A solid sphere of \(100 \mathrm{~kg}\) and radius \(10 \mathrm{~m}\) moving in a space becomes a circular…
A solid sphere of \(100 \mathrm{~kg}\) and radius \(10 \mathrm{~m}\) moving in a space becomes a circular disc of radius \(20 \mathrm{~m}\) in one hour. Then the rate of change of moment of inertia in the process is
Given, mass of solid sphere, \(M_s=100 \mathrm{~kg}\)
radius of solid sphere, \(R_s=10 \mathrm{~m}\)
radius of circular disc, \(R_c=20 \mathrm{~m}\) and time \(=1\) hour \(=60\) minute \(=60 \times 60 \mathrm{sec}\)
Moment of inertia of the solid sphere,
\(I_s=\frac{2}{5} M_s R_s^2=\frac{2}{5} \times 100 \times(10)^2=4000 \mathrm{~kg}-\mathrm{m}^2\)
Similarly,
moment of inertia of the disc, \(I_c=\frac{1}{2} M_c R^2\)
\(=\frac{1}{2} \times 100 \times(20)^2=20,000 \mathrm{~kg}-\mathrm{m}^2\)
Rate of change of moment of inertia \(=\frac{I_c-I_s}{t}\)
$\begin{aligned}
& =\frac{20000-4000}{60 \times 60}=\frac{16000}{60 \times 60}=\frac{160}{36} \\
& =\frac{40}{9} \mathrm{~kg} \cdot \mathrm{m}^{2} \cdot \mathrm{s}^{-1}
\end{aligned}$