Four spheres, each of mass 'M' and radius 'r' are situated at the four corners o

Question

Four spheres, each of mass 'M' and radius 'r' are situated at the four corners of square of side 'R'. The moment of inertia of the system about an axis perpendicular to the plane of square and passing through its centre will be

Options:

A . 52M(4r2+5R2)

B . 25M(4r2+5R2)

C . 25M(4r2+5r2)

D . 52M(4r2+5r2)

Answer: Option B
:
B
M. I. of sphere A about its diameter

I_{O^{'}} = \frac{2}{5} M r^{2}

Now M.I. of sphere A about an axis perpendicular to the plane of square and passing through its centre will be

I_{O} = I_{O^{'}} + M {(\frac{R}{\sqrt{2}})}^{2} = \frac{2}{5} M r^{2} + \frac{M R^{2}}{2}

Moment of inertia of system (i.e. four sphere)

= 4 I_{O} = 4 [\frac{2}{5} M r^{2} + \frac{M R^{2}}{2}] = \frac{2}{5} M [4 r^{2} + 5 R^{2}]

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