A wheel is rotating freely with an angular speed ω on a shaft. The...

From angular momentum conservation
$I\omega + 0 = I{\omega}_{C} + 3I{\omega}_{C}$
$\omegac = \dfrac{\omega}{4}$

Loss of kinetic energy $= \dfrac{1}{2}I{\omega}^{2} - \dfrac{1}{2} ( I + 3I ) {\left(\dfrac{\omega}{4}\right)}^{2}$

$= \dfrac{1}{2}I{\omega}^{2} - \dfrac{1}{2}I\dfrac{{\omega}^{2}}{4}$

$= \dfrac{3}{8}I{\omega}^{2}$

Fractional loss $= \dfrac{3}{4}$