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There are many physical problems
in which a body undergoes
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motion about a central point.
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And when that happens, there's
a natural coordinate system
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to describe that motion,
which in two dimensions
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is polar coordinates.
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So let's consider the
orbit of an object.
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For instance, the best
example is a circular orbit.
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If we have a circular
orbit of an object,
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there is a central point,
which we'll call P.
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Now given this type of
motion with an object,
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it naturally makes sense to
choose a coordinate system
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called polar coordinates.
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The way that coordinate
system works is as follows.
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First off, we need to
choose a reference angle.
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And so we'll choose
a horizontal line.
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And we'll draw a ray.
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And we'll show a direction
of increasing reference angle
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theta.
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In this example, theta
will go from 0 to 2 pi.
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Along with the reference
angle, we have a distance
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from the central point.
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And that distance
we'll refer to as r.
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So the coordinates of our
point are r and theta.
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Now the variable r is
always greater than r
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and can go to infinity--
greater than 0.
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So this is our polar
coordinate system.
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When you have a
coordinate system,
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remember, at every
point in space,
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there has to be unit vectors.
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So at this point right here,
how do we choose unit vectors
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for polar coordinates?
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We always choose
the unit vectors
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to point in the increasing
direction of the coordinate.
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Take the r-coordinate.
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That increases radially outward.
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So our unit vectors
here will have a r hat
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pointing radially outward.
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What about the theta direction?
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Tangential to the circle,
in this particular case.
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Because theta is increasing
in this direction,
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we choose our
tangential unit vector,
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which we're going to call theta
hat, which is at right angles
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to r hat, to point in the
direction of increasing theta.
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And so at this point, we now
have a set of unit vectors.
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Now one has to be very
careful in polar coordinates
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for the following reason.
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Suppose that you're at
another point over here.
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Now because we have
two different points,
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let's just give some
names to these points.
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We'll call this s1
and this point s2.
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And the unit vector's over
here where r1 and theta were.
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When we're at the
point as 2, we have
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to choose unit vectors
exactly the same way.
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r hat 2 points in the direction
of increasing r, and theta hat
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2 points in the direction
of increasing theta.
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So what we see in
polar coordinates
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is that r hat 1 is
not equal to r hat 2.
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Why are they not equal?
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They both are unit
vectors, so they both have
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the same magnitude, but they
point in opposite directions,
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in the same way that theta hat
1 is not equal to theta hat 2.
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So unlike Cartesian coordinates,
in which at every single point
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had the same unit vectors,
in polar coordinates,
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the unit vectors depend
on where you are in space.
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And that will make our
analysis on polar coordinates
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a little bit more complicated.