Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D${}_1$ has a 2 kg mass, 0.2 m radius, and an initial angular velocity of 50 rad s${}^{-1}$. Disc D${}_2$ has 4 kg mass, 0.1 m radius, and initial angular velocity of 200 rad s${}^{-1}$. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad.s${}^{-1}$) of the system will be:

1. 60

2. 100

3. 120

4. 40

A square lamina is as shown in the figure. The moment of inertia of the frame about the three axes is $I_1,$ $I_{\mathrm{2,}}$ $and$ $I_3$ respectively. Select the correct alternative.

     
1. $I_2=I_3>I_1$
2. $I_1>I_2>I_3$
3. $I_2=I_3<I_1$
4. $I_1<I_2<I_3$

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 passes 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:
    
1.$\frac{11}{5}$mr²

2. 3mr²

3. $\frac{16}{5}$mr²

4. 4mr²

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 is R = 2m and its moment of inertia about the axle is 200 kg m². Initially, it is at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at a speed of 1 m s^–1 relative to the ground. The time taken by the man to complete one revolution is:
1. $\mathrm{\pi } \sec$                 
2. $\frac{3\mathrm{\pi }}{2}\sec$
3. $2\mathrm{\pi } \sec$               
4. $\frac{\mathrm{\pi }}{2}\sec$

A man of 50 kg mass is standing in a gravity free space at a height of 10 m above the floor. He throws a stone of 0.5 kg mass downwards with a speed of $2$ $m{s}^{-1}.$ When the stone reaches the floor, the distance of the man above the floor will be: 

A thin wire of length L and uniform linear mass density $\rho$ is bent into a circular loop with the centre at O as shown. The moment of inertia of the loop about the axis XX' is

1. $\frac{\rho L^3}{8{\pi }^2}$
2. $\frac{\rho L^3}{16{\pi }^2}$
3. $\frac{5\rho L^3}{16{\pi }^2}$
4. $\frac{3\rho L^3}{8{\pi }^2}$

The one-quarter sector is cut from a uniform circular disc of radius R. This sector has a mass M. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation will be: 

A billiard ball of mass m and radius r, when hit in a horizontal direction by a cue at a height h above its centre, acquires a linear velocity $v_0$ . The angular velocity ${\omega }_0$ acquired by the ball will be:

1. $\frac{5v_0{r}^2}{2h}$

2. $\frac{2v_0{r}^2}{5h}$

3. $\frac{2v_{0}h}{5r^2}$

4. $\frac{5v_{0}h}{2r^2}$

A ladder is leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents the path followed by its center of mass?

1.		2.
3.		4.