \\\qquad \begin{array}{rl} I_{C D}=I_{C}+m d^{2} & I_{C}=I \text { about untreof Rod } \\ d= & \text { Distance of centre } \\\text { from the axis. }\end{array} \\\Rightarrow I_{C D}=\frac{m L^{2}}{12}+m\left(\frac{\sqrt{5} L}{2}\right)^{2}=\frac{4}{3} m L^{2} \\\Rightarrow I_{C D}=I_{A B}=\frac{4}{3} m L^{2} \text { (Both are similar) } \\\Rightarrow I_{\text {Total }}=I_{A B}+I_{O B}+I_{0 C}+I_{C D}=\frac{10}{3} m L^{2}\end{array}$$
Question 10
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The figure shows four objects rotating about an axis passing through CM. Which of these has angular momentum about the center of mass in direction of angular velocity

A
Semi Cone

B
Bottle

C
Arbitrary shape Potato

D
All of the above