Use spherical coordinates to find the moment of inertia of the solid homogeneous hemisphere of radius 5 and density 1 about a diameter of its base. Select the correct answer.
A) $205.13$
B) $2616.67$
C) $195.22$
D) $213.5$
E) $198.08$

Question

Quizplus · Accepted Answer

The moment of inertia of a solid hemisphere of radius $R$ and density $\rho$ about an axis passing through its center and perpendicular to its base is $\frac{2}{5}\rho R^5$. However, we need to find the moment of inertia about a diameter of its base, which by the perpendicular axis theorem, is $\frac{2}{3}$ times the moment of inertia about an axis passing through its center and perpendicular to its base. Therefore, the moment of inertia we want is $\frac{4}{15}\rho R^5 = \frac{4}{15}(5^5)(1) = 2616.67$. Thus, the answer is B.

Use Spherical Coordinates to Find the Moment of Inertia of the Solid