Question

A disc having mass M and radius R is rotating with angular velocity $ \omega $ , another disc of mass 2M and radius R/2 is placed coaxially on the first disc gently. The angular velocity of system will now be:

• A. $ \frac{4\omega }{5} $
• B. $ \frac{2\omega }{5} $
• C. $ \frac{3\omega }{2} $
• D. $ \frac{2\omega }{3} $

Answer 1

Answer: (D)

First, we find out the moment of inertia of the disc about the axis passing the centre and normal to through the plane is $ I=\frac{M{{R}^{2}}}{2} $ So, for first disc $ {{I}_{1}}=\frac{M{{R}^{2}}}{2} $ Similarly, for second disc $ {{I}_{2}}=\frac{2M}{2}{{\left( \frac{R}{2} \right)}^{2}}=\frac{M{{R}^{2}}}{4} $ Total moment of inertia of the whole system $ I={{I}_{1}}+{{I}_{2}}=\frac{M{{R}^{2}}}{2}+\frac{M{{R}^{2}}}{4}=\frac{3}{4}M{{R}^{2}} $ According to the conservation of angular momentum $ {{I}_{1}}\omega =I\omega $ $ \frac{M{{R}^{2}}}{2}\times \omega =\frac{3}{4}M{{R}^{2}}\times \omega $ So, $ \omega =\frac{2}{3}\omega $