Two coaxial discs, having moments of inertia $I_1$ and $\dfrac{I_1}{2}$, are rotating with respective angular velocities $\omega_1$ and $\dfrac{\omega_1}{2}$, 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 $E_i$ are the final and initial total energies, then $(E_f - E_i)$ is:
Answer: (D) $-\dfrac{I_1\omega_1^2}{24}$
Angular momentum is conserved:
$$I_1\omega_1 + \frac{I_1}{2}\cdot\frac{\omega_1}{2} = \frac32I_1\,\omega \;\Rightarrow\; \omega = \frac56\omega_1$$
$$E_i = \frac12I_1\omega_1^2 + \frac12\cdot\frac{I_1}{2}\cdot\frac{\omega_1^2}{4} = \frac{9}{16}I_1\omega_1^2,\qquad E_f = \frac12\cdot\frac32I_1\cdot\frac{25}{36}\omega_1^2 = \frac{25}{48}I_1\omega_1^2$$
$$E_f - E_i = \frac{25-27}{48}I_1\omega_1^2 = -\frac{I_1\omega_1^2}{24}$$
Solution by Sreeraj P, M.Sc Physics