System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 67

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Question 1
1 / -0

The radius of gyration of a disc about its axis passing through its centre and perpendicular to its plane is

A
$${R}/{\surd 2}$$

B
$${R}/{2}$$

C
$${\surd 2}R$$

D
$$2R$$

Solution

Moment of Inertia of disc about center of mass is $$I=\cfrac{mR^{2}}{2}$$
Let a line parallel to axis at a distance $$d$$.
Radius of Gyration$$=R$$
Then, $$\cfrac{mR^{2}}{2}=mR^{2}$$
$$\Rightarrow\,R^{2}=\cfrac{R^{2}}{2}$$
$$\Rightarrow R=\cfrac{R}{\sqrt 2}$$
Question 2
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A plot of angular momentum of a rigid body (along y axis) about an axis with time (along x axis) gives rise to a straight line, whose equation is given by y = 3x+1. Find the torque acting on the body

A
8 N-m

B
3 N-m

C
30 N-m

D
0.3 N-m

Solution

We know that torque= dL/dt. Integrating, we get $$L=L_0+\tau t$$.

This equation resembles the equation of a straight line y= mx +C. Comparing with y=3x+1, we get the torque = 3 N-m

The correct option is (b)
Question 3
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Figure below shows the variation of the moment of inertia of a uniform rod about an axis normal to its length with the distance of the axis from the end of the rod. The moment of inertia of the rod about an axis passing through its centre and perpendicular to the its length is then

A
$$0.5 kg-m^2$$

B
$$0.15 kg-m^2$$

C
$$0.10 kg-m^2$$

D
$$0.05 kg-m^2$$

Solution

$$\dfrac{mL^2}{3} = 0.2$$
$$\dfrac{mL^2}{12} = I$$
$$\dfrac{I}{0.2} = \dfrac{1}{4} \Rightarrow I = 0.05$$
Question 4
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A ball is made of a material of density $$\rho$$ when $${ \rho }_{ oil }<{ \rho }<{ \rho }_{ water }$$ with $${ \rho }_{ oil }$$ and $${ \rho }_{ water }$$ representing the densities of oil and water, respectively. The oil and water are immiscible. If the above ball is in equilibrium in a mixture of this oil and water, which of the following pictures represents its equilibrium position?

A

B

C

D

Solution

$$\rho_)<\rho<\rho_\omega$$
$$\rho_\omega\rightarrow$$ Density of water
$$\rho_0\rightarrow$$ Density of oil
$$\rho\rightarrow$$ Density of ball
The principal is thathigher density materials settels down and lower density floats over it So$$\rho_0>\rho$$ ball will go down the oil and $$\rho_\omega>\rho$$ so ball will float above the water.
Question 5
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Four particles each of mass $$'m'$$ are placed at the corners of a square of side $$'L''$$. The radius of gyration of the system about an axis normal to the square and passing through its centre.

A
$$\dfrac { L }{ 2 }$$

B
$$\dfrac { L }{ \sqrt { 2 } }$$

C
$$L$$

D
$$\sqrt { 2 }\ L$$

Solution

I=m$$(\dfrac{l}{\sqrt{2}})^2\times 4=2ml^2$$
$$2ml^2=4mr^2$$
Radius of gyration r=$$\dfrac{l}{\sqrt{2}}$$
Question 6
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Two identical rods each of moment of inertia $$'I'$$ about a normal axis through centre are arranged in the from of a cross. The $$M.I$$. of the system about an axis through centre and perpendicular to the plane of system is:

A
$$2I$$

B
$$I$$

C
$$2.5\ I$$

D
$$6I$$

Solution

$$I_z=I_x=I_y$$
Where x and y and z are mutually perpendicular axes
$$I_0=I+I=2I$$
Question 7
1 / -0

Moment of inertia of a uniform rod of length $$L$$ and mass $$M$$, about an axis passing through $$L/4$$ from one end and perpendicular to its length is

A
$$\dfrac { 7 }{ 36 } { ML }^{ 2 }$$

B
$$\dfrac { { ML }^{ 2 } }{ 12 }$$

C
$$\dfrac { 7 }{ 48 } { ML }^{ 2 }$$

D
$$\dfrac { 11 }{ 48 } { ML }^{ 2 }$$

Solution

Moment of inertia through centre of rod=$$\dfrac{Ml^2}{12}$$
Moment of inertia through a point at a distance of $$\dfrac{l}{4}$$ from an end is I=$$\dfrac{Ml^2}{12}+\dfrac{Ml^2}{16}=\dfrac{7Ml^2}{48}$$
Question 8
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A small ball strikes a stationary uniform rod, which is free to rotate, in gravity-free space. The ball does not stick to the rod The rod will rotate about:

A
its centre of mass

B
the centre of mass of 'rod plus ball'

C
the point of impact on the ball on the rod

D
the point about which the moment of inertia of the rod plus ball' is minimum

Solution
Question 9
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Four rods of equal length l and mass m each form a square as shown in figure. The moment of inertia of the system about the axis 1 and axis 2 is

A
$$\dfrac { 2 }{ 3 } { ml }^{ 2 }$$,$$\dfrac { 5 }{ 3 } { ml }^{ 2 }$$,

B
$$\dfrac { 5 }{ 3 } { ml }^{ 2 }$$,$$\dfrac { 10 }{ 3 } { ml }^{ 2 }$$,

C
$$\dfrac { 1 }{ 3 } { ml }^{ 2 }$$,$$\dfrac { 5 }{ 3 } { ml }^{ 2 }$$,

D
$$\dfrac { 10 }{ 3 } { ml }^{ 2 }$$,$$\dfrac { 2 }{ 3 } { ml }^{ 2 }$$,

Solution
Question 10
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Find the $$M.I$$. of a thin uniform rod about an axis perpendicular to tis length and passing through a point which is at a distance $$"\dfrac { l }{ 3 } "$$ from one end. Then the radius of gyration about that axis.

A
$$\dfrac { l }{ 3 }$$

B
$$\dfrac { l }{ 2 }$$

C
$$\dfrac { l }{ 4 }$$

D
$$\dfrac { l }{ 6 }$$

Solution

Let R be radius of gyration
Moment of inertia through centre of rod is $$\dfrac{Ml^2}{12}$$
Moment of inertia through a point at a distance is $$\dfrac{l}{3}$$ from one end is equal to a distance of $$\dfrac{l}{6}$$ from the centre of rod is
$$\dfrac{Ml^2}{12}+M(\dfrac{l}{6})^2=\dfrac{Ml^2}{9}$$
$$\dfrac{Ml^2}{9}=MR^2$$
$$R=\dfrac{l}{3}$$