13.2 Gravitational force between point masses

For points outside a spherically symmetric mass distribution, gravity acts as if the total mass were concentrated at the sphere’s centre.

Use centre-to-point distance r, not distance from the surface, when applying point-mass equations.

The external gravitational field of a spherical planet can be calculated with its total mass and the distance from its centre.

The point-mass equivalence does not apply inside a nonuniform shell or to an arbitrary irregular body.

Two point masses attract with F=Gm₁m₂/r² along the line joining their centres, where G is the universal gravitational constant.

Use centre-to-centre separation and treat the force as attractive. For several masses, add force vectors.

Doubling one mass doubles force; doubling separation reduces force to one quarter.

The inverse-square law is not inverse distance, and the force pair acts on both masses with equal magnitude in opposite directions.

For a circular orbit, gravitational attraction provides F=mv²/r, so the orbital speed satisfies GMm/r²=mv²/r.

Use the distance r from the attracting body’s centre and remember that gravity is inward while velocity is tangential.

A lower circular orbit has greater speed because v=√(GM/r) increases as r decreases.

Gravity is not absent in orbit; an orbiting object is continuously falling while its tangential motion carries it around.

A geostationary satellite stays above one longitude because it orbits eastward in Earth’s equatorial plane with the same angular speed and period as Earth’s rotation.

All three conditions matter: circular orbit, equatorial plane and synchronous period. Its apparent fixed position is relative to the rotating Earth.

Communications dishes can point at a geostationary satellite without tracking it across the sky.

Any satellite with a 24-hour period is not automatically geostationary if its orbit is tilted or elliptical.