A thin uniform circular disc of mass \(M\) and radius \(R\) is rotating in a horizontal plane about an axis passing through its center and perpendicular to its plane with an angular velocity $\omega$. Another disc of the same dimensions but of mass \(\frac{1}{4}M\) is placed gently on the first disc co-axially. The angular velocity of the system will be:

1.	$\frac{2}{3}\mathrm{\omega }$	2.	$\frac{4}{5}\mathrm{\omega }$
3.	$\frac{3}{4}\mathrm{\omega }$	4.	$\frac{1}{3}\mathrm{\omega }$

When a torque acting upon a system is zero, then which of the following will be constant 

A wheel whose moment of inertia is 12 ${\mathrm{Kgm}}^2$ has an initial angular velocity of 40 rad/sec. A constant torque of 20 Nm acts on the wheel. The time in which the wheel is accelerated to 100 rad/sec is

1.  72 seconds

2.  16 seconds

3.  8 seconds

4.  36 seconds

A flywheel is in the form of a uniform circular disc of radius 1 m and mass 2 kg. The work which must be done on it to increase its frequency of rotation from 5 rev ${\mathrm{s}}^{-1}$ to 10 rev ${\mathrm{s}}^{-1}$ is approximately

1.  1.5 x ${10}^2$ J

2.  3.0 x ${10}^2$ J

3.  1.5 x ${10}^3$ J

4.  3.0 x ${10}^3$ J

A constant torque acting on a uniform circular wheel changes its angular momentum from ${\mathrm{A}}_0$ to $4 {\mathrm{A}}_0$ in 4s. The magnitude of this torque is 

1. $\frac{3{\mathrm{A}}_0}{4}$

2. $4 {\mathrm{A}}_0$

3. ${\mathrm{A}}_0$

4. $12 {\mathrm{A}}_0$

A body rolls down an inclined plane without slipping. The fraction of total energy associated with its rotation will be

$1. {\mathrm{K}}^2+ {\mathrm{R}}^2$
$2. \frac{{\mathrm{K}}^2}{{\mathrm{R}}^2}$
$3. \frac{{\mathrm{K}}^2}{{\mathrm{K}}^2+ {\mathrm{R}}^2}$
$4. \frac{{\mathrm{R}}^2}{{\mathrm{K}}^2+ {\mathrm{R}}^2}$

Where k is radius of gyration of the body about an axis passing through centre of mass and R is the radius of the body. 

In the following figure, a weight W is attached to a string wrapped round a solid cylinder of mass M mounted on a frictionless horizontal axle at O.

If the weight starts from rest and falls a distance h, then its speed at that instant is

1. Proportional to \(\text R\)

2. Proportional to \(1 \over R\)

3. Proportional to \(1 \over R^2\)

4. Independent of \(\text R\)

A solid cylinder rolls down an inclined plane that has friction sufficient to prevent sliding. The ratio of rotational energy to total kinetic energy is

1.  $\frac{1}{2}$

2.  $\frac{1}{3}$

3.  $\frac{2}{3}$

4.  $\frac{3}{4}$

A swimmer while jumping into water from a height easily forms a loop in the air, if