Gravitation and spherically symmetric Bodies

We have stated the law of gravitation in terms of the interaction between two
particles. It turns out that the gravitational interaction of any two bodies having
spherically symmetric mass distributions (such as solid spheres or spherical
shells) is the same as though we concentrated all the mass of each at its center, as
in Fig. 13.2. Thus, if we model the earth as a spherically symmetric body with
mass mE, the force it exerts on a particle or on a spherically symmetric body
with mass m, at a distance r between centers, is

provided that the body lies outside the earth. A force of the same magnitude is exerted on the earth by the body. (We will prove these statements in Section 13.6.)

At points inside the earth the situation is different. If we could drill a hole to the center of the earth and measure the gravitational force on a body at various depths, we would find that toward the center of the earth the force decreases, rather than increasing as 1/r² . As the body enters the interior of the earth (or other spherical body), some of the earth’s mass is on the side of the body opposite from the center and pulls in the opposite direction. Exactly at the center, the earth’s gravitational force on the body is zero.

Spherically symmetric bodies are an important case because moons, planets, and stars all tend to be spherical. Since all particles in a body gravitationally attract each other, the particles tend to move to minimize the distance between them. As a result, the body naturally tends to assume a spherical shape, just as a lump of clay forms into a sphere if you squeeze it with equal forces on all sides. This effect is greatly reduced in celestial bodies of low mass, since the gravitational attraction is less, and these bodies tend not to be spherical (Fig. 13.3).

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