WEBVTT
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When we're looking
at polar coordinates,
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one of the important issues is
to understand the unit vectors.
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Let's describe our
coordinate system, again.
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We have a point where
there is an object.
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And at this point, we have a
pair of unit vectors, r hat
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and theta hat.
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Now those unit vectors
will change, depending
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on where you are in space.
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We now want to
address the question,
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how do I compare
polar coordinates
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and Cartesian coordinates.
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In Cartesian coordinates,
at this point
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here, we would have-- let's say
we choose a Cartesian plus x
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and plus y, then we would have
a unit vector i hat and j hat.
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And how are these
vectors related?
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Remember our angle theta, so
we have the angle theta here
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and the angle theta there.
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And now I would like to apply
a simple vector decomposition
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to r hat and theta hat and
express each of these unit
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vectors in terms of the unit
vectors i hat and j hat.
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Let's begin with r hat.
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As you can see in
the diagram, r hat
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has a horizontal component
and a vertical component.
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So what we have is r hat.
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It's a unit vector,
so its length is 1.
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Its horizontal component
is adjacent to the angle,
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so that's cosine of theta i
hat plus sine of theta j hat.
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The vertical component
is opposite the angle.
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In a similar fashion,
theta hat-- well, theta hat
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has a component in the
negative i hat direction,
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which is opposite the angle.
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And it has a component in
the positive j hat direction,
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which is adjacent to the angle.
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So we have minus sine theta i
hat plus cosine theta j hat.
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And that's how we can
decompose our unit vectors r
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hat and theta hat in terms
of Cartesian coordinates.
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Now why is this significant?
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Because if we're describing
the motion of an object,
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for instance, an object
that's going around a circle,
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then our polar coordinate
theta is a function of time.
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And so these unit vectors are
actually changing in direction.
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You saw that before.
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Over here, r hat and theta hat
point in different directions.
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So what we actually have as
functions of time is r hat of t
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equals cosine theta of t i hat.
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Now the unit
vectors don't change
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in Cartesian coordinates.
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At every single point, you
have the same Cartesian unit
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vectors.
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And so this vector
is time dependent.
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Now the significance of that
is our first important vector
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in kinematics is
the position vector.
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The position vector
is a vector that
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goes from the origin
to where the object is.
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We'll call that r of t.
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So this position vector r of t
can be expressed as a length r.
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And its direction is
in the r hat direction,
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which is a function of time.
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So we have r cosine
theta of t i hat
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plus r sine theta of t j hat.
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Now we can now define the
velocity of this object
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where the velocity is the
derivative of the position
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vector.
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When you differentiate,
remember, r is a constant,
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so we get r.
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Now what is the
derivative with respect
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to time of cosine theta t.
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Because the argument of
theta is a function of t,
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we need to use the chain rule.
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So the derivative is minus sine
theta of t, d theta dt i hat.
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And the derivative of
the sine is cosine theta
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of t d theta dt j hat.
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Now notice that I can pull out
the common term d theta dt.
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So I have r d theta dt.
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And I have minus sine theta
of t i hat plus cosine
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theta of t j hat.
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And if you'll notice, this is
exactly the unit vector theta
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hat.
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So we can write
our velocity vector
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for this object that's moving in
a circle as r d theta dt theta
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hat.
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When we write a
vector like this,
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it's pointing tangentially,
the theta hat direction,
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and this part is the component.
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So often we can use a
notation v theta theta
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hat, where v theta is the
component r d theta dt.
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Now this component can be
positive or negative or 0.
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For example, if d theta dt is
positive, what does that mean?
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That means that
our angle theta is
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increasing so the object is
moving the way I indicate
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with my finger.
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If d theta dt is 0, then
the angle is not changing,
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so the object is at rest.
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And finally, if d
theta dt is negative,
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then the angle
theta is decreasing,
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and so the object is
moving in this direction.
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So this is our velocity for
a circular motion expressed
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in polar coordinates.