A thin circular ring of mass \(M\) and radius \(R\) is rotating about its axis with a constant angular velocity \(\omega\)$$.  Four objects of mass \(m\) are held gently at the opposite ends of the ring's two perpendicular diameters. The angular velocity of the ring will be:
1. \(\dfrac{M\omega}{M+4m}\)
2. \(\dfrac{(M+4m)\omega}{M}\)
3. \(\dfrac{(M-4m)\omega}{M+4m}\)
4. \(\dfrac{M\omega}{4m}\)

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:

Three-point masses each of mass \(m,\) are placed at the vertices of an equilateral triangle of side \(a.\) The moment of inertia of the system through a mass \(m\) at \(O\) and lying in the plane of \(COD\) and perpendicular to \(OA\) is:

The moment of inertia of a uniform circular disc of radius '\(R\)' and mass '\(M\)' about an axis touching the disc at its diameter and normal to the disc will be:
1. \(\frac{3}{2} M R^{2}\)
2. \(\frac{1}{2} M R^{2}\)
3. \(M R^{2}\)
4. \(\frac{2}{5} M R^{2}\)${\mathrm{}}^{}$

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. \(157~\text N\) 

If a body is moving in a circular path with decreasing speed, then: (symbols have their usual meanings):

1.  \(\overset{\rightarrow}{r} . \overset{\rightarrow}{\omega}=0\) 
2.  \(\overset{\rightarrow}{\tau} . \overset{\rightarrow}{v}=0\) 
3.  \(\overset{\rightarrow}{a} . \overset{\rightarrow}{v}<0\) 
4.  All of these