Question:
What is the moment of inertia of a solid sphere of density \( \rho \) and radius \( R \) about its diameter?
What is the moment of inertia of a solid sphere of density \( \rho \) and radius \( R \) about its diameter?
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The moment of inertia of a solid sphere about its diameter is proportional to the square of the radius and the density.
Updated On: Mar 25, 2026
- \( \frac{105}{176} R^5 \rho \)
- \( \frac{105}{176} R^2 \rho \)
- \( \frac{176}{105} R^2 \rho \)
- \( \frac{176}{105} R^5 \rho \)
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The Correct Option is C
Solution and Explanation
Step 1: Formula for moment of inertia of a solid sphere.
The moment of inertia of a solid sphere about its diameter is given by: \[ I = \frac{2}{5} M R^2 \] where \( M \) is the mass of the sphere and \( R \) is its radius.
Step 2: Substitute mass.
The mass of the sphere is related to its density \( \rho \) by \( M = \rho \frac{4}{3} \pi R^3 \). So, substituting this in the formula for \( I \): \[ I = \frac{2}{5} \left( \rho \frac{4}{3} \pi R^3 \right) R^2 = \frac{176}{105} R^2 \rho \] Thus, the correct answer is (3).
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