question_answer

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:

[JEE MAIN Held on 09-01-2020 Morning]

A) \[\frac{23}{13}\]

B) \[\frac{15}{13}\]

C) \[\frac{13}{15}\]

D) \[\frac{13}{23}\]

Correct Answer: D

Solution :

[d]

\[{{I}_{o}}=3{{I}_{1}}\] \[{{I}_{1}}=\frac{2}{5}m{{\left( \frac{d}{2} \right)}^{2}}+m{{\left( AO \right)}^{2}}\] \[AO=\frac{d}{\sqrt{3}}\]

\[\Rightarrow {{I}_{O}}=\frac{13}{10}M{{d}^{2}}\]

\[{{I}_{A}}=2\left[ \frac{2}{5}M{{\left( \frac{d}{2} \right)}^{2}}+M{{d}^{2}} \right]+\frac{2}{5}M{{\left( \frac{d}{2} \right)}^{2}}\]

\[=\frac{23}{10}M{{d}^{2}}\] \[\frac{{{I}_{O}}}{{{I}_{A}}}=\frac{13}{23}\]