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Q.

Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centre form an equilateral triangle of side of length d. The ratio $I_{0} / I_{A}$ of moment of inertia, where $I_{0}$ is MOI of the system about an axis passing the centroid and $I_{A}$ is MOI of the system about centre of any of the spheres and perpendicular to the plane of the triangle, is:

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a

$\frac{23}{13}$

b

$\frac{13}{23}$

c

$\frac{13}{15}$

d

$\frac{15}{13}$

answer is B.

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Detailed Solution

MOI about centroid $I_{O} = 3 [\frac{2}{5} m {(\frac{d}{2})}^{2} + m {(\frac{d}{\sqrt{3}})}^{2}] = \frac{13}{10} m d^{2} [h e r e O A = O B = O C = \frac{d}{\sqrt{3}}]$
MOI about axis passing through ‘r’ and $⊥^{l r}$ to plane
$I_{A} = 2 [\frac{2}{5} m {(\frac{d}{2})}^{2} + m {(d)}^{2}] + \frac{2}{5} m {(\frac{d}{2})}^{2} = \frac{23}{10} m d^{2}$

$∴ R a t i o = \frac{I_{O}}{I_{A}} = \frac{13}{23}$

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