Questions 35–42 come from the moment of inertia experiment

The disk $x^2+y^2\le a^2$ has $I_0={\int }_0^a{r}^{2}2\pi rdr=\frac{1}{2}\pi a^4$. Why is this different from $Iy$ in Problem 20? Find the radius of gyration $\bar{r}=\sqrt{I_0/M}$. (The rotational energy $\frac{1}{2}{I}_0{\omega }^2$ equals $\frac{1}{2}M{r}^2{\omega }^2$—when the whole mass is turning at radius $\bar{r}$.)

Questions 35–42 come from the moment of inertia experiment.

This exercise builds on another exercise in its source book, which isn't shown here.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §8.5 Masses and Moments, #34