A solid sphere of mass M and radius R is in pure rolling with angular speed $\omega$ on a horizontal plane as shown. The magnitude of angular momentum of the sphere about origin O is:

1.  $\frac{7}{5}M{R}^2\omega$

2.  $\frac{3}{2}M{R}^2\omega$

3.  $\frac{1}{2}M{R}^2\omega$

4.  $\frac{2}{3}M{R}^2\omega$

A thin circular ring \(\mathrm{M}\) and radius \(\mathrm{r}\) is rotating about its axis with a constant angular velocity ω. Four objects, each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be:

1. $\frac{\mathrm{M\omega }}{4\mathrm{m}}$

2. $\frac{\mathrm{M\omega }}{\mathrm{M}+4\mathrm{m}}$

3. $\frac{(\mathrm{M}+4\mathrm{m})\mathrm{\omega }}{\mathrm{M}}$

4. $\frac{(\mathrm{M}+4\mathrm{m})\mathrm{\omega }}{\mathrm{M}+4\mathrm{m}}$

A disc is rotating with angular speed \(\omega.\) If a child sits on it, what is conserved here?

Two discs, each having moment of inertia 5 kg-${\mathrm{m}}^2$ about their central axis, rotating with speeds 10 rad ${\mathrm{s}}^{-1}$ and 20 rad ${\mathrm{s}}^{-1}$ in the same direction, are brought in contact face to face with their axes of rotation coinciding. The loss of kinetic energy in the process is:

1. 2 J

2. 5 J

3. 125 J

4. 0 J

A rigid body has moment of inertia 12 kgm${}^2$ about
a fixed axis and the body is rotating about it, such
that angular displacement $\mathrm{\theta }<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\left(2\mathrm{t}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>\frac{{\mathrm{t}}^2}{3}\right)$ rad at time
t second. Kinetic energy of body at t = 3 second

1. Zero

2. 15 J

3. 20 J

4. 30 J

When a mass is rotating in a plane about a fixed point, its angular momentum is directed along:

1. a line perpendicular to the plane of rotation

2. the line making an angle of 45^o to the plane of rotation

3. the radius

4. the tangent to the orbit

A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity $\omega .$If two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity -

(1) $\frac{\omega \left(M-2m\right)}{M+2m}$                                           

(2) $\frac{\omega M}{M+2m}$

(3) $\frac{\omega \left(M+2m\right)}{M}$                                         

(4) $\frac{\omega M}{M+m}$