Point masses \(m_1\) and \(m_2,\) are placed at the opposite ends of a rigid rod of length \(L\) and negligible mass. The rod is set into rotation about an axis perpendicular to it. The position of a point \(P\) on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity \(\omega_0\) is minimum is given by:
     

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 in the figure.
   
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. \( \frac{1}{2} m v_0^2\)

Three identical spherical shells, each of mass \(m\) and radius \(r\) are placed as shown in the figure. Consider an axis \(XX',\) which is touching two shells and passing through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about the \(XX'\) axis is:
           

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. \(1\)

An automobile moves on a road with a speed of \(54~\text{kmh}^{-1}.\)  The radius of its wheels is \(0.45\) m and the moment of inertia of the wheel about its axis of rotation is \(3~\text{kg-m}^2.\) If the vehicle is brought to rest in \(15\) s, the magnitude of average torque transmitted by its brakes to the wheel is:
1. \(6.66~\text{kg-m}^2\text{s}^{-2}\)
2. \(8.58~\text{kg-m}^2\text{s}^{-2}\)
3. \(10.86~\text{kg-m}^2\text{s}^{-2}\)
4. \(2.86~\text{kg-m}^2\text{s}^{-2}\)

A solid cylinder of mass \(50~\text{kg}\) and radius \(0.5~\text{m}\) is free to rotate about the horizontal axis. A massless string is wound around the cylinder with one end attached to it and the other end hanging freely. The tension in the string required to produce an angular acceleration of \(2~\text{rev/s}^2\) will be:
1. \(25~\text N\) 
2. \(50~\text N\) 
3. \(78.5~\text N\) 
4. \(157~\text N\) 

A rod \(\mathrm{PQ}\) of mass \(M\) and length \(L\) is hinged at end \(\mathrm{P}\). The rod is kept horizontal by a massless string tied to point \(\mathrm{Q}\) as shown in the figure. When the string is cut, the initial angular acceleration of the rod is: