A small mass attached to a string rotates on a frictionless table top as shown. If the tension on the string is increased by pulling the string causing the radius of the circular motion to decrease by a factor of \(2,\) the kinetic energy of the mass will

                              

1.	Increase by a factor of \(4\)
2.	Decrease by a factor of \(2\)
3.	Remain constant
4.	Increase by a factor of \(2\)

One-quarter sector is cut from a uniform circular disc of radius \(R.\)  This sector has mass \(M.\)  It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc.  Its moment of inertia about the axis of rotation is:

       
1. \(\frac{1}{2} M R^2\)
2. \(\frac{1}{4} M R^2\)
3. \(\frac{1}{8} M R^2\)
4. \(\sqrt{2} M R^2\)

A wheel is rotating at the rate of \(33~ \text{rev/min}\)  If it comes to stop in \(20 ~\text{s.}\) Then, the angular retardation will be
1. \(\pi \frac{\text{rad}}{\text{~s}^2}\)
2. \(11 \pi ~\text{rad} / \text{s}^2\)
3. \(\frac{\pi}{200} ~\text{rad} / \text{s}^2 \)
4. \(\frac{11 \pi}{200}~\text{rad} / \text{s}^2\)

A solid sphere is rotating about a diameter at an angular velocity \(w.\) If it cools so that its radius reduces to$\frac{\mathrm{}}{}$\(\frac1n\) of its  original value, its angular velocity becomes
1. \(\frac wn\) 
2. \(\frac{w}{{n}^2}\)
3. \(nw\)
4. \(n^2w\)

A horizontal platform is rotating with uniform angular velocity around the vertical axis passing through its centre. At some instant of time a viscous fluid of mass 'm' is dropped at the centre and is allowed to spread out and finally fall. The angular velocity during this period

1. Decreases continuously

2. Decreases initially and increases again

3. Remains unaltered

4. Increases continuously

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. \(\frac{L}{\sqrt{2}}\)

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  

The moments of inertia of two freely rotating bodies A and B are $I_A and I_B$ respectively. $I_A>I_B$ and their angular momenta are equal. If $K_A and K_B$ are their kinetic energies, then

1. $K=K_B$

2. $K_A>K_B$

3. $K_A<K_B$

4. $K_A=2K_B$

A body of mass \(M\) and radius \(R\) is rolling horizontally without slipping with speed \(v.\) It then rolls up a hill to a maximum height \(h.\) If \(h=\frac{5v^{2}}{6g},\) what is the moment of inertia of the body?
1. \(\frac{MR^{2}}{2}\)
2. \(\frac{2MR^{2}}{3}\)
3. \(\frac{3MR^{2}}{4}\)
4. \(\frac{2MR^{2}}{5}\)

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