Mrs. Perkins's Electric Quilt by Paul J. Nahin

Author:Paul J. Nahin [Nahin, Paul J.]
Language: eng
Format: epub
ISBN: 978-1-4008-3346-7
Published: 2009-01-14T16:00:00+00:00

where

Thus,

or

where c(=4π2/k) is a constant. Now, the centripetal orbital acceleration of the planet is and so the gravitational force F (due to the Sun) on the planet, with m as the mass of the planet, is

where C is the constant mc. And so there you have it—the inverse square law. (The centrality of the force, for this special case, follows from the high degree of symmetry of the assumed circular orbit—the gravitational force on the planet is directed toward the Sun at the center of the orbit.) Newton’s Principia actually has a more general derivation, for an assumed elliptical orbit that may not be simply circular.3

Let me now show you a neat use of Kepler’s third law that quickly solves a problem we considered in an earlier discussion. Recall from Discussion 6 the problem of calculating how long it would take the Earth, if suddenly stopped in its orbit, to fall into the Sun. That analysis was fairly complicated; Kepler’s third law, however, makes the analysis almost a one-liner. But first let me review the geometry of the elliptical orbit of the Earth around the sun.

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