Find the polar moment of inertia of a homogeneous

Find the polar moment of inertia of a homogeneous circular plate of radius $R$ with respect to the center.

Step 1: Set up

\[I_O=\int_0^{2\pi}\int_0^R\rho^2\cdot\rho\,d\rho\,d\theta = 2\pi\int_0^R\rho^3\,d\rho = 2\pi\cdot\frac{R^4}4\]

\[\boxed{I_O=\frac{\pi R^4}2}\]

Step 2: Cross-check via the perpendicular axis theorem

From Problem 146, $I_x=I_y=\dfrac{\pi R^4}4$, so $I_O=I_x+I_y=\dfrac{\pi R^4}2$. ✓

Step 3: Radius of gyration

\[k_O=\sqrt{I_O/A}=\sqrt{\frac{\pi R^4/2}{\pi R^2}}=\frac R{\sqrt2}\approx0.707R\]