Consider a sphere of radius a of constant density where the vectors \(\vec{r}\), \(\vec{r}_0\), \(\vec{r}\) are defined in \(\mathbb{R}^3\) as:
\(\vec{r} = x\hat{x} + y\hat{y} + z\hat{z}\)
\(\vec{r}_0 = x'\hat{x} + y'\hat{y} + z'\hat{z}\)
\(\vec{r} = \vec{r} - \vec{r}_0\)
Show that the gravimetric potential of the sphere U(P), observed at the point P(x, y, z), is:
\(U(P) = \frac{4\pi\gamma\rho a^3}{3r}\)
where: \(r = |\vec{r}|\), and \(\gamma\) is the universal gravitation constant Note: the gravimetric potential is defined as:
\(U(P) = \int_V \frac{\rho(x', y', z')}{|\vec{r} - \vec{r}_0|} dv'\)
