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

For \(L = 3.0~\text{m,}\) the total torque about pivot A provided by the forces as shown in the figure is:

For the same total mass, which of the following will have the largest moment of inertia about an axis passing through the centre of gravity and perpendicular to the plane of the body?
1. A disc of radius \(a\)
2. a ring of radius \(a\)
3. a square lamina of side \(2a\)
4. four rods forming square of side \(2a\)

The radius of gyration of a uniform rod of length \(L\) about an axis passing through its centre of mass is
1. \(\frac{L}{2 \sqrt{3}}\)
2. \(\frac{L^2}{12}\)
3. \(\frac{L}{\sqrt{3}}\)
4. \(\frac{L}{\sqrt{2}}\)