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. $\frac{L}{\sqrt{2}}$

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 inertia of the cross about an axis passing through the point at which the two rods are joined and are 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. $\frac{M{l}^2}{3}$

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

1.  50%

2.  100%

3.  125%

4.  25%

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  

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%

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

One circular ring and one circular disc both having the same mass and radius. The ratio of their moments of inertia about the axes passing through their centres and perpendicular to planes will be

1. 1:1

2. 2:1

3. 1:2

4. 4:1

A wheel of radius R rolls on the ground with a uniform velocity v. The velocity of topmost point relative to the bottommost point is

1. v

2. 2v

3. v/2

4. zero

The figure shows a uniform solid block of mass M and edge lengths a, b and c. Its M.O.I. about an axis through one edge and perpendicular (as shown) to the large face of the block is 

1. $\frac{M}{3}\left(a^2+b^2\right)$

2. $\frac{M}{4}\left(a^2+b^2\right)$

3. $\frac{7M}{12}\left(a^2+b^2\right)$

4. $\frac{M}{12}\left(a^2+b^2\right)$

If the net external forces acting on the system of particles is zero, then which of the following may vary ?

1. Momentum of the system

2. Velocity of centre of mass

3. Position of centre of mass

4. None of the above