From a circular disc of radius \(R\) and mass \(9M,\) a small disc of mass \(M\) and radius \(R/3\) is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is:
1. \(MR^{2}\)
2. \(4MR^{2}\)
3. \(\frac{4}{9}MR^{2}\)
4. \(\frac{40}{9}MR^{2}\)

Given the following statements:

Which one of the following pairs of statements is correct?

A thin circular ring of mass \(M\) and radius \(r\) is rotating about its axis with constant angular velocity ω. Two objects each of mass \(m\) are attached gently to the opposite ends of the diameter of the ring. The ring now rotates with angular velocity given by:

1. $\frac{2M\omega }{M+2m}$

2. $\frac{(M+2m)\omega }{M}$

3. $\frac{M\omega }{M+2m}$

4. $\frac{(M+2m)\omega }{2m}$

Three masses are placed on the \(x\)-axis: \(300~\text{g}\) at origin, \(500~\text{g}\) at \(x= 40~\text{cm}\) and \(400~\text{g}\) at \(x= 70~\text{cm}.\) The distance of the centre of mass from the origin is:
1. \(45~\text{cm}\)
2. \(50~\text{cm}\)
3. \(30~\text{cm}\)
4. \(40~\text{cm}\)

A circular platform is mounted on a frictionless vertical axle. Its radius \(R = 2~\text{m}\) and its moment of inertia about the axle is \(200~\text{kg m}^2\). It is initially at rest. A \(50~\text{kg}\) man stands on the edge of the platform and begins to walk along the edge at the speed of \(1~\text{ms}^{-1}\) relative to the ground. The time taken by man to complete one revolution is:
1. \(\frac{3\pi}{2}\text{s}\)
2. \(2\pi~\text{s}\)
3. \(\frac{\pi}{2}\text{s}\)
4. \(\pi~\text{s}\)

The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through:

         
1. \(C\)
2. \(D\)
3. \(A\)
4. \(B\)

A uniform rod of length \(l\) and mass \(M\) is free to rotate in a vertical plane about \(A\). The rod, initially in the horizontal position, is released. The initial angular acceleration of the rod is: (Moment of inertia of the rod about \(A\) is \(\frac{Ml^2}{3}\))
1. \(\frac{3g}{2l}\)
2. \(\frac{2l}{3g}\)
3. \(\frac{3g}{2l^2}\)
4. \(\frac{Mg}{2}\)

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 is:
1. $M{R}^2$

2. $\frac{2}{5}M{R}^2$

3. $\frac{3}{2}M{R}^2$

4. $\frac{1}{2}M{R}^2$