Question

We have been discussing the parallel between Newton’s Universal Law of Gravity and Coulomb’s force law:

• Both laws define force. Coulomb's law describes the force between electric charges whereas Newton’s law describes the force between masses.

• Both are inverse square laws. The forces are proportional to the inverse square of the distance between masses for Newton’s law and the inverse square of the distance between charges for Coulomb’s law.

• The forces defined by both laws are central forces meaning that the forces act along a line joining two charges in Coulomb’s law and along the line joining two masses in Newton’s law.

• Both force laws are conservative forces meaning the work done by these forces on any object is independent of the path followed by the object. It only depends in the initial and final positions of the object in which the forces act.

A consequence of these similarities is that all of the mathematical machinery we have been learning for charges should also apply to masses. Consequently, there should be a Gauss’s Law for Gravity. The purpose of this exercise will be to develop Gauss’s Law for Gravity:

a. The electric field was defined as ?⃑ = ? ? ⁄ and we used this to find the electric field for a point charge. Using analogous reasoning, infer the gravitational field ? of a point mass. Write your answer using the unit vector ?̂, but be careful to include the correct sign. Remember that the gravitational force between two like masses is attractive not repulsive.

b. Using this same reasoning, infer an analogous Gauss’s Law for Gravity. Use the symbol Φ? to represent the gravitational flux, ? for the gravitational field, and ??? for the enclosed mass. How should the gravitational constant “big G” be included, 4p?

c. Consider a spherical planet of total mass M, radius R and uniform density ?. Using Gauss’s Law for Gravity, determine the gravitational field ? at points ? > ? and ? ≤ ?. Be sure to draw a picture showing where your Gaussian surface is located in both situations and label any quantities of interest.

d. Make a plot of g versus r and graph your solutions from parts (c) and (d). What is the maximum value of the gravitational field? Where does this maximum occur?

e. Suppose a spherical planet is discovered with mass M, radius R and a mass density that varies with radius as ? = ?0 (1 − ? ? ). Where ?0 is the mass density at the center of the planet. This function decreases linearly as you move away from the center of the planet. Determine ?0 in terms of M and R. Hint: divide the planet into infinitesimal shells of thickness ?? and sum the masses.

f. Using Gauss’s Law for Gravity determine the gravitational field ? at points ? ≤ ?. Be sure to draw a picture showing where your Gaussian surface is located and label any quantities of interest.

g. Make sense of your solution. Make sure that your solution is continuous with the field for points ? > ? outside a sphere from part c. That is, make sure you get the expected value for ? = ? at the surface of the planet.

Answer 1