Problem 11. It turns out that Gauss's Law applies for gravity, too! Last semester, we saw that the gravitational field from a point mass is $\vec{g} = -\frac{Gm}{r^2}\hat{r}$, where $G = 6.67 \times 10^{-11} Nm^2/kg^2$ is the gravitational constant.
(a) Following the same process we did in class to introduce Gauss's Law, determine what Gauss's Law for gravity should be (i.e., what is the net gravitational flux, $\Phi_g$, through a closed surface equal to?).
(b) Let's approximate the Earth as a uniform sphere with density $\rho$ and radius $R$. If we were to drill a small hole straight through the center of the Earth and drop an object into the hole, use Gauss's Law to show that the object would undergo simple harmonic motion.
(c) Given that the radius of the Earth is $6.38 \times 10^6 m$ and the mass of the Earth is $5.97 \times 10^{24} kg$, find the period of the resulting oscillations.
