Question

A body standing on the platform which is free to rotate about its axis. The boy holds an open umbrella in his hand. The axis of the umbrella coincides with that of the platform. The moment of inertia of ''the platform plus the boy system" is $3.0\times {10}^{-3}kg-m^2$ and that of the umbrella is $2.0\times {10}^{-3}kg-m^2$. The boy starts spinning the umbrella about an axis at an angular speed o $2.0rev/s$ with respect to himself. Find the angular velocity imparted to the platform.

Answer 1

$I-1=2\times {10}^{-3}kg-m^2;I_2=3\times {10}^{-3}kg-m^2;{\omega }_1=2rad/s$
From the earth reference the umbrella has a angular velocity $({\omega }_1-{\omega }_2)$
And the angular velocity of the man will be ${\omega }_2$
Therefore $I_1({\omega }_1-{\omega }_2)=I_2{\omega }_2$
$\Rightarrow 2\times {10}^{-3}(2-{\omega }_2)=3\times {10}^{-3}\times {\omega }_2$
$\Rightarrow 5{\omega }_2=4\Rightarrow {\omega }_2=0.8rad/s$.