Q. Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities $\omega_1$ and $\omega_2$ . They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is:-

Solution:

COAM : $I \omega_1 + I \omega_2 = 2 I \omega\, \Rightarrow \, \omega = \frac{\omega_1 + \omega_2}{2}$
$ (K.E.)_i = \frac{1}{2} I \omega_1^2 + \frac{1}{2} I \omega_2^2$
$(K.E)_f = \frac{1}{2} \times 2I\omega^2 = I \left(\frac{\omega_1 + \omega_2}{2} \right)^2$
Loss in $K.E. = (K.E.)_i - (K.E)_f = \frac{I}{4} \left( \omega_1 - \omega_2\right)^2$

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