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$\frac1n$ of its  original value, its angular velocity becomes
1. $\frac wn$ 
2. $\frac{w}{{n}^2}$
3. $nw$
4. $n^2w$