The ratio of the acceleration for a solid sphere (mass \(m\) and radius \(R\)) rolling down an incline of angle \(\theta\) without slipping and slipping down the incline without rolling is:
1. \(5:7\)
2. \(2:3\)
3. \(2:5\)
4. \(7:5\) 

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. \(0.75~\text{m}\) 

A solid cylinder of mass \(3\) kg is rolling on a horizontal surface with a velocity of \(4\) ms^-1. It collides with a horizontal spring of force constant \(200\) Nm^-1. The maximum compression produced in the spring will be:
1. \(0.5\) m
2. \(0.6\) m
3. \(0.7\) m
4. \(0.2\) 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:
        
1. \(F_1+F_2\)
2. \(F_1-F_2\)
3. \(\frac{F_1+F_2}{2}\)
4. \(2F_1+F_2\)

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