A round disc of moment of inertia $I_2$ about its axis perpendicular to its plane and passing through its cent

Question

A round disc of moment of inertia $I_2$ about its axis perpendicular to its plane and passing through its center is placed over another disc of moment of inertia $I_1$ rotating with an angular velocity $\omega$ about the same axis. The final angular velocity of the combination of discs is:

Accepted Answer

When the second disc is placed over the first disc, there is no external torque acting on the system about the axis of rotation. Therefore, the angular momentum of the system is conserved.
Initial angular momentum, $L_i = I_1\omega_1 + I_2\omega_2 = I_1\omega + I_2(0) = I_1\omega$
Let the common final angular velocity be $\omega'$. The final moment of inertia of the system is $(I_1 + I_2)$.
Final angular momentum, $L_f = (I_1 + I_2)\omega'$
By conservation of angular momentum:
$L_i = L_f$
$I_1\omega = (I_1 + I_2)\omega'$
$\omega' = \frac{I_1\omega}{I_1+I_2}$

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