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$°$ 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.  $\frac{g}{2}$

2. $\frac{g}{4}$

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 v₁ and that of the second is v₂ at a certain instant. The speed of the centre of mass is:

1. \({v_1 + v_2} \over 2\)

2. \({2v_1 + 3v_2} \over 5\)

3. \({3v_1 + 2v_2} \over 5\)

4. zero

An electric fan rotating at 1200 rpm is switched off. If the fan stops after 10 seconds, the number of revolutions completed by the fan before it stops will be: (assume uniform retardation)

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: