A thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement($s$) is/are correct, when the rod makes an angle $60^{\circ}$ with vertical ? [ $g$ is the acceleration due to gravity]

$(1)$ The radial acceleration of the rod's center of mass will be $\frac{3 g }{4}$

$(2)$ The angular acceleration of the rod will be $\frac{2 g }{ L }$

$(3)$ The angular speed of the rod will be $\sqrt{\frac{3 g}{2 L}}$

$(4)$ The normal reaction force from the floor on the rod will be $\frac{ Mg }{16}$

The ratio of kinetic energies of two spheres rolling with equal centre of mass velocities is $2 : 1$. If their radii are in the ratio $2 : 1$; then the ratio of their masses will be

A flywheel is making $\frac{3000}{\pi}$ revolutions per minute about its axis. If the  moment of inertia of the flywheel about that axis is $400\, kgm^2$, its rotational kinetic  energy is 

A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is $\pi: 22$ then, the value of its angular speed will be $………..\,rad / s$.

The rotational kinetic energy of a solid sphere of mass $3 \;kg$ and radius $0.2\; m$ rolling down an inclined plane of height $7\; m$ is