Question

Four spheres, each of radius a and mass m, are placed with their centre on four corners of a square of side b. Calculate the moment of inertia of the arrangement about any diagonal of the square.

Answer 1

$D=2a$

$R=\frac{2a}{2}=a$

m is the mass of the sphere

$momentofinertiaofthesphereaxisofrotationisgivenby\frac{2}{5}\left({MR}^2\right)$

$AB=\frac{2}{5}\left({MR}^2\right)+\frac{2}{5}\left({MR}^2\right)$

$\frac{4}{5}\left({MR}^2\right)$

$CD=2\times \left(\frac{2}{5}{MR}^2\right)=4/5{MR}^2$

$ABistheonefixedsideforrotationhenceparallelaxis$

$I_{AB}=I_{CD}+\Sigma mi{ri}^2$

$\frac{4}{5}{MR}^2+(m_1+m_2)b_2$

$\frac{4}{5}{MR}^2+2{Mb}^2$

$\frac{4}{5}{MR}^2+\frac{4}{5}{MR}^2+2{Mb}^2$

$\frac{4}{5}{Ma}^2+\frac{4}{5}M{a}^2+2{Mb}^2$

$\frac{8}{5}{Ma}^2+2{Mb}^2$