Question

Moment of inertia of a solid sphere about its diameter is $I$. It is then cast into $27$ small spheres of same diameter. The moment of inertia of each new sphere is

• A. $\dfrac{I}{243}$
• B. $\dfrac{I}{122}$
• C. $\dfrac{I}{31}$
• D. $\dfrac{I}{62}$

Hint:

When a body is divided uniformly, mass scales with number of parts and radius scales with cube root.

Answer 1

The Correct Option is A

Step 1: Moment of inertia of a solid sphere.
For a solid sphere about its diameter:
\[ I = \frac{2}{5}MR^2 \]

Step 2: Effect of casting into 27 identical spheres.
Mass of each small sphere:
\[ m = \frac{M}{27} \] Radius of each small sphere:
\[ r = \frac{R}{3} \]

Step 3: Moment of inertia of each small sphere.
\[ I' = \frac{2}{5} m r^2 = \frac{2}{5} \times \frac{M}{27} \times \frac{R^2}{9} \] \[ I' = \frac{2}{5}MR^2 \times \frac{1}{243} \]

Step 4: Express in terms of $I$.
\[ I' = \frac{I}{243} \]

Step 5: Conclusion.
The moment of inertia of each new sphere is $\dfrac{I}{243}$.