Question

A disc of the moment of inertia $I_{1}$ is rotating in a horizontal plane about an axis passing through a centre and perpendicular to its plane with constant angular speed $\omega _{1}$ Another disc of the moment of inertia $I_{2}$ having zero angular speed is placed coaxially on a rotating disc. Now both the disc are rotating with the constant angular speed $\omega _{2}$ . The energy lost by the initial rotating disc is

• A. $\frac{1}{2}\left[\frac{I_{1} + I_{2}}{I_{1} I_{2}}\right]\omega _{\text{1}}^{2}$
• B. $\frac{1}{2}\left[\frac{I_{1} I_{2}}{I_{1} - i_{2}}\right]\omega _{\text{1}}^{2}$
• C. $\frac{1}{2}\left[\frac{I_{1} - I_{2}}{I_{1} I_{2}}\right]\omega _{\text{1}}^{2}$
• D. $\frac{1}{2}\left[\frac{I_{1} I_{2}}{I_{1} + I_{2}}\right]\omega _{\text{1}}^{2}$

Answer 1

Answer: (D)

$I_{1}\left(\omega \right)_{1}=\left(I_{1} + I_{2}\right)\left(\omega \right)_{2}$
$\frac{\omega _{2}}{\omega _{1}}=\frac{I_{1}}{I_{1} + I_{2}}$
$E_{1}-E_{2}$
$=\frac{1}{2}I_{1}\omega _{1}^{2}-\frac{1}{2}\left(\right.I_{1}\left(+ I\right)_{2}\left.\right)\omega _{2}^{2}$
$=\frac{1}{2}\omega _{1}^{2}\left[I_{1} - \left(I_{1} + I_{2}\right) \frac{\omega _{2}^{2}}{\omega _{1}^{2}}\right] \, $
$=\frac{1}{2}\omega _{1}^{2}\left[I_{1} - \left(I_{1} + I_{2}\right) \frac{I_{1}^{2}}{\left(I_{1} + I_{2}\right)^{2}}\right] \, $
$=\frac{1}{2}\omega _{1}^{2}\left[\frac{I_{1}^{2} + I_{1} I_{2} - I_{1}^{2}}{I_{1} + I_{2}}\right]=\frac{1}{2}\left[\frac{I_{1} I_{2}}{I_{1} + I_{2}}\right]\omega _{1}^{2} \, \, $