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 length d . Find the ratio I 0 / I A , where I 0 is

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 length d . Find the ratio I 0 / I A , where I 0 is the moment of inertia of the system about an axis passing through the centroid and I A is the moment of inertia of the system about an axis passing through the centre of any sphere and perpendicular to the plane of the triangle.

TestHub · Accepted Answer

The moment of inertia of the system about the point $O$ is

$I_{0} = 3 [\frac{2}{5} M {(\frac{d}{2})}^{2} + M {(\frac{d}{\sqrt{3}})}^{2}] = \frac{13}{10} M d^{2}$

The moment of inertia of the system about the point $A$ is

$I_{A} = 2 [\frac{2}{5} M {(\frac{d}{2})}^{2} + M (d)^{2}] + \frac{2}{5} M {(\frac{d}{2})}^{2} = \frac{23}{10} M d^{2}$

So, the ratio $= \frac{13}{23}$

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