A small mass attached to a string rotates on a frictionless table top as shown. If the tension on the string is increased by pulling the string causing the radius of the circular motion to decrease by a factor of \(2,\) the kinetic energy of the mass will

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. \(n^2w\)