System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 69

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Question 1
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A thin rod of length $$4l$$, mass $$4m$$ is bent at the points as shown in figure. What is the moment of inertia of the rod about the axis passing point O and perpendicular to the plane of the paper?

A
$$\cfrac{m{l}^{2}}{3}$$

B
$$\cfrac{10m{l}^{2}}{3}$$

C
$$\cfrac{m{l}^{2}}{12}$$

D
$$\cfrac{m{l}^{2}}{24}$$

Solution

Since the mass of rod is 54m and length is 4l.
so massC = m
and length of AB = BO=OC= CD = l
WE, know moment of Inertia of a rod about to end = $$\dfrac{ml^2}{3}$$

So, moment of Inertia of AB,BO,OC,CD about B,O,O,C respectively = $$\dfrac{ml^2}{3}$$

FRom parallel axis theorem

Moment of Inertia of AB about O.

$$= \dfrac{ml^2}{3} + ml^2 = \dfrac{4ml^2}{3}$$

Similerly od CD about O $$= \dfrac{4ml^2}{3}$$

SO moment of Inertia Rod about O

$$= \dfrac{ml^2}{3} +\dfrac{ml^2}{3} + \dfrac{4ml^2}{3} + \dfrac{4ml^2}{3}$$

$$= \dfrac{10ml^2}{3}$$

Hence option(B) is correct.
Question 2
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Three identical thin rods each of length $$L$$ and mass $$m$$ are welded perpendicular to one another as shown in the figure. The assembly is rotated about an axis that passes through the end of one rod and is parallel to another. Find moment of inertia of this structure.

A
$$\dfrac { 11 }{ 12 } m{ L }^{ 2 }$$

B
$$\dfrac { 9 }{ 16 } m{ L }^{ 2 }$$

C
$$\dfrac { 8 }{ 11 } m{ L }^{ 2 }$$

D
$$2m{ L }^{ 2 }$$

Solution
Question 3
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Radius of gyration of a body about an axis at a distance 6 cm from its center of mass is 10 cm. Its radius of gyration about a parallel axis through its center of mass is :

A
6 cm

B
8 cm

C
10 cm

D
12 cm

Solution
Question 4
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A man of mass $$80kg$$ is riding on a small cart of mass $$40kg$$ which is rolling along a level floor at a speed of $$2m/s$$. He is running on the cart so that his velocity relative to the cart is $$3m/s$$ in the direction opposite to the motion of cart .What is the speed of the centre of mass of the system :-

A
$$1.5m/s$$

B
$$1 m/s$$

C
$$3m/s$$

D
$$0$$

Solution
Question 5
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A rigid of mass $$M$$ slides along semi circular track (in vertical plane) followed by a flat track. At the given instant velocity of end $$'B'$$ is $$V$$ along the horizontal plane. Then at the given instant

A
Angular speed of the rod $$\dfrac {v}{r} $$

B
$$ V_{CM} = \dfrac {V} { \sqrt{3}} $$

C
$$ \vec {L} $$ at $$ 'O'$$ is $$\dfrac {4}{3} mvr$$

D
$$\dfrac {KE_r}{KE_t} = \dfrac {4}{3} mvr $$

Solution
Question 6
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A wheel of grind stone has applied at its axle $$2\ cm$$ in radius, a constant tangential force of $$600\ N$$. Calculate the torque acting on it, and angular momentum acquired by it after $$8\ sec$$ starting from rest.

A
$$24 m^2/s$$

B
$$108 m^2/s$$

C
$$72 m^2/s$$

D
$$96 m^2/s$$

Solution
Question 7
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Three identical thin rods, each of mass m and length L are joined to form an equilateral triangle. Find the moment of inertia of the triangle about one of its sides:

A
$$\dfrac{mL^2}{6}$$

B
$$\dfrac{3mL^2}{2}$$

C
$$\dfrac{mL^2}{3}$$

D
$$\dfrac{mL^2}{2}$$

Solution
Question 8
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A rod of length L is pivoted at an end. The linear mass density of the rod $$\lambda$$ varies with the distance x from end as $$\lambda = ax^2 + b\ kgm^{-1}$$, where a and b are positive constants. Find the moment of inertia of the rod about the axis passing through this end and perpendicular to its length.

A
$$\dfrac{aL^5}{5} + \dfrac{bL^3}{3}$$

B
$$\dfrac{2aL^5}{3} + \dfrac{2bL^3}{5}$$

C
$$\dfrac{aL^5}{7} + \dfrac{bL^3}{3}$$

D
$$\dfrac{aL^5}{5} + \dfrac{2bL^3}{3}$$

Solution
Question 9
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The time period of a bar pendulum when suspended at distances $$30\ cm$$ and $$50\ cm$$ from its centre of gravity comes out to be the same. If the mass of the body is $$2kg$$. Find out its moment of inertia about an axis passing through first point.

A
$$0,24\ kg-m^{2}$$

B
$$0.72\ kg-m^{2}$$

C
$$0.48\ kg-m^{2}$$

D
Data insufficient

Solution

For point $$-1$$
$$\tau =mg \left (\dfrac {30}{100}\right)\sin \Theta$$
As $$\Theta $$ is very small
$$\tau =\dfrac {3\ mg}{10}\Theta$$
$$C_1=\dfrac {3\ mg}{10}$$
where $$t$$ point $$2$$
$$\tau =\dfrac {5}{10}mg \Theta $$
$$C_2=\dfrac {5}{10}mg$$
As the period of $$SHM(T)=\Theta n\sqrt {\dfrac {I}{C}}$$
Let at $$COM\ I$$ is the moment of inertia
By parallel axis theorem
$$I_1=I+(2)\left (\dfrac {3}{10}\right)^2$$
$$I_2=I+2\left (\dfrac {5}{10}\right)^2$$
Because time period of blue points are equal
$$2\pi \sqrt {\dfrac {I_1}{c_1}}=2\pi \sqrt {\dfrac {T_2}{c_2}}\ \Rightarrow \dfrac {I_1}{I_2}=\dfrac {c_1}{c_2}$$
$$\Rightarrow \ \dfrac {I+2\left (\dfrac {3}{10}^2 \right)}{I+2\left (\dfrac {5}{10}\right)^2 }=\dfrac {\dfrac {3\ mg}{10}}{\dfrac {5\ mg}{10}}=\dfrac {3}{5}$$
$$\Rightarrow \ I=0.30\ kg\ m^2$$
At point $$-1, I_1=0.30+2\left (\dfrac {3}{10}\right)^2 =0.48\ kg\ m^2$$
Option $$C$$ is correct
Question 10
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Two small sphere of masses $$10kg$$ and $$30kg$$ are joined by a rod of length $$0.5m$$ and of negligible mass . The M.I of the system about a normal axis through centre of mass of the system is

A
$$1.875kgm^2$$

B
$$245kgm^2$$

C
$$0.75khm^2$$

D
$$1.75kgm^2$$

Solution