Find the moment of inertia of a circular plate of radius

Find the moment of inertia of a circular plate of radius $R$ with respect to the coordinate axes.

Explanation of Method

Since both coordinate axes are diameters of the circle, apply the diameter result from Problem 146 directly, and verify with the perpendicular-axis theorem against the polar moment from Problem 149.

Step 1: Apply the diameter result

Since the $x$- and $y$-axes both pass through the center and are diameters, the result of Problem 146 applies to each:

\[I_x=I_y=\frac{\pi R^4}4\]

Step 2: Cross-check with the polar moment

From Problem 149, $I_O=\dfrac{\pi R^4}2$, and indeed $I_x+I_y=\dfrac{\pi R^4}4+\dfrac{\pi R^4}4=\dfrac{\pi R^4}2$. ✓

Step 3: Radii of gyration

\[k_x=k_y=\sqrt{I_x/A}=\frac R2\]

Result

Final answer:

\[\boxed{I_x = I_y = \dfrac{\pi R^4}{4}}\]