Kinetic Energy: The energy associated with the body in motion is called kinetic energy.
Linear Kinetic Energy: The kinetic energy associated with the linear speed of an object is called linear kinetic energy. It is given by
\(K_{l}=\frac{1}{2} m v^{2}\)
\(\mathrm{m}\) is mass and \(\mathrm{v}\) is the speed of the body
Rotational Kinetic Energy: The kinetic energy associated with the rotational motion of the body is called rotational kinetic energy.
\(K_{r}=\frac{1}{2} I \omega^{2}\)
\(I\) is the moment of inertia, and \(\omega\) is the angular velocity.
Moment of Inertia: The inertia associated with the rotational motion of an object is called the moment of inertia. It is measured along a particular axis.
Moment of Inertia of a sphere with radius \(r\) is given as
\(I=\frac{2}{5} m r^{2}-(1)\)
Angular velocity: The rate of angular displacement with respect to time is called angular velocity. For a body in rotational motion with radius \({r}\), angular velocity is given as
\(w=\frac{v}{r} \cdots(2)\)
Rotational Kinetic Energy of ball
\(K_{R}=\frac{1}{2} I_{\omega}^{2}\)
\(K_{R}=\frac{1}{2} \frac{2}{5} m r^{2} \times\left(\frac{v}{r}\right)^{2} \text { using Eq (1) and (2) }\)
\(K_{R}=\frac{1}{5} m v^{2}-(3)\)
The total energy of the body in rotational motion is equal to the sum of liner kinetic energy and rotational kinetic energy.
\(E=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2}-(4)\)
Putting Equation (3) in (4)
\(E=\frac{1}{2} m v^{2}+\frac{1}{5} m v^{2}\)
\(E=\frac{5+2}{10} m v^{2}=\frac{7}{10} m v^{2}\)
So, the total energy associated with the ball is
\(E=\frac{7}{10} m v^{2}-(5)\)
Now, to get the energy ratio \(\mathrm{R}\) of rotational kinetic energy to the total energy we divide (3) by (5)
\(R=\frac{\frac{1}{5} m v^{2}}{\frac{7}{10} m v^{2}}\)
\(R=\frac{1}{5} \times \frac{10}{7}\)
\(R=\frac{2}{7}\)
So, the required fraction is \(\frac{2}{7}\).