First slide

Moment of inertia

Question

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 $I_{0}$ of the system about an axis passing the centroid and about centre of any of the spheres $I_{A}$ and perpendicular to the plane of the triangle is :

Moderate

A

$\frac{23}{13}$
B

$\frac{13}{23}$
C

$\frac{13}{15}$
D

$\frac{15}{13}$

Solution

MOI about centroid $= 3 [\frac{2}{5} m {(\frac{d}{2})}^{2} + m {(\frac{d}{\sqrt{3}})}^{2}] = \frac{13}{10} m d^{2} [h e r e O A = O B = O C = \frac{d}{\sqrt{3}}]$
MOI about axis passing through ‘r’ and $⊥^{l r}$ to plane
$= 2 [\frac{2}{5} m {(\frac{d}{2})}^{2} + m {(d)}^{2}] + \frac{2}{5} m {(\frac{d}{2})}^{2} = \frac{23}{10} m d^{2} ∴ R a t i o = \frac{13}{23}$

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