Question

Three identical spherical shells, each of mass $2 \,kg$ and radius $10 \,cm$ are placed as shown in figure. Consider an axis $XX '$ which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about $XX'$ axis is _______$kg\, m ^{2}$.

Answer 1

Using parallel axes theorem,
$I _{1}=\frac{2}{3} mr ^{2}+ mr ^{2}$ and $I _{1}= I _{2} $
$I _{3}=\frac{2}{3} m ^{2}$
$\therefore M.I$. of system about $XX ^{\prime}$ is
$I = I _{1}+ I _{2}+ I _{3} $
$=2\left(\frac{2}{3} mr ^{2}+ mr ^{2}\right)+\frac{2}{3} mr ^{2}$
$=4 mr ^{2}=4 \times 2 \times\left(10 \times 10^{-2}\right)^{2}$
$=8 \times 10^{-2}$
$\therefore I =0.08 \,kg\, m ^{2}$