A uniform sphere of mass m and radius r rolls without slipping down an inclined plane,  inclined at an angle of 45° to the horizontal. The minimum value of the coefficient of friction between the sphere and the inclined plane will be:

1. 1/6

2. 2/7

3. 1/5

4. 1/7

A disc hangs from an ideal string wrapped around it. If it is allowed to fall, the acceleration of the disc will be:

A block of mass m slides down on a smooth inclined plane and reaches the bottom with speed v. If the same mass is in the form of a ring which rolls down on an identical inclined plane, where friction is sufficient for pure rolling, the speed of the ring at the bottom will be:

1. $v$

2. $v\sqrt{2}$

3. $\frac{v}{\sqrt{2}}$

4. $v\sqrt{\frac{2}{5}}$

A hollow sphere of mass m is rolling with a speed v on a smooth horizontal surface and strikes a massless spring of force constant k attached to a massless smooth platform. The maximum compression of the spring will be:
            
1. $\sqrt{\frac{m}{k}}v$

2. $\sqrt{\frac{2m}{3k}}v$

3. $\sqrt{\frac{m}{5k}}v$

4. $\sqrt{\frac{5m}{3k}}v$

A solid cylinder of mass \(20\) kg rotates about its axis with angular speed \(100\) rad s^-1. The radius of the cylinder is \(0.25\) m. The kinetic energy associated with the rotation of the cylinder is:
1. \(3000\) J
2. \(3125\) J
3. \(2528\) J
4. \(2100\) J

A hoop of radius \(2\) m weighs \(100\) kg. It rolls along a horizontal floor so that its center of mass has a speed of \(20\) cm/s. How much work has to be done to stop it?
1. \(10\) J
2. \(9\) J
3. \(4\) J
4. \(6\) J

A solid cylinder rolls up an inclined plane of the angle of inclination 30°. At the bottom of the inclined plane, the centre of mass of the cylinder has a speed of 5 m/s. How far will the cylinder go up the plane?

1. 4.9 m
2. 1.3 m
3. 4.7 m
4. 3.8 m

A disc rotating about its axis with angular speed ${\omega }_0$ is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is R. The linear velocities of the points A, B, and C on the disc (as shown in the figure) respectively are:
     
1. \(\omega_0 R, \omega_0 R, \frac{\omega_0 R}{2}\)
2. \(\frac{\omega_0 R}{2}, \omega_0 R, \omega_0 R\)
3. \(\omega_0 \mathrm{R}, \frac{\omega_0 \mathrm{R}}{2}, \omega_0 \mathrm{R}\)
4. \(\omega_0 R, \omega_0 R, 0\)

A cylinder of mass \(10\) kg and radius of \(15\) cm is rolling perfectly on a plane of inclination \(30^\circ.\) The coefficient of static friction \(\mu_s=0.25.\) How much is the force of friction acting on the cylinder?
1. \(16.3\) N
2. \(15.5\) N
3. \(19.1\) N
4. \(20.0\) N

Choose the incorrect statement.

1.  The centre of mass of a two-particle system lies on the line joining the two particles, being closer to the heavier particle.

2.  In rolling, the point of contact of the rolling body remains at rest relative to the surface on which it is rolling.

3.  The parallel axis theorem is applicable only for laminar bodies.

4.  A particle moving on a straight line may have non-zero angular momentum about a point.