Did you know the moon is in constant freefall towards Earth? Here’s how Sir Isaac Newton figured this out and discovered Universal Gravitation in the process.

The apple’s role

Action-at-a-distance. His contemporaries thought it impossible. Not so Newton. He embraced the concept. The falling apple, the orbiting moon, perhaps governed by the same, unseen force?  As Newton later recalled,  “I began to think of gravity extending to the orb of the moon.” [1] Alas, the extent to which a falling apple inspired this tremendous insight is not known.

Modeling the moon’s orbit – framing the problem

With critical help from Robert Hooke (here), Newton hypothesized that a central attractive gravitational force causes the moon’s circular motion by continually diverting its straight-line inertial motion into a closed orbit around Earth. He then applied his advanced mathematical skills together with his Laws of Motion to test his hypothesis. 

So how did Newton do the math?

For all the difficulty of philosophy seems to consist in this—from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena – Sir Isaac Newton [2]

Both Joseph-Louis Lagrange and Pierre-Simon, Marquis de Laplace regretted that there was only one fundamental law of the universe, the law of universal gravitation, and that Newton had lived before them, foreclosing them from the glory of its discovery – I. Bernard Cohen and Richard S. Westfall [3]

Newton suggested that a rapid decrease in gravitational attraction with distance must occur in order to explain the differences between the “fall” of the moon and the fall of the apple.  But what was the exact relationship and how could he prove it?

Newton used the available data, i.e., Earth’s radius, distance to moon, moon’s period of revolution, to calculate the moon’s acceleration toward Earth. He then proposed that the gravitational attraction must be proportional to the inverse-square of distance, a proposal consistent with his thinking that gravitational intensity falls off with distance just as does the brightness of light, which decreases with the increasing area of the sphere through which it passes.  Since the area increases with r², then the intensity must decrease with 1/r². 

When Newton finished this calculation for the moon’s acceleration and then multiplied it by (60)², 60:1 being the ratio of the distances of the moon and the apple to the Earth’s center, he arrived at an estimate of the acceleration of the apple at the Earth’s surface of 32.2 ft/sec², a value in excellent agreement with Christiaan Huygens’s independent measurement (32.2 ft/sec²) of acceleration at the Earth’s surface using pendulums.

Universal Gravitation

The implication was stunning.  The moon orbits for the same reason the apple falls.  By identifying that the same cause attracts both moon and apple, Newton united the Heavens and the Earth.  He did not speculate.  He did the math.  He proposed that Earth exerts a gravitational attraction that decreases with 1/r².  His quantification was his proof.  The numbers fell into place.  Like Lagrange and Laplace, I would have loved to have been the one to do this calculation and see it work.

Explore more about Newton’s life and accomplishments in Chapter 8 of my book, Block by Block – The Historical and Theoretical Foundations of Thermodynamics.

References

[1] Gleick, James. 2003. Isaac Newton. 1st ed. New York: Pantheon Books, p. 55.

[2] Newton, Isaac. 2006. The Mathematical Principles of Natural Philosophy: Volume 1. Translated by Andrew Motte. Symonds, 1803, p. x.

[3] Cohen, I. Bernard, and Richard S. Westfall, eds. 1995. Newton: Texts, Backgrounds, Commentaries. 1st ed. A Norton Critical Edition. New York, NY: W.W. Norton, p. xiv-xv.