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

The moment of inertia of a horizontal ring about its vertical axis through the centre is mR². The moment of inertia about its tangent parallel to the plane is:

1.  $\frac{3{\mathrm{mR}}^2}{2}$

2.  $\frac{{\mathrm{mR}}^2}{4}$

3.  $\frac{{\mathrm{mR}}^2}{2}$

4.  $\frac{3{\mathrm{mR}}^2}{4}$

A uniform rod of length l is hinged at one end and is free to rotate in the vertical plane. The rod is released from its position, making an angle $\theta$ with the vertical. The acceleration of the free end of the rod at the instant it is released is:
    

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

An insect, initially on the circumference of a disc, starts moving along a chord of the disc, rotating about an axis passing through the center and perpendicular to the plane of the disc. Its angular speed: 

For a rigid body rotating about a fixed axis, which of the following quantities is the same at an instant for all the particles of the body?

A body of mass M is moving on a circular track of radius r in such a way that its kinetic energy K depends on the distance travelled by the body s according to relation K = $\beta$s, where $\beta$ is a constant. The angular acceleration of the body is:

1.  $\frac{\beta r}{M^2}$

2.  $\sqrt{\frac{\beta r}{M}}$

3.  $\frac{M{r}^2}{\beta }$

4.  $\frac{\beta }{Mr}$

If a particle moves in a circle with a constant angular speed ($\mathrm{\omega }$) about point O, then its angular speed about point A will be:
           
1.  2$\mathrm{\omega }$

2.  $\frac{\mathrm{\omega }}{2}$

3.  $\mathrm{\omega }$

4.  $\frac{\mathrm{\omega }}{4}$