A solid sphere is rotating freely about its axis of symmetry in free space. The radius of the sphere is increased keeping its mass the same. Which of the following physical quantities would remain constant for the sphere?

Two discs of the same moment of inertia are rotating about their regular axis passing through centre and perpendicular to the plane of the disc with angular velocities \(\omega_1\) ${\mathrm{}}_{}$and \(\omega_2\). They are brought into contact face to face with their axis of rotation coinciding. The expression for loss of energy during this process is:
1. \(\frac{1}{4}I\left(\omega_1-\omega_2\right)^2\)
2. \(I\left(\omega_1-\omega_2\right)^2\)
3. \(\frac{1}{8}I\left(\omega_1-\omega_2\right)^2\)
4. \(\frac{1}{2}I\left(\omega_1-\omega_2\right)^2\)

Two rotating bodies \(A\) and \(B\) of masses \(m\) and \(2m\) with moments of inertia \(I_A\) and \(I_B\)  \(\left(I_B>I_A\right)\) have equal kinetic energy of rotation. If \(L_A\) and \(L_B\) be their angular momenta respectively, then:
1. \(L_A = \frac{L_B}{2}\)
2. \(L_A = 2L_B\)
3. \(L_B>L_A\)
4. \(L_A>L_B\)

A mass \(m\) moves in a circle on a smooth horizontal plane with velocity \(v_0\) at a radius \(R_0\). The mass is attached to a string that passes through a smooth hole in the plane, as shown.
   
The tension in the string is increased gradually and finally, \(m\) moves in a circle of radius \(\frac{R_0}{2}\). The final value of the kinetic energy is:
1. \( m v_0^2 \)
2. \( \frac{1}{4} m v_0^2 \)
3. \( 2 m v_0^2 \)
4. \( \frac{1}{2} m v_0^2\)

A force $\stackrel{}{}$\(\vec{F}=\alpha \hat{i}+3 \hat{j}+6 \hat{k}\) is acting at a point  \(\vec{r}=2 \hat{i}-6 \hat{j}-12 \hat{k}\). The value of \(\alpha\) for which angular momentum about the origin is conserved is:
1. \(-1\)
2. \(2\)
3. zero
4. \(1\)

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

A circular platform is mounted on a frictionless vertical axle. Its radius \(R = 2~\text{m}\) and its moment of inertia about the axle is \(200~\text{kg m}^2\). It is initially at rest. A \(50~\text{kg}\) man stands on the edge of the platform and begins to walk along the edge at the speed of \(1~\text{ms}^{-1}\) relative to the ground. The time taken by man to complete one revolution is:
1. \(\frac{3\pi}{2}\text{s}\)
2. \(2\pi~\text{s}\)
3. \(\frac{\pi}{2}\text{s}\)
4. \(\pi~\text{s}\)

A circular disk of a moment of inertia \(\mathrm{I_t}\) is rotating in a horizontal plane, about its symmetric axis, with a constant angular speed \(\omega_i.\) Another disk of a moment of inertia \(\mathrm{I_b}\) is dropped coaxially onto the rotating disk. Initially, the second disk has zero angular speed. Eventually, both the disks rotate with a constant angular speed \(\omega_f.\) The energy lost by the initially rotating disc due to friction is:
1. \( \frac{1}{2} \frac{\mathrm{I}_{\mathrm{b}}^2}{\left(\mathrm{I}_{\mathrm{t}}+\mathrm{I}_{\mathrm{b}}\right)} \omega_{\mathrm{i}}^2\)

2. \( \frac{1}{2} \frac{\mathrm{I}_{\mathrm{t}}^2}{\left(\mathrm{I}_{\mathrm{t}}+\mathrm{I}_{\mathrm{b}}\right)} \omega_{\mathrm{i}}^2\)

3. \( \frac{1}{2} \frac{\mathrm{I}_{\mathrm{b}}-\mathrm{I}_{\mathrm{t}}}{\left(\mathrm{I}_{\mathrm{t}}+\mathrm{I}_{\mathrm{b}}\right)} \omega_{\mathrm{i}}^2 \)

4. \( \frac{1}{2} \frac{\mathrm{I}_{\mathrm{b}} \mathrm{I}_{\mathrm{t}}}{\left(\mathrm{I}_{\mathrm{t}}+\mathrm{I}_{\mathrm{b}}\right)} \omega_{\mathrm{i}}^2 \)

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. $\frac{2M\omega }{M+2m}$

2. $\frac{(M+2m)\omega }{M}$

3. $\frac{M\omega }{M+2m}$

4. $\frac{(M+2m)\omega }{2m}$

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 ω. If two objects each of mass m are attached gently to the opposite ends of the diameter of the ring, the ring will then rotate with an angular velocity: