Question

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 :

• A. $\frac{15}{13}$
• B. $\frac{13}{15}$
• C. $\frac{13}{23}$
• D. $\frac{23}{13}$

Answer 1

Answer: (C)

$\frac{13}{23}$

Answer 2

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