Write the formula for rotational kinetic energy.

A disc of mass $3 \,kg$ rolls down an inclined plane of height $5 \,m$. The translational kinetic energy of the disc on reaching the bottom of the inclined plane is .......... $J$

A solid sphere rolls down without slipping on an inclined plane, then percentage of rotational kinetic energy of total energy will be ........ $\%.$

A thin uniform rod of length $l$ and mass $m$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega $. Its centre of mass rises to a maximum height of

Two discs of moments of inertia $I_1$ and $I_2$ about their respective axes (normal to  the disc and passing through the centre), and rotating with angular speed $\omega _1$ and $\omega _2$ are  brought into contact face to face with their axes of rotation coincident. What is the  loss in kinetic energy of the system in the process ?

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 $\frac{ Mg }{16}$