Microworld: Calculus
in Action: Chapter 2, Section 1: Polar Coordinates
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Author:James
E. White
Before
we can study the dynamics of satellite orbits, we must pay some attention
to the kinematics, that is the abstract geometric description
of projectile motion. The fact is that we never launch our satellites vertically
away from the Earth. When the boosters stop burning and the satellite is a
projectile, it is not pointing straight up. If it were, then, as we shall
see in Harmony of the Spheres, the satellite would either fall back
to the Earth, or escape. We will take up the question of escape velocity,
the speed it would need never to fall back to the Earth on its vertical trajectory
later in this chapter.
The
satellite will have two components to its motion: a radial
component that describes how far from the center it is, and an angular
component that describes how its angle with a fixed line from the Earth is
changing. (For vertical motion, the angular velocity component will
be 0.) The picture of a satellite's trajectory may in general resemble the
one at the top of the page.
The
length of the light blue arrow is the positive radial velocity component at
any time. The angular velocity component is a little more difficult to visualize.
In fact, we will abandon the Cartesian representation (once we explain what
it is) to follow an ingenious idea of Newton to represent these components
of motion. We will establish a "moving frame" of polar coordinates,
attached to the satellite, that will give us a clear and simple way to understand
how these components change. Finally, the tiny red arrow in the picture that
points to the Earth represents the gravitational acceleration. We will say
much more about that later.
In
this microworld, we will introduce a style of description that will serve
us well throughout the book. It is called polar coordinates. As it happens,
and we shall prove in Harmony of the Spheres, any satellite moving
under the influence of a central force (like Newton's law of gravity) must
remain in a plane. If its orbit is singular (straight up and down) then in
fact it remains in a line, but we shall generally assume that the angular
velocity component is not 0, and so this will not happen. Therefore we will
be justified in using two polar coordinates (instead of three) to describe
the motion.
After
we discuss polar coordinates as a way of describing satellite motion, then
we will, in the next microworld, return to the principle of reciprocity or
conservation of energy for satellite orbits in terms of these coordinates.
The
following is the Table of Contents for this 6 page Microworld
- Cartesian to Polar Coordinates
- Conic Sections
- Polar Curves and their Moving Frames
- Motion in a Gravitational Field
- Symbolic Calculator
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
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