Chapter 24, Problem 56EAP

Newton's law of gravity and Coulomb's law are both inverse-

square laws. Consequently, there should be a "Gauss's law for

gravity."

a. The electric field was defined as $\stackrel{\to }{E}={\stackrel{\to }{F}}_{onq}/q$ , and we used this

to find the electric field of a point charge. Using analogous

reasoning, what is the gravitational field $\stackrel{\to }{g}$ of a point mass?

Write your answer using the unit vector $\hat{r}$ , but be careful with

signs; the gravitational force between two "like masses" is

attractive, not repulsive.

b. What is Gauss's law for gravity, the gravitational equivalent

of Equation 24.18? Use ${\Phi }_G$ for the gravitational flux, $\stackrel{\to }{g}$ for the

gravitational field, and M_in for the enclosed mass.

c. A spherical planet is discovered with mass M, radius R, and

a mass density that varies with radius as $\rho ={\rho }_0(1-r/2R)$ ,

where ${\rho }_0$ is the density at the center. Determine ${\rho }_0$ in terms

of M and R.

Hint: Divide the planet into infinitesimal shells of thickness dr,

then sum (i.e., integrate) their masses.

d. Find an expression for the gravitational field strength inside

the planet at distance r < R.