Next: The Clouds Gather Up: Isaac Newton Previous: Optics

## Gravitation.

One of Newton's greatest achievements was on the field of celestial mechanics where he produced the first synthesis in the theories describing Nature: he realized that the same force that makes things fall, gravity, is responsible for the motion of the Moon around the Earth and the planets around the Sun.

He reasoned (more or less) as follows. Suppose I let an apple fall form a very high tower, it will take, say, t seconds to reach the ground. Now suppose I throw it very hard, then again it will take t seconds to reach the ground provided I assume the Earth is flat. But the Earth isn't flat and has curved from beneath the apple! Hence the apple will take longer to hit the ground. By throwing the apple with increasing force one reaches a point where the apple never hits the ground as the distance it falls equals the distance the earth has curved under it: the apple is in orbit! (see Fig. 4.12)

With this thought experiment Newton convincingly argued that an apple can behave in the same way as the Moon, and, because of this it is the very same force, gravity, which makes the apple fall and the Moon orbit the Earth. This is consistent with the hypothesis that gravitation is universal. In a way it represents the unification a several physical effects which appear unrelated at first sight: the falling of apples and the orbiting of planets.

Having realized this he then used the results of Kepler and showed that if the planets and the sun are assumed to be point-like, the gravitational force drops as the inverse distance squared: the gravitational force between two bodies of masses m and M separated by a distance r is attractive and directed along the line joining the bodies, its value is

where G is a universal constant, in words,

 All matter attracts all other matter with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

Having discovered this Newton was able to explain a wide range of previously unrelated phenomena: the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis, and motion of the Moon as perturbed by the gravity of the Sun. It also predicts the position of the planets for thousands of years so that the occurrence of eclipses can be foretold with exquisite accuracy, Moon landings can be planned without uncertainties, etc.

Consider now the application of the second law to the case of the gravitational force.

so that the factors of m cancel (!) This implies that the motion of a body generated by the gravitational force is independent of the mass of the body (!!), (just as Galileo had observed). This unique feature results from Fgrav being precisely proportional to m. So m is seen made to play two roles:

• On the one hand m in F = m a is a measure of how strongly is a body accelerated by a given force: it is a measure of the body's inertia. In this role m is called the inertial mass.
• On the other hand m in Fgrav is a measure of how strongly is a body affected by the force of gravity and also how strong a gravitational force is generated by m; in this role it is called the gravitational mass.

These two quantities refer to different properties of a body and need not be equal. Extremely precise measurements, however, indicate that they are equal (at least to one part in ten parts per trillion). Newton just stated that this was the way of the world and kept going. Einstein, in contrast, noted this as a very important fact of nature, which he used to give birth to his General Theory of Relativity (Chap. 7).

To give an idea of the trust and excellent successes of Newtonian gravity consider the story of the discovery of Neptune. In 1843 a young astronomer at Cambridge, J.C. Adams discovered an anomaly in the orbit of Uranus and by the end of 1845 had concluded that this was due, not to a failure of Newton's law of gravity, but to the presence of a new planet. Adams submitted his results to G. Airy, his boss, who was unconvinced and dropped the matter. Meanwhile U. Leverrier in France had done a similar set of calculations independently, he published in 1846. This spurred Airy into action, but the Cambridge Observatory lacked an up to date chart of the region of the sky were the new planet was supposed to have resided at the time. During that time Leverrier wrote to J.G. Galle at the Berlin Observatory who promptly located the new planet. After much discussion this planet was called Neptune.

Next: The Clouds Gather Up: Isaac Newton Previous: Optics

Jose Wudka
9/24/1998