Of Particular Significance

# The Impossible Commentary: Newton, Gravity, and the Speed of the Moon

#### POSTED BY Matt Strassler

ON 10/24/2023

Additional supplementary material for the upcoming book; your comments/corrections are welcome. This entry has to do with how Newton realized that weight and mass aren’t the same thing — that the pull of Earth’s gravity depends on how far you are from the Earth’s center.

• (Quote) Newton knew right away that if the force of gravity were as powerful out by the Moon as it is at Earth’s surface—if the Moon accelerated toward the Earth at the same rate that your dropped keys do—then motion and gravity would be wildly out of balance [and so the Moon would have fallen and crashed into the Earth.]
• (Endnote) To avoid disaster, the Moon’s orbital speed would need to be 40 miles per second, leading it to circle Earth twice a day.

Here I’ll explain why this is true, using a little math. (If you already know something about Kepler’s laws of planetary orbits, additional relevant discussion can be found in this post from 2022.)

### Circular Motion

An object traveling in circular motion, making a circle of radius r, must be pulled inward toward the center of the circle to keep it from rushing away. The inward acceleration a caused by whatever force keeps it in circular motion is related to the circle’s radius, and to the speed of the object v , by a simple formula recognized in the 17th century by Christoph Huygens. The formula can be written in various equivalent ways:

The first form is particularly instructive: it says that in circular motion, acceleration is to velocity as velocity is to radius. [The reasons behind this relation are discussed at the end of this post from 2022.]

Furthermore, the speed of an object in steady circular motion is the circle’ s circumference 2πr divided by the time T (the period) that it takes to complete one cycle. This allows us to obtain a relation between the period, the radius and the acceleration:

The radius of the Earth is about 4000 miles (6400 km); the circular orbit of a near-Earth satellite, just above the atmosphere, would have an orbital radius only a few percent larger, perhaps 4200 miles (6700 km). As for the acceleration at or near Earth’s surface, that was measured by Galileo: in the absence of air resistance, the speed of a falling object increases every second by almost 10 meters per second (about 32 feet per second). Equations (1) and (2) allow us to calculate the speed v of the orbiting satellite: about 5 miles per second (8 km/second). The period T of the orbit can then be obtained from equation (2): it takes about 90 minutes = 1.5 hours = 1/16 days for the satellite to orbit the Earth. Even though there were no near-Earth satellites in Newton’s day, he could imagine them and calculate what their speed and period would be.

### If weight were constant

Suppose that the acceleration a due to Earth’s gravity were a constant, independent of an object’s distance r from the Earth’s center. (This would be true if weight were the same as mass, since an object’s mass is intrinsic to the object, and thus independent of its distance from Earth or anything else.) Then both v and T , according to equations (1) and (2), would grow as the square root of r. We can write this using the “is proportional to” symbol, “∝”:

But this would lead to a wrong conclusion. The Moon is roughly 60 times further from Earth’s center compared to us (or to a near-Earth satellite.) The square root of 60 is about 7.5. If equation (3) were correct, then the Moon’s speed in its orbit would be 7.5 x (5 miles per second), which is about 40 miles per second (64 km/second) — and it would then orbit the Earth in an amount of time equal 7.5 x 1.5 hours, which is just under half a day. This prediction is a total failure! The Moon orbits the Earth about once a month.

There are two possible responses to this revelation.

1. the gravity that holds us to the ground is not the force that keeps the Moon in its orbit.
2. the acceleration a of an object due to Earth’s gravity changes with the object’s distance r.

Newton guessed the second option. It was a guess because no experiments available at Newton’s time could actually move far enough away from the Earth’s surface, or measure gravity precisely enough, to say whether this was true or not. Let’s see that it was correct.

### If weight decreased with distance from Earth

If instead a decreases as 1/r2, then the relation between v, T and r would become

This last relation between period and radius was familiar to Newton: it was Kepler’s law for the orbits of the planets [a law which you can confirm through do-it-yourself astronomy, as I have explained in this sequence of posts: 1, 2, 3 ; see also here] . So if gravity’s force indeed causes an acceleration that falls as the inverse of distance-squared, and if the Sun’s gravity holds the planets in orbit, and if Earth’s gravity holds the Moon, then equation (4) should apply to all objects orbiting the Earth. That means that the Moon’s speed can be inferred from that of a near-Earth satellite (real or imagined.) The speed should be (1/7.5) x (5 miles per second), or about 2/3 of a mile per second (1 km/second). Meanwhile its period would be given by multiplying 1.5 hours = 1/16 days by 603/2 = (7.5)3 = 465. This works!! 465 x (1/16 days) is roughly one month!

That is how Newton obtained evidence that

• the Earth’s gravitational pull holds the Moon in orbit,
• the Sun’s gravitational pull holds the planets in orbit,
• gravity’s pull decreases as the square of the inverse of the distance from the pulling object (specifically, if the object is ball-shaped, from its center),
• and so an object’s weight (which depends on its location) is conceptually different from its mass (which is independent of its location)

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### 6 Responses

1. Friedel Hartmann says:

Beautiful. The fundamental and classical result of Newton put into perspective. Before the apple fell from a tree there was Johannes Kepler and Christoph Huygens. The Nobel Prize should have gone to all three.

2. Angelos says:

Prof, in a previous response you stated the gravitional waves and a real force because they moved LIGO and gravitons are the quanta of that gravitational force.

Ok, that observation is definitely true, but is it the 4th force of nature, gravity, or some other force, one of the other 3 forces which so well are described by the Standard Model?

Here’s one view, albeit a layman’s view who never derived any part of the Standard Model, why may be the graviton doesn’t exist, at least not in the same way the photon, gluon, or the W & Z bosons exist. Suppose there are only the 3 forces, how could any one of these 3 mechanism create gravity as we observe and measure it?

IMHO, it would be either the weak and/or the strong forces that could produce gravitational waves because there are already contribute to massive particles. So, what if a gravitational wave is the residue from an inbalance, an instability in the protons and neutrons that emit these gravitational waves?

Will the g-2 experiment at the FermiLab give any insight on this outrageous theory from a layman?

Thanks.

3. Nicely presented! So, Newton’s critical intuitive guess was not that the moon is pulled towards Earth like an apple, but that the pull on the moon fades as 1/r^2 where the unit of distance is the earth’s radius.

That is a vastly better description of what happened than the dumbed-down, no-math version! I hope it catches on in schools as an example of how knowing math can give you insights that you otherwise might never have had.

1. That’s exactly right, Terry… the key was not the idea that gravity might be the source of the pull holding the Moon in orbit, but the recognition that mass and weight are not the same, and that the Moon’s weight might be much less than its mass would naturally imply to an Earthling who’d never left the ground.

The fact that F_gravity ~ 1/r^2 would give Kepler’s third law must have been known to multiple people who understood both Huygens’ formula and the basic F=ma relation. Certainly Robert Hooke knew this too. It is not clear to me which people might have understood the Moon’s orbit around the same time that Newton did. In any case, Newton was not alone in exploring these ideas. But it seems that he was first in getting them all straight at the same time.

As for your last paragraph concerning how “knowing math can give you insights that you otherwise might never have had”, I think you would enjoy reading this post about Kepler’s third law (if you haven’t already), which gives an even more remarkable example.

4. Spivak’s Calculus has a chapter on the orbit of the planets. I am going to work through Spivak this coming year, and am excited to learn this stuff thoroughly!

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