To solve the problem, it will be necessary to define the rotational and translational kinetic energy in order to determine the relationship between the two. Rotational energy is defined as,
[tex]KE_{Rotational} = \frac{1}{2} I\omega^2[/tex]
Here,
I = Moment of Inertia
[tex]\omega[/tex] = Angular velocity
Now the translational energy will be,
[tex]KE_{Translational} = \frac{1}{2} mv^2[/tex]
Here,
m = Mass
v = Velocity
Therefore the relation between them will be,
[tex]\frac{KE_{Rotational} }{KE_{Translational}} = \frac{\frac{1}{2} I\omega^2 }{\frac{1}{2} mv^2 }[/tex]
Applying the moment of inertia of a sphere we have,
[tex]\frac{KE_{Rotational} }{KE_{Translational}} = \frac{\frac{1}{2} (\frac{2}{5}mr^2)\omega^2 }{\frac{1}{2} mv^2 }[/tex]
[tex]\frac{KE_{Rotational} }{KE_{Translational}} = \frac{2}{5} \frac{r^2\omega^2}{v^2}[/tex]
[tex]\frac{KE_{Rotational} }{KE_{Translational}} = \frac{2}{5} \frac{(2.42*10^{-2})^2(158)^2}{23.3^2}[/tex]
[tex]\frac{KE_{Rotational} }{KE_{Translational}} = 0.01077[/tex]
Therefore the ratio will be 0.01077
