A solid sphere slides from the top to the bottom of a smooth inclined plane. It takes time T. If it rolls on a rough inclined plane of the same dimension, then the time taken by it to reach from top to the bottom will be:

1.  $\sqrt{3}T$

2.  $\sqrt{\frac{2}{5}}T$

3.  $\sqrt{\frac{5}{3}}T$

4.  $\sqrt{\frac{7}{5}}T$

 A disc is set in pure roll on an inclined plane. Let us take three points on the disc as shown in the figure. Then v_B : v_C : v_D equals: (v_B, v_c and v_D are corresponding speeds)

1.  $1$ $:$ $1$ $:$ $1$

2.  $1$ $:$ $2$ $:$ $\sqrt{2}$

3.  $2$ $:$ $1$ $:$ $\sqrt{2}$

4.  $1$ $:$ $\sqrt{2}$ $:$ $\sqrt{2}$

The coefficient of friction between a solid cylinder and a rough inclined plane is just sufficient for the pure rolling of the cylinder. Which of the following bodies can move with pure rolling on the inclined plane?

1.  Hollow cylinder

2.  Solid sphere

3.  Hollow sphere

4.  Ring

A disc rolls without slipping, climbs on a rough incline, and comes back. During the motion on the incline,

1.  Friction is downward during the upward journey.

2.  Friction is upward during the upward journey.

3.  Friction is zero.

4.  Information is insufficient to predict.

A ring and a disc have the same mass and roll without slipping at the same linear velocity v. If the total kinetic energy of the ring is 8 J, then the total kinetic energy of the disc will be:

1. 8 J

2. 6 J

3. 16 J

4. 4 J

A solid homogenous sphere is moving on a rough horizontal surface, partially rolling and partially sliding. During this motion of this sphere:

1. The total kinetic energy is conserved.

2. The angular momentum of the sphere about the point of contact is conserved

3. Only the rotational kinetic energy about the centre of mass is conserved

4. Angular momentum about the centre of mass is conserved

A string of negligible thickness is wrapped several times around a cylinder kept on a rough horizontal surface. A man standing at a distance l from the cylinder holds one end of the string and pulls the cylinder towards him. There is no slipping anywhere. The length of the string that passed through the hand of the man while the cylinder reaches his hands is-[Assume radius of the cylinder is negligible compared to length 'l' of string]
       

A solid sphere is rolling without slipping such that the velocity of its centre of mass is v. Ratio of speeds  of horizontal extreme points A and B is
                             
1. 1:1

2. $\sqrt{2}$:1

3. 2:1

4. 1:$\sqrt{2}$

The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle  θ without slipping and slipping down the incline without rolling will be:
1. 5:7
2. 2:3
3. 2:5
4. 7:5

The disc is placed on a rough horizontal surface and given only angular velocity ${\omega }_0$. The velocity of the centre of mass when it starts rolling is: