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Now let's consider the case
where an object is undergoing
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circular motion, but
in this motion let's
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again introduce our polar
coordinate system, r-hat
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and theta-hat.
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We now want to consider the
case where d theta dt is not
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constant.
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And what does that mean?
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That means that if d theta
dt is positive, for instance,
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the object is speeding
up in this direction,
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or if d theta dt is
negative, the object
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is slowing down
in this direction.
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That's one type of case.
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So in this instant,
we always know
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that there is the radial
component at any instant given
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by minus r d theta dt squared.
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But because it's speeding
up and slowing down,
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there is now a non-zero
tangential component
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to the acceleration.
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Let's see where that comes from.
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So again, if we write our
velocity vector as r d theta dt
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theta-hat, this is the
product of two terms.
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And because it's a
product of two terms,
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we need the product
rule from calculus
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in when we take a derivative.
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So the derivative
will be the derivative
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of the first term
times the second term
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plus the first term times the
derivative of the second term.
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Now we've already
analyzed this piece
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and this was precisely
minus r d theta dt
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quantity squared r-hat.
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That was the always the
non-zero radial acceleration.
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But now let's analyze
this piece separately.
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r, for our circular
motion, is a constant.
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So it's only d theta dt
that is no longer constant.
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So we simply take a
second derivative.
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And so we get r times d squared
theta dt squared theta-hat.
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And that is our acceleration.
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Notice that it has
two components.
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We'll write the first
component, a theta,
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that's its tangential
component, theta-hat
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plus the radial component ar.
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That's again, the component.
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And because this
is a vector, r-hat
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where the a theta is now the
second derivative of d theta
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squared dt squared.
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And just to remind you that ar
is minus ar d theta dt squared.
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So when d squared theta
dt squared is positive,
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it means d theta
dt is increasing.
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And so if this object is going
in this clockwise direction,
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we call that speeding up.
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In a similar fashion,
it's easy to understand
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that when d squared theta
dt squared is negative,
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then d theta dt is decreasing.
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And so it can be slowing down.
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Or if it slows
down and stops, it
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can start to move in
the other direction.
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So again, the acceleration
has two components,
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a tangential component,
and that depends
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on the type of circular
motion we're talking about,
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whether d theta dt
is constant or not.
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It always has a non-zero
inward radial component
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given by the component
minus r d theta dt squared,
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regardless of whether it's
speeding up or slowing down.