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Two coaxial discs ($I_1, I_2$ with angular velocities $\omega_1, \omega_2$) are suddenly coupled. **Energy dissipated** in the process is:
AZero (angular momentum conserved)
B$\dfrac{I_1 I_2}{2(I_1 + I_2)}(\omega_1 - \omega_2)^2$
C$\tfrac{1}{2}(I_1 + I_2)(\omega_1 + \omega_2)^2$
D$\tfrac{1}{2}(I_1\omega_1^2 + I_2\omega_2^2)$
Answer & Solution
Correct answer: B. $\dfrac{I_1 I_2}{2(I_1 + I_2)}(\omega_1 - \omega_2)^2$
Angular-momentum conservation: $\omega_f = (I_1\omega_1 + I_2\omega_2)/(I_1+I_2)$. $\Delta KE = \tfrac{1}{2}(I_1\omega_1^2 + I_2\omega_2^2) - \tfrac{1}{2}(I_1+I_2)\omega_f^2$. Algebra yields $\dfrac{I_1 I_2}{2(I_1 + I_2)}(\omega_1 - \omega_2)^2$ — analogous to the reduced-mass energy loss in inelastic collisions.
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