Centres Of Four Solid Spheres Of Diameter ‘2a’ And Mass ‘m’ Make Square Of Side ‘b’. Calculate The Moment Of Inertia Of The System About One Side Of The Square.

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Ans.

Four spheres, each of mass m and radius a, are placed at the corners of the square of side b, as shown in Fig.1.

Fig. 1

We know that the moment of inertia of the sphere about its diameter is \( \frac{2}{5}ma^2 \), [ To know the derivation of the moment of inertia of a sphere about its diameter (CLICL HERE) ].

We want to calculate the moment of inertia of the system about the axis EH.

The moment of inertia of each of the sphere E and H about the axis EH is
\( \frac{2}{5}ma^2 \).

The moment of inertia of each of the sphere F and G about he axis EH is
\( \left(\frac{2}{5}ma^2+mb^2\right) \).

So the moment of inertia of the whole system about the side EH is

\( 2\times{\frac{2}{5}ma^2}+2\times{\left(\frac{2}{5}ma^2+mb^2\right)} \)
= \( \frac{8}{5}m{a^2}+2mb^2 \)
= \( \frac{2}{5}m\left(4a^2+5b^2\right) \)

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