A small mass attached to a string rotates on a frictionless tabletop as shown. If the tension in 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:-
 

$ABC$ is an equilateral triangle with $O$ as its centre. ${\overrightarrow{\mathit{F}}}_1, {\overrightarrow{\mathit{F}}}_2$and ${\overrightarrow{\mathit{F}}}_3$ represent three forces acting along the sides $AB$, $BC$ and $AC$ respectively. If the total torque about $O$ is zero then the magnitude of $\overrightarrow{F_3}$ is:-

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

A mass $m$ moves in a circle on a smooth horizontal plane with velocity $v_0$ at a radius $R_0$. The mass is attached to a string which passes through a smooth hole in the plane as shown.
 
The tension in the string is increased gradually and finally $m$moves in a circle of radius $\frac{R_0}{2}$. The final value of the kinetic energy is:-