Transcript Kepler`s Third Law

Kepler's First Law
The orbits of the planets are ellipses,
with the Sun at one focus of the
ellipse.
The Sun is not at the center of the ellipse, but is
instead at one focus (generally there is nothing at
the other focus of the ellipse).
Kepler's Second Law:
II. The line joining the planet to the Sun
sweeps out equal areas in equal times as the
planet travels around the ellipse
The line joining the Sun and planet sweeps out equal areas
in equal times, so the planet moves faster when it is nearer
the Sun.
Kepler's Third Law:
III. The ratio of the squares of the revolutionary
periods for two planets is equal to the ratio of the
cubes of their semimajor axes:
In this equation P represents the period of
revolution for a planet and R represents the
length of its semimajor axis. The subscripts
"1" and "2" distinguish quantities for planet 1
and 2 respectively. The periods for the two
planets are assumed to be in the same time
units and the lengths of the semimajor axes
for the two planets are assumed to be in the
same distance units.
Kepler's Third Law implies that the period for a planet to orbit the Sun
increases rapidly with the radius of its orbit. Thus, we find that Mercury,
the innermost planet, takes only 88 days to orbit the Sun but the
outermost planet (Pluto) requires 248 years to do the same.
What Really Happened with the Apple?
The apple is
accelerated, since
its velocity
changes from zero
as it is hanging on
the tree and moves
toward the ground.
Thus, by Newton's
2nd Law there
must be a force
that acts on the
apple to cause this
acceleration. Let's
call this force
"gravity",
Sir Isaac's Most Excellent Idea
Now came Newton's
truly brilliant insight: if
the force of gravity
reaches to the top of
the highest tree, might
it not reach even
further; in particular,
might it not reach all
the way to the orbit of
the Moon!
If we increase the muzzle velocity of
an imaginary cannon, the projectile
will travel further and further
before returning to earth. Newton
reasoned that if the cannon
projected the cannon ball with
exactly the right velocity, the
projectile would travel completely
around the Earth, always falling in
the gravitational field but never
reaching the Earth, which is curving
away at the same rate that the
projectile falls. That is, the cannon
ball would have been put into orbit
around the Earth. Newton concluded
that the orbit of the Moon was of
exactly the same nature
the Moon continuously "fell" in its path around the Earth because of
the acceleration due to gravity, thus producing its orbit.
By such reasoning, Newton came to the conclusion that any two objects in
the Universe exert gravitational attraction on each other, with the force
having a universal form: