A circular ring of mass M and radius R is rotating about its axis with constant angular velocity ω. Two particles, each of mass m are attached gently to the opposite ends of the diameter of the ring. The angular velocity of the ring will now become:

1. \(\frac{m \omega}{M   +   2 m}\)

2. \(\frac{m \omega}{M   -   2 m}\)

3.  \(\frac{M \omega}{M   +   2 m}\)

4. \(\frac{M   +   2 m}{M \omega}\)

A disc is rotating with angular speed \(\omega.\) If a child sits on it, what is conserved here?
1. Linear momentum
2. Angular momentum
3. Kinetic energy
4. Potential energy

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 round disc of the moment inertia ${\mathrm{I}}_2$ about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia ${\mathrm{I}}_1$ rotating with an angular velocity ω about the same axis. The final angular velocity of the combination of discs is:

1. ω

2. $\frac{{\mathrm{I}}_1\omega }{{\mathrm{I}}_1+{\mathrm{I}}_2}$

3. $\frac{({\mathrm{I}}_1+{\mathrm{I}}_2)\omega }{{\mathrm{I}}_1}$

4. $\frac{{\mathrm{I}}_2\omega }{{\mathrm{I}}_1+{\mathrm{I}}_2}$

Two bodies have their moments of inertia I and 2I, respectively, about their axis of rotation. If their kinetic energies of rotation are equal, their angular momentum will be in the ratio:

1. 1 : 2

2. $\sqrt{2}$ :1

3. 1: $\sqrt{2}$

4. 2 : 1