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.  \(\dfrac{7}{5} M R^{2} \omega\)

2.  \(\dfrac{3}{2} M R^{2} \omega\)

3.  \(\dfrac{1}{2} M R^{2} \omega\)

4.  \(\dfrac{2}{3} M R^{2} \omega\)

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

A thin circular ring of mass \(M\) and radius \(r \) is rotating about its axis with constant angular velocity \(ω.\) Two objects each of mass \(m\) are attached gently to the opposite ends of the diameter of the ring. The ring now rotates with angular velocity given by:

1. \(\dfrac{2 M \omega}{M + 2 m}\)

2. \(\dfrac{\left(\right. M + 2 m \left.\right) \omega}{M}\)

3. \(\dfrac{M \omega}{M + 2 m}\)

4. \(\dfrac{\left(\right. M + 2 m \left.\right) \omega}{2 m}\)

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are \(0.1~\text{kg-m}^2\) and \(10~\text{rad s}^{–1}\) respectively while those for the second one are \(0.2~\text{kg-m}^2\) and \(5~\text{rad s}^{–1}\) respectively. At some instant, they get stuck together and start rotating as a single system about their common axis with some angular speed \(\omega.\) Then \(\omega\) is:
1. \(\dfrac{5}{3}~\text{rad/s}\)
2. \(\dfrac{10}{3}~\text{rad/s}\)
3. \(\dfrac{15}{3}~\text{rad/s}\)
4. \(\dfrac{20}{3}~\text{rad/s}\)

The angular momentum about any point of a single particle moving with constant velocity: