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:

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

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

Two persons of masses \(55~\text{kg}\) and \(65~\text{kg}\) respectively, are at the opposite ends of a boat. The length of the boat is \(3.0~\text{m}\) and weighs \(100~\text{kg}.\) The \(55~\text{kg}\) man walks up to the \(65~\text{kg}\) man and sits with him. If the boat is in still water, the centre of mass of the system shifts by:
1. \(3.0~\text{m}\) 
2. \(2.3~\text{m}\) 
3. zero
4. \(0.75~\text{m}\) 

\(\mathrm{ABC}\) is an equilateral triangle with \(O\) as its centre. \(F_1,\) \(F_2,\) and \(F_3\) represent three forces acting along the sides \({AB},\) \({BC}\) and \({AC}\) respectively. If the total torque about \(O\) is zero, then the magnitude of \(F_3\) is:

The moment of inertia of a thin uniform rod of mass \(M\) and length \(L\) about an axis passing through its mid-point and perpendicular to its length is \(I_0\). Its moment of inertia about an axis passing through one of its ends and perpendicular to its length is:
1. \(I_0+\frac{ML^2}{4}\)
2. \(I_0+2ML^2\)
3. \(I_0+ML^2\)
4. \(I_0+\frac{ML^2}{2}\)

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