A Solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (See figure). Both roll without slipping all throughout. The two climb maximum heights \(h_s\) and \(h_c\) on the incline. The ratio \(\frac{h_{s}}{h_{c}}\) is given by:

          
1. \( \frac{2}{\sqrt{5}} \)
2. \( \frac{14}{15} \)
3. \(\frac{4}{5} \)
4. \( 1\)

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of \(\theta,\) where \(\theta\) is the angle by which it has rotated, is given as \(k\theta^2\) (where \(k\) is constant). If its moment of inertia is \(I,\) then the angular acceleration of the disc is:
1. \(\frac{k}{I} \theta\)

2. \(\frac{k}{2 I} \theta\)

3. \(\frac{k}{4 I} \theta\)

4. \(\frac{2 k}{I} \theta\)$$

The moment of inertia of a body about a given axis is \(1.5\) kg-m². Initially, the body is at rest. In order to produce rotational kinetic energy of \(1200\) J, the angular acceleration of \(20\) rad/s² must be applied about the axis for a duration of:
1. \(5\) s
2. \(3\) s
3. \(2.5\) s
4. \(2\) s

A cord is wound around the circumference of the wheel of radius \(r.\) The axis of the wheel is horizontal and the moment of inertia about it is \(I.\) A weight \(mg\) is attached to the cord at the end. The weight falls from rest. After falling through a distance \('h',\) the square of the angular velocity of the wheel will be:
1. \( \dfrac{2 m g h}{I+2 m r^2} \)

2. \( \dfrac{2 m g h}{I+m r^2} \)

3. \( 2 g h\)

4. \( \dfrac{2 g h}{I+m r^2} \)