WEBVTT
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For a particle that's
moving in a circle,
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we found that when it's moving
at a constant rate of d theta
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dt--- and let's recall what
we meant by theta of t--
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and here's our particle,
and we introduced our polar
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coordinates r hat and theta hat,
then we found that the velocity
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was r d theta dt theta hat.
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And so let's assume that
this quantity is positive,
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in which case the velocity is
pointing in the positive theta
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direction.
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And that means that
everywhere in the circle,
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the velocity is
tangential to the circle,
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and the magnitude is a constant.
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So for this case of
uniform circular motion,
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we calculated that
the acceleration
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was equal to minus r d theta dt
quantity squared r hat, which
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means that at every point, the
acceleration vector is pointing
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towards the center.
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Now we can write that
acceleration vector
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as a component a of r-- r hat--
where this component is given
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by r times d theta dt squared.
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It it's always negative, because
when you square this quantity,
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it's always a positive quantity.
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The minus sign, just
to remember-- that
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means that the acceleration
is pointing inward.
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Now how can we think about that?
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Well, if we look at the velocity
vector, what's happening here
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is the velocity is
not changing magnitude
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but changing direction.
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And if you compare
two points-- and let's
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just pick two arbitrary points.
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So let's remove this
acceleration for a moment
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and consider two arbitrary
points-- say, a time t1 and t2.
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So our velocity
vectors are tangent.
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The length of these
vectors are the same.
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And if we move them tail
to tail-- [? Vt2-- ?]
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and take the difference,
delta v, where
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delta v is equal to v
of t2 minus V of t1,
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then we can get an understanding
why the acceleration
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is pointing inward,
because recall
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that acceleration
by definition is
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a limit as delta t goes to 0.
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That means as this
point approaches
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that point of the change
in velocity over time.
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And so when we
look at this limit
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as we shrink down our time
interval between t2 and t1,
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then this vector will
point towards the center
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of the circle.
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And that's why
the direction of a
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is in the minus r hat direction.
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Again, let's just recall
that this is the case
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for we called uniform
circular motion, which
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is defined by the condition
that d theta dt is a constant.