Question

Two coaxial disc, having moments if inertia $I_1$ and $\frac{I_1}{2}$, are roatating with respective angular velocities ${\omega }_1$ and $\frac{{\omega }_1}{2}$, about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If $E_f$ and $E_i$ the final and initial total energies, then $(E_f-E_i)$ is :

• A. $\frac{I_1{\omega }_1^2}{12}$
• B. $\frac{3}{8}{I}_1{\omega }_1^2$
• C. $\frac{I_1{\omega }_1^2}{6}$
• D. $\frac{-I_1{\omega }_1^2}{24}$

Answer 1

The correct option is D $\frac{-I_1{\omega }_1^2}{24}$

$E=\frac{1}{2}{I}_1\times {\omega }_1^2+\frac{1}{2}\frac{I_2}{2}\times \frac{{\Omega }_1^2}{4}$
$\frac{I_1{\omega }_1^2}{2}\left(\frac{9}{8}\right)=\frac{9}{16}{I}_1{\omega }_1^2$
$I_1{\omega }_1+\frac{I_1{\omega }_1}{4}=\frac{3I_1}{2}\omega$
$\frac{5}{4}{I}_1{\omega }_1=\frac{3I_1}{2}\omega$
$\omega =\frac{5}{6}{\omega }_1$
$E_f=\frac{1}{2}\times \frac{3I_1}{2}\times \frac{25}{36}{\omega }_1^2$
$=\frac{25}{48}{I}_1{\omega }_1^2$
$\Rightarrow E_f-E_i=I_1{\omega }_1^2(\frac{25}{48}-\frac{9}{16})=\frac{-2}{48}{I}_1{\omega }_1^2$
$=\frac{-I_1{\omega }_1^2}{24}$