A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity ω. If two objects each of mass m are attached gently to the opposite ends of the diameter of the ring, the ring will then rotate with an angular velocity:

1.	\(\frac{\omega(M-2 m)}{M+2 m} \)	2.	\(\frac{\omega M}{M+2 m} \)
3.	\(\frac{\omega(M+2 m)}{M} \)	4.	\(\frac{\omega M}{M+m}\)

A wheel of radius R rolls without slipping on the ground with a uniform velocity v. The relative acceleration of the topmost point of the wheel with respect to the bottommost point is:

1. $\frac{{\mathrm{v}}^2}{\mathrm{R}}$

2. $\frac{2{\mathrm{v}}^2}{\mathrm{R}}$

3. $\frac{{\mathrm{v}}^2}{2\mathrm{R}}$

4. $\frac{4{\mathrm{v}}^2}{\mathrm{R}}$

Five uniform circular plates, each of diameter \(D\) and mass \(m,\) are laid out in a pattern shown. Using the origin shown, the \(y\text-\text{coordinate}\) of the centre of mass of the ''five–plate'' system will be:

A solid cylinder of mass \(50~\text{kg}\) and radius \(0.5~\text{m}\) is free to rotate about the horizontal axis. A massless string is wound around the cylinder with one end attached to it and the other end hanging freely. The tension in the string required to produce an angular acceleration of \(2~\text{rev/s}^2\) will be:
1. \(25~\text N\) 
2. \(50~\text N\) 
3. \(78.5~\text N\) 
4. \(157~\text N\) 

A disc and a solid sphere of the same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?

1. Sphere

2. Both reach at the same time

3. Depends on their masses

4. Disc

A solid sphere is in rolling motion. In rolling motion, a body possesses translational kinetic energy (K_t) as well as rotational kinetic energy (K_r) simultaneously. The ratio K_t : (K_t + K_r) for the sphere will be:

1. 7:10

2. 5:7

3. 10:7

4. 2:5

A uniform rod of mass 2M is bent into four adjacent semicircles, each of radius r, all lying in the same plane. The moment of inertia of the bent rod about an axis through one end A and perpendicular to the plane of the rod is:
      

1. 22 ${\mathrm{Mr}}^2$

2. 88 ${\mathrm{Mr}}^2$

3. 44 ${\mathrm{Mr}}^2$

4. 66 ${\mathrm{Mr}}^2$