A solid sphere of mass \(M\) and the radius \(R\) is in pure rolling with angular speed \(\omega\) on a horizontal plane as shown in the figure. The magnitude of the angular momentum of the sphere about the origin \(O\) is:
                     

1. \(\frac{7}{5} M R^{2} \omega\)
2. \(\frac{3}{2} M R^{2} \omega\)
3. \(\frac{1}{2} M R^{2} \omega\)
4. \(\frac{2}{3} M R^{2} \omega\)$$

Two particles of mass, \(2\) kg and \(4\) kg, are projected from the top of a tower simultaneously, such that \(2\) kg of mass is projected with a speed \(20\) m/s at an angle \(30^{\circ}\) $$above horizontal and \(4\) kg is projected at \(40\) m/s horizontally. The acceleration of the centre of mass of the system of two particles will be:
1. \(\dfrac{g}{2}\)
2. \(\dfrac{g}{4}\)
3. \(g\)
4. \(2g\)

Two particles of masses \(2\) kg and \(3\) kg start to move towards each other due to mutual forces of attraction. The speed of the first particle is \(v_1\) and that of the second is \(v_2\) at a certain instant. The speed of the centre of mass is:

A boy is standing on a disc rotating about the vertical axis passing through its centre. He pulls his arms towards himself, reducing his moment of inertia by a factor of m. The new angular speed of the disc becomes double its initial value. If the moment of inertia of the boy is I₀ , then the moment of inertia of the disc will be:

1.  $2{\mathrm{I}}_{0}m$

2.  ${\mathrm{I}}_0\left(1-\frac{2}{m}\right)$

3.  ${\mathrm{I}}_0\left(1-\frac{1}{m}\right)$

4.  $\left(\frac{{\mathrm{I}}_0}{2m}\right)$

Four masses are joined to light circular frames as shown in the figure. The radius of gyration of this system about an axis passing through the center of the circular frame and perpendicular to its plane would be:
(where '\(a\)' is the radius of the circle)
                        
1. \(\frac{a}{\sqrt{2}}\)
2. \(\frac{a}{{2}}\)
3. \(a\)
4. \(2a\)

Five particles of mass \(2\) kg each are attached to the circumference of a circular disc of a radius of \(0.1\) m and negligible mass. The moment of inertia of the system about the axis passing through the centre of the disc and perpendicular to its plane will be:
1. \(1\) kg-m²
2. \(0.1\) kg-m²
3. \(2\) kg-m²
4. \(0.2\) kg-m²

When a uniform rod of length \(l,\) hinged at one end is released from rest while making an angle \(\theta\) with the vertical, what will be the acceleration of its free end at that instant?

On a rough horizontal surface (coefficient of friction $\mu$), a cubical block of side 'a' and mass m is projected horizontally. The net torque on the block about its centre of mass till the block stops is equal to:

1.  zero

2.  $\frac{1}{2}\mu mga$

3.  $\mu$mga

4.  mga