# the Gravitational force Between spherical Mass distributions

Any spherically symmetric mass distribution can be thought of as a combination of concentric spherical shells. Because of the principle of superposition of forces, what is true of one shell is also true of the combination. So we have proved half of what we set out to prove: that the gravitational interaction between any spherically symmetric mass distribution and a point mass is the same as though all the mass of the spherically symmetric distribution were concentrated at its center

The other half is to prove that two spherically symmetric mass distributions interact as though both were points. That’s easier. In Fig. 13.23a the forces the two bodies exert on each other are an action–reaction pair, and they obey Newton’s third law. So we have also proved that the force that m exerts on sphere M is the same as though M were a point. But now if we replace m with a spherically symmetric mass distribution centered at m’s location, the resulting gravitational force on any part of M is the same as before, and so is the total force. This completes our proof.

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