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  4. Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d. The ratio I0/IA of moment of inertia I0 of the system about an axis passing the centroid and about center of any of the spheres IA and perpendicular to the plane of the triangle is :

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Q. Three solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side of length $d$. The ratio $I_0/I_A$ of moment of inertia $I_0$ of the system about an axis passing the centroid and about center of any of the spheres $I_A$ and perpendicular to the plane of the triangle is :Physics Question Image

JEE MainJEE Main 2020System of Particles and Rotational Motion

Solution:

From parallel axis theorem
$I_{0}=3\times\left[\frac{2}{5}M\left(\frac{d}{2}\right)^{2}+M\left(\frac{d}{\sqrt{3}}\right)^{2}\right]=\frac{13}{10}Md^{2}$
$I_{A}=I_{0}+3M\left(\frac{d}{\sqrt{3}}\right)^{2}$
$=\frac{13}{10}Md^{2}+Md^{2}$
$=\frac{23}{10}Md^{2}$
$\frac{I_{0}}{I_{A}}=\frac{13}{23}$