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.	\(\frac{\omega M}{M+m}\)

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 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 ${\mathrm{Mr}}^2$

A horizontal heavy uniform bar of weight \(W\) is supported at its ends by two men. At the instant, one of the men lets go off his end of the rod, the other feels the force on his hand changed to:

The moment of inertia of a thin uniform circular disc about one of its diameter is I. Its moment of inertia about an axis perpendicular to the circular surface and passing through its center will be:

1. $\sqrt{2}I$

2. 2 l

3. $\frac{I}{2}$

4. $\frac{I}{\sqrt{2}}$