Newton's proof of the connection between elliptical orbits and inverse-square forces ranks among the “top ten” calculations in the history of science. This time-honored calculation is a highlight in an upper-level mechanics course. It would be worthwhile if students in introductory physics could prove the relation elliptical orbit ⇒ 1/r2 force without having to rely on upper-level mathematics. We introduce a simple procedure—Newton's Recipe—that allows students to readily and accurately deduce the algebraic form of force laws from a geometric analysis of orbit shapes.

Topics
History of science, Educational aids
1.
Isaac Newton, The Principia, Mathematical Principles of Natural Philosophy, a new translation by I. Bernard Cohen and Anne Whitman (University of California, Berkeley, 1999).
2.
J. Bruce Brackenridge, The Key to Newton's Dynamics (University of California, Berkeley, 1995).
3.
The general solution of F = ma for any force F is r − ro = vot + (Fo/2m)t2, where the interval of time is small enough for the force to be approximately constant (t ≈0, F ≈ Fo). Using Newton's notation, this vector equation of displacements is PQ = PR + RQ. The orbital displacement is PQ = r − ro. The inertial displacement due to vo alone is PR = vot. The falling displacement due to Fo alone is RQ = (Fo/2m)t2.
4.
Newton's force measure, QR/(SP × QT)2, is introduced in Proposition 6, Book 1 of the Principia (Ref. 1, p. 453). Newton derived theoretical force functions F(r) by calculating QR/QT2 as a function of SP = r using Euclid's propositions. We derive “experimental” force functions by measuring QR, SP, and QT using a ruler.
5.
The inverse ratio (SP × QT)2/QR has units of volume. Indeed, Newton associates centripetal force with a hypothetical solid. In Proposition 6, Corollary 1 (Ref. 1, p. 454), Newton writes “… the centripetal force will be inversely as the solid (SP2 × QT2)/QR, provided that the magnitude of the solid is always taken as that which it has ultimately when the points P and Q come together.”
6.
What does “close” mean? In theory, the force formula is exact only for infinitesimal deviations. In our experiment, we assume that a deviation is “infinitesimal” if the length of the deviation QR is less that 10% of the Sun-planet distance SP. Given the size of the orbits drawn in our class (30 cm < SP < 90 cm), we suggest to the students that they choose future points Q around P so that the deviations QR are about 4 cm, 3 cm, 2 cm, and 1 cm. Measuring smaller distances with a ruler involve larger relative errors.
7.
How do you take the “calculus limit?” If you are in the calculus regime of infinitesimal deviations (approximated by QR< 0.1SP) and parabolic arcs, then the values of the force measures in the force sequence should be roughly constant or slowly approaching a limiting value. We have the students estimate the limit by simply looking at the values, noticing a trend, and performing a qualitative extrapolation (or average).
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© 2007 American Association of Physics Teachers.
2007
American Association of Physics Teachers
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