A cylinder of mass 'M' is suspended by two strings wrapped around it as shown. The acceleration 'a' and the tension T when the cylinder falls and the string unwinds itself are, respectively,
             

1.  $\mathrm{a} = \mathrm{g}, \mathrm{T} = \frac{\mathrm{Mg}}{2}$

2.  $\mathrm{a} = \frac{g}{2}, \mathrm{T} = \frac{\mathrm{Mg}}{2}$

3.  $\mathrm{a} = \frac{g}{3}, \mathrm{T} = \frac{\mathrm{Mg}}{3}$

4.  $\mathrm{a} = \frac{2g}{3}, \mathrm{T} = \frac{\mathrm{Mg}}{6}$

A sphere can not roll on:

1.  a smooth horizontal surface 

2.  a rough horizontal surface 

3.  a smooth inclined surface 

4.  a rough inclined surface 

A solid cylinder and a solid sphere, both having the same mass and radius, are released from a rough inclined plane. Both roll without slipping. Then,

1.  the force of friction that acts on the two is the same

2.  the force of friction is greater in the case of a sphere than in a cylinder

3.  the force of friction is greater in the case of a cylinder than in a sphere

4.  the force of friction will depend on the nature of the surface of the body that is moving and that of the inclined surface and is independent of the shape and size of the moving body

The speed of a uniform spherical shell after rolling down an inclined plane of vertical height h from rest is:

1.  \(\sqrt{\frac{10   g h}{7}}\)

2.    \(\sqrt{\frac{6   g h}{5}}\)

3.  \(\sqrt{\frac{4   g h}{5}}\)

4.  \(\sqrt{2   g h}\)

A solid sphere is in rolling motion. In rolling motion, a body possesses translational kinetic energy (K_t) as well as rotational kinetic energy (K_r) simultaneously. The ratio K_t : (K_t + K_r) for the sphere will be:

1. 7:10

2. 5:7

3. 10:7

4. 2:5

A disc and a solid sphere of the same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?

1. Sphere

2. Both reach at the same time

3. Depends on their masses

4. Disc

The ratio of the acceleration for a solid sphere (mass \(m\) and radius \(R\)) rolling down an incline of angle \(\theta\) without slipping and slipping down the incline without rolling is:
1. \(5:7\)
2. \(2:3\)
3. \(2:5\)
4. \(7:5\) 

A solid cylinder of mass \(3\) kg is rolling on a horizontal surface with a velocity of \(4\) ms^-1. It collides with a horizontal spring of force constant \(200\) Nm^-1. The maximum compression produced in the spring will be:
1. \(0.5\) m
2. \(0.6\) m
3. \(0.7\) m
4. \(0.2\) m

A wheel of radius R rolls without slipping on the ground with a uniform velocity v. The relative acceleration of the topmost point of the wheel with respect to the bottommost point is:

1. $\frac{{\mathrm{v}}^2}{\mathrm{R}}$

2. $\frac{2{\mathrm{v}}^2}{\mathrm{R}}$

3. $\frac{{\mathrm{v}}^2}{2\mathrm{R}}$

4. $\frac{4{\mathrm{v}}^2}{\mathrm{R}}$