Three-point masses 'm' each, are placed at the vertices of an equilateral triangle of side a. Moment of inertia of the system about axis COD is-

          

1. $2m{a}^2$

2. $\frac{2}{3}m{a}^2$

3. $\frac{5}{4}m{a}^2$

4. $\frac{7}{4}m{a}^2$

A particle is moving in a circular orbit with constant speed. Select wrong alternate

One solid sphere A and another hollow sphere B are of same mass and same outer radii. Their moment of inertia about their diameters are respectively \(\text{I}_{A}\) and \(\text{I}_{B}\) such that
1. \(\text{I}_{\text{A}}=\text{I}_{\text{B}}\)
2. \(\text{I}_{\text{A}}>\text{I}_{\text{B}}\)
3. \(\text{I}_{\text{A}}<\text{I}_{\text{B}}\)
4. \(\frac{\text{I}_{\text{A}}}{\text{I}_{\text{B}}}=\frac{d_A}{d_B}\)

A couple produces:

1. Purely linear motion

2. Purely rotational motion

3. Linear and rotational motion

4. No motion

A particle of mass \(1 ~\text{kg}\) is kept at (1m, 1m, 1m). \((1~\text{m},~1~\text{m},~1~\text{m}),\) The moment of inertia of this particle about \(z-\)axis would be
1. \(1~\text{kg}-\text{m}^2\)
2. \(2~\text{kg}-\text{m}^2\)
3. \(3~\text{kg}-\text{m}^2\)
4. None of these

One-quarter sector is cut from a uniform circular disc of radius \(R.\)  This sector has mass \(M.\)  It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc.  Its moment of inertia about the axis of rotation is:

       
1. \(\frac{1}{2} M R^2\)
2. \(\frac{1}{4} M R^2\)
3. \(\frac{1}{8} M R^2\)
4. \(\sqrt{2} M R^2\)

A wheel is rotating at the rate of \(33~ \text{rev/min}\)  If it comes to stop in \(20 ~\text{s.}\) Then, the angular retardation will be
1. \(\pi \frac{\text{rad}}{\text{~s}^2}\)
2. \(11 \pi ~\text{rad} / \text{s}^2\)
3. \(\frac{\pi}{200} ~\text{rad} / \text{s}^2 \)
4. \(\frac{11 \pi}{200}~\text{rad} / \text{s}^2\)

A solid sphere is rotating about a diameter at an angular velocity \(w.\) If it cools so that its radius reduces to$\frac{\mathrm{}}{}$\(\frac1n\) of its  original value, its angular velocity becomes
1. \(\frac wn\) 
2. \(\frac{w}{{n}^2}\)
3. \(nw\)
4. \(n^2w\)

A horizontal platform is rotating with uniform angular velocity around the vertical axis passing through its centre. At some instant of time a viscous fluid of mass 'm' is dropped at the centre and is allowed to spread out and finally fall. The angular velocity during this period

1. Decreases continuously

2. Decreases initially and increases again

3. Remains unaltered

4. Increases continuously