question_answer

A thin uniform circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity \[\omega \]. Another disc of the same dimensions but of mass M/4 is placed gently on the first disc co-axially. Show that angular velocity of the system is \[4\omega /5\].

Answer:

                Initial angular momentum of one disc. \[L=I\omega =\frac{1}{2}M{{R}^{2}}\omega \] When another disc. of mass\[\frac{M}{4}\] and radius R is placed gently on it, total moment of inertia of the combination is \[I'=\frac{1}{2}M{{R}^{2}}+\frac{1}{2}\left( \frac{M}{4} \right){{R}^{2}}=\frac{5}{8}M{{R}^{2}}\] As no external torque has been applied, angular momentum is conserved. \[\therefore \]  \[I'\omega '=I\omega ,\]              \[\omega '=\frac{I\omega }{I'}\frac{\frac{1}{2}M{{R}^{2}}\omega }{\frac{5}{8}M{{R}^{2}}}=\frac{4}{5}\omega \]