Question

A boy is standing on a platform which is free to rotate about its axis. The boy holds an open umbrella in his hands. 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 × 10⁻³ kg-m² and that of the umbrella is 2⋅0 × 10⁻³ kg-m². The boy starts spinning the umbrella about the axis at an angular speed of 2⋅0 rev/s with respect to himself. Find the angular velocity imparted to the platform.

Answer 1

Given:
Moment of inertia of umbrella = I₁ = 2 × 10⁻³ kg-m²
Moment of inertia of the system = I₂ = 3 × 10⁻³ kg-m²
Angular speed of the umbrella with respect to the boy = ω₁ = 2 rev/s
Let the angular velocity imparted to the platform be ω₂.
Taking the Earth as the reference, we have:
Angular velocity of the umbrella = (ω₁ − ω₂)
Applying conservation of angular momentum, we get:
$$\begin{gathered} I_1\left({\omega }_1-{\omega }_2\right)=I_2\left({\omega }_2\right) \\ \Rightarrow 2\times {10}^{-3}\left(2-{\omega }_2\right)=3\times {10}^{-3}{\omega }_2 \\ \Rightarrow 5{\omega }_2=4 \\ \Rightarrow {\omega }_2=0.8\mathrm{rev}/\mathrm{s} \end{gathered}$$