Question

The moments of inertia of two rotating bodies $A$ and $B$ are $I_A$ and $I_B (I_A > I_B)$. If their angular momenta are equal, then

• A. Kinetic energy of $A =$ Kinetic energy of $B$
• B. Kinetic energy of $A >$ Kinetic energy of $B$
• C. Kinetic energy of $A < $ Kinetic energy of $B$
• D. Kinetic energy of the two bodies cannot be compared with the given data

Answer 1

Answer: (C)

$I_{A} \omega_{A} = I_{B}\omega_{B} $ (Given)
$ \therefore \frac{\omega_{A}}{\omega_{B}} = \frac{I_{B}}{I_{A}} \quad...\left(i\right) $
Kinetic energy $= \frac{1}{2} I\omega^{2} $
$ \therefore \frac{\left(K.E\right)_{A}}{\left(K.E\right)_{B}} = \frac{\frac{1}{2}I_{A}\omega_{A}^{2}}{\frac{1}{2} I_{B} \omega_{B}^{2}} $ (Using $(i)$)
$= \frac{I_{A}}{I_{B}} \times\left(\frac{I_{B}}{I_{A}}\right)^{2} $
$ = \frac{I_{B}}{I_{A}}$
As $I_{A} > I_{B} $ (Given)
$\therefore \left(K. E\right)_{A} < \left(K .E\right)_{B}$