A constant torque acting on a uniform circular wheel changes its angular momentum from \(A_0\) to \(4A_0\) in \(4~\text{s}\). The magnitude of this torque is:
1. \(\dfrac{3A_0}{4}\)
2. \(4A_0\)
3. \(A_0\)
4. \(12A_0\)

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

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:

Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D${}_1$ has 2 kg mass and 0.2 m radius and initial angular velocity of 50 rad s${}^{-1}$. Disc D${}_2$ has 4 kg mass, 0.1 m radius and initial angular velocity of 200 rad s${}^{-1}$. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad.s${}^{-1}$) of the system is

1. 60

2. 100

3. 120

4. 40

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

$1.<mspace\; width="1em"></mspace>{\mathrm{K}}^2+<mspace\; width="1em"></mspace>{\mathrm{R}}^2$
$2.<mspace\; width="1em"></mspace>\frac{{\mathrm{K}}^2}{{\mathrm{R}}^2}$
$3.<mspace\; width="1em"></mspace>\frac{{\mathrm{K}}^2}{{\mathrm{K}}^2+<mspace\; width="1em"></mspace>{\mathrm{R}}^2}$
$4.<mspace\; width="1em"></mspace>\frac{{\mathrm{R}}^2}{{\mathrm{K}}^2+<mspace\; width="1em"></mspace>{\mathrm{R}}^2}$

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. 

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 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

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 \(\text R\)