Gauss’ Law for gravitation The gravitational force due to

QUESTION:

Gauss' Law for gravitation The gravitational force due to a point mass M is proportional to \(\mathbf{F}=G M \mathbf{r} /|\mathbf{r}|^{3}\), where \(\mathbf{r}=\langle x, y, z)\) and G is the gravitational constant.

a. Show that the flux of the force field across a sphere of radius a centered at the origin is \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi G M\).

b. Let S be the boundary of the region between two spheres centered at the origin of radius a and b with a < b. Use the Divergence Theorem to show that the net outward flux across S is zero.

c. Suppose there is a distribution of mass within a region D containing the origin. Let p(.r. _v. _) be the mass density (mass per unit volume). Interpret the sta        tement that

\(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi G \iiint_{D} \rho(x, y, z) d V\)

d. Assuming F satisfies the conditions of the Divergence Theorem, conclude from pan (c) that \(\nabla \cdot \mathbf{F}=4 \pi G \rho\).

e. Because the gravitational force is conservative, it has a potential function \(\varphi\). From pan (d) conclude that \(\nabla^{2} \varphi=4 \pi G \rho\).

Text Transcription:

F = GMr|r|^3

r = langle x, y, z rangle

iint_S F cdot n dS = 4piGM

iint_S f cdot n dS = 4piG iint_D p(x,y, z) dV

Nabla cdot F = 4piGp

Varphi

nabla^2 varhpi = 4piGp