If $L, M$,and $P$ are the angular momentum,mass,and linear momentum of a particle respectively,which of the following represents the kinetic energy of the particle when the particle rotates in a circle of radius $R$?

$A$ solid sphere of mass $m$ and radius $R$ is rotating about its diameter. $A$ solid cylinder of the same mass and same radius is also rotating about its geometrical axis with angular speed twice that of the sphere. The ratio of the kinetic energy of the sphere to the kinetic energy of the cylinder will be:

$A$ body having a moment of inertia about its axis of rotation equal to $3 \ kg \ m^{2}$ is rotating with an angular velocity of $3 \ rad \ s^{-1}$. The kinetic energy of this rotating body is the same as that of a body of mass $27 \ kg$ moving with velocity $v$. The value of $v$ is: (in $m \ s^{-1}$)

If the angular momentum of a body is increased by $200\,\%$,then the increase in its rotational kinetic energy will be ....... $\%$.

$A$ solid cylinder of mass $M$ and radius $R$ is rotating about its geometrical axis. $A$ solid sphere of the same mass and same radius is also rotating about its diameter with an angular speed half that of the cylinder. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder will be

$A$ thin uniform rod of length $2 \ m$,cross-sectional area $A$,and density $d$ is rotated about an axis passing through its center and perpendicular to its length with angular velocity $\omega$. If the value of $\omega$ in terms of its rotational kinetic energy $E$ is $\sqrt{\frac{\alpha E}{Ad}}$,then the value of $\alpha$ is $...........$