(20) Newton's theory of "Universal Gravitation"

The Moon orbits around the Earth. Since its size does not appear to change, its distance stays about the same, and hence its orbit must be close to a circle. To keep the Moon moving in that circle--rather than wandering off--the Earth must exert a pull on the Moon, and Newton named that pulling force gravity.

Was that the same force which pulled all falling objects downward?

Supposedly, the above question occured to Newton when he saw an apple falling from a tree. John Conduitt, Newton's assistant at the royal mint and husband of Newton's niece, had this to say about the event when he wrote about Newton's life:

In the year 1666 he retired again from Cambridge ... to his mother in Lincolnshire & while he was musing in a garden it came into his thought that the power of gravity (which brought an apple from a tree to the ground) was not limited to a certain distance from earth, but that this power must extend much further than was usually thought. Why not as high as the Moon thought he to himself & that if so, that must influence her motion & perhaps retain her in her orbit, whereupon he fell a-calculating what would be the effect of that superposition...

If it was the same force, then a connection would exist between the way objects fell and the motion of the Moon around Earth, that is, its distance and orbital period. The orbital period we know--it is the lunar month, corrected for the motion of the Earth around the Sun, which also affects the length of time between one "new moon" and the next. The distance was first estimated in ancient Greece--see here and here.

To calculate the force of gravity on the Moon, one must also know how much weaker it was at the Moon's distance. Newton showed that if gravity at a distance R was proportional to 1/R² (varied like the "inverse square of the distance"), then indeed the acceleration g measured at the Earth's surface would correctly predict the orbital period T of the Moon.

Newton went further and proposed that gravity was a "universal" force, and that the Sun's gravity was what held planets in their orbits. He was then able to show that Kepler's laws were a natural consequence of the "inverse squares law" and today all calculations of the orbits of planets and satellites follow in his footsteps.

Nowadays students who derive Kepler's laws from the "inverse-square law" use differential calculus, a mathematical tool in whose creation Newton had a large share. Interestingly, however, the proof which Newton published did not use calculus, but relied on intricate properties of ellipses and other conic sections. Richard Feynman, Nobel-prize winning maverick physicist, rederived such a proof (as have some distinguished predecessors); see reference at the end of the section.

Here we will retrace the calculation, which linked the gravity observed on Earth with the Moon's motion across the sky, two seemingly unrelated observations. If you want to check the calculation, a hand-held calculator is helpful.

Calculating the Moon's Motion

We assume that the Moon's orbit is a circle, and that the Earth's pull is always directed toward's the Earth's center. Let R_E be the average radius of the Earth (first estimated by Erathosthenes)

R_E= 6 371 km

The distance R to the Moon is then about 60 R_E. If a mass m on Earth is pulled by a force mg, and if Newton's "inverse square law" holds, then the pull on the same mass at the Moon's distance would be 60² = 3600 times weaker and would equal

mg/3600

If m is the mass of the Moon, that is the force which keeps the Moon in its orbit. If the Moon's orbit is a circle, since R = 60 R_E its length is

2 p R = 120 p R_E

Suppose the time required for one orbit is T seconds. The velocity v of the motion is then

v = distance/time = 120 p R_E/T

(Please note: gravity is not what gives the Moon its velocity. Whatever velocity the Moon has was probably acquired when it was created. But gravity prevents the Moon from running away, and confines it to some orbit.)

The centripetal force holding the Moon in its orbit must therefore equal

mv²/R = mv²/(60 R_E)

and if the Earth's gravity provides that force, then

mg/3600 = mv²/(60 R_E)

dividing both sides by m and then multiplying by 60 simplifies things to

g/60 = v²/R_E = (120 p R_E)²/(T² R_E)

Canceling one factor of R_E , multiplying both sides by 60 T² and dividing them by g leaves

T² = (864 000 p² R_E)/g = 864 000 R_E (p²/g)

Providentially, in the units we use g ~ 9.81 is very close to p² ~ 9.87, so that the term in parentheses is close to 1 and may be dropped. That leaves (the two parentheses are multiplied)

T² = (864 000) (6 371 000)

With a hand held calculator, it is easy to find the square roots of the two terms. We get (to 4-figure accuracy)

864 000 = (929.5)²       6 371 000 = (2524)²

Then

T @ (929.5) (2524) = 2 346 058 seconds

To get T in days we divide by 86400, the number of seconds in a day, to get

T = 27.153 days

pretty close to the accepted value

T = 27.3217 days

The Formula for the Force of Gravity

Newton rightly saw this as a confirmation of the "inverse square law". He proposed that a "universal" force of gravitation F existed between any two masses m and M, directed from each to the other, proportional to each of them and inversely proportional to the square of their separation distance r. In a formula (ignoring for now the vector character of the force):

Suppose M is the mass of the Earth, R its radius and m is the mass of some falling object near the Earth's surface. Then one may write

From this

The capital G is known as the constant of universal gravitation. That is the number we need to know in order to calculate the gravitational attraction between, say, two spheres of 1 kilogram each. Unlike the attraction of the Earth, which has a huge mass M, such a force is quite small, and the number G is likewise very, very small. Measuring that small force in the lab is a delicate and difficult feat.

It took more than a century before it was first achieved. Only in 1796 did Newton's countryman Henry Cavendish actually measure such weak gravitational attraction, by noting the slight twist of a dumbbell suspended by a long thread, when on of its weights was attracted by the gravity of a third heavy object. A century later (as already noted) the Hungarian physicist Roland Eötvös greatly improved the accuracy of such measurements.