The radius of gyration of a uniform rod of length L about an axis passing through its centre of mass is

1. $\frac{L}{2\sqrt{3}}$

2. $\frac{{L}^{2}}{12}$

3. $\frac{L}{\sqrt{3}}$

4. $\frac{L}{\sqrt{2}}$

Concept Questions :-

Moment of inertia
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Two uniform, thin identical rods each of mass M and length l are joined together to form a cross. What will be the moment of inertial of the cross about an axis passing through the point at which the two rods are joined and perpendicular to the plane of the cross

1. $\frac{M{l}^{2}}{12}$

2. $\frac{M{l}^{2}}{6}$

3. $\frac{M{l}^{2}}{4}$

4. $\frac{M{l}^{2}}{3}$

Concept Questions :-

Moment of inertia
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If the linear momentum is increased by 50%, the kinetic energy will increase by

1.  50%

2.  100%

3.  125%

4.  25%

Concept Questions :-

Linear momentum
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A solid sphere, a hollow sphere, and a disc, all having the same mass and radius, are placed at the top of a smooth incline and released. Least time will be taken in reaching the bottom by

1.  the solid sphere

2.  the hollow sphere

3.  the disc

4.  all will take the same time

Concept Questions :-

Rolling motion
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A bullet of mass m and velocity v is fired into a block of mass M and sticks to it. The final velocity of the system equals

1.

2.

3.

4.  None of these

Concept Questions :-

Linear momentum
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If the kinetic energy of a body increases by 0.1%, the percent increase of its momentum will be

1.  0.05%

2.  0.1%

3.  1.0%

4.  10%

Concept Questions :-

Linear momentum
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A bullet of mass 5 g is fired at a velocity of 900 m${s}^{-1}$ from a rifle of mass 2.5 kg.  What is the recoil velocity of the rifle?

(1) 0.9 $m{s}^{-1}$                                     (2) 180 $m{s}^{-1}$

(2) 900 $m{s}^{-1}$                                     (4) 1.8 $m{s}^{-1}$

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A particle P moving with speed v undergoes a head-on elastic collision with another particle Q of identical mass but at rest.  After the collision-

(1) both P and Q move forward with speed $\frac{v}{2}$.

(2) both P and Q move forward with speed $\frac{v}{\sqrt{2}}$.

(3) P comes to rest and Q moves forward with speed v.

(4) P and Q move in opposite directions with speed $\frac{v}{\sqrt{2}}$.

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The moments of inertia of two freely rotating bodies A and B are  respectively. ${I}_{A}>{I}_{B}$ and their angular momenta are equal. If  are their kinetic energies, then

(1) $K={K}_{B}$

(2) ${K}_{A}>{K}_{B}$

(3) ${K}_{A}<{K}_{B}$

(4) ${K}_{A}=2{K}_{B}$

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A solid cylinder and a hollow cylinder, both of the same mass and same external diameter are released from the same height at the same time on an inclined plane. Both roll down without slipping. Which one will reach the bottom first?

1.  Both together only when the angle of inclination of the plane is ${45}^{0}$

2.  Both together

3.  Hollow cylinder

4.  Solid cylinder