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Newton's Universal Law of Gravitation

The next celebrity in the story of gravity is Sir Isaac Newton. Newton was
born on Christmas Day in 1642. He was known as a reclusive, short-tempered
man, who did not take criticism very well. Because of his disdain for
criticism, his most famous work, "Principia", was only published after
strong persuasion by Halley (yes, of Halley's comet), about 20 years after
he had developed the ideas presented in the work. In Principia Newton
presented his three laws of motion, he invented a new system of
mathematics which is called calculus today, and he came up with his
Universal Law of Gravitation. Principia was primarily written as an answer
to a
challenge set forth to physicists by Christopher Wren, who offered a cash
prize for the best solution. The challenge was to explain, not just
describe, the motions of the planets. Newton was the first person to
associate gravity with the orbits of planets. People were familiar with
gravity here on Earth, but the idea that this same force could explain
the motions of heavenly bodies was new.

Newton used Kepler's Laws in deriving his Universal Law of Gravitation.
Deriving Newton's law is fairly simple for the case of a circle, the simplest
form of an ellipse.

Derivation of the Universal Law of
Gravitation
The very important conclusion from Newton's Law of Universal Gravitation
is that the force due to gravity is proportional to one over the
separation squared (the 1 / r² law). It is called a universal
law, because it applies to any two masses, whether they are clusters of
galaxies, planets, or elementary particles. The force of gravity operates
over all distances, as far as we can tell: in fact, it is the only
significant force affecting the movement of large-scale structures in
space. Take, for example, clusters of galaxies, which are held together by
gravity over distances of ten million light years. A light year is the
distance traveled in one year, if going at the speed of light (one light
year is about 9.5 * 10^{12} km: about 10 trillion kilometers). To
put this in perspective, a cluster of
galaxies is a group of galaxies that are held together by gravity; a galaxy
is a group of stars that are held together by gravity (for example, the
milky way); a star (like our sun) may have a solar system associated with
it, which is held together by gravity.
We tend not to notice gravity so much on the smaller scale; when you are
walking past someone in the hall, you probably don't worry about being
pulled together by your gravitational attraction to each other. And when
we talk about atoms, we generally don't talk about the force of gravity
holding them together. That is because there are electromagnetic
forces which act over these short distances, and they are much stronger
than gravitational forces. On very large scales (like galaxies), these
electromagnetic forces tend to cancel each other out over relatively short
distances, so they don't really affect the motions of stars and planets.
Gravity, on the other hand, is always a positive force; it is additive,
and it can never cancel itself out.

There are many interesting similarities between electromagnetic forces and
gravitational forces. The laws for electromagnetism were derived much more
recently, but just like gravity, the strength of the force is proportional
to one over the separation squared. In fact, you will notice that the
formulas are very similar, except that it is the charges, not the masses,
that cause the electromagnetic forces to exist.

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