The instantaneous angular position of a point on the rotating wheel is given by the equation, $\theta \left(t\right)=\left(2t^3-6t^2\right).$ The torque on the wheel becomes zero at

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: