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Now we'd like to discuss
angular acceleration

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for circular motion.

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So suppose we have our angle
theta, radius r, and r hat

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and theta hat.

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Recall that we described
the angular velocity

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as the derivative of d theta dt,
and we made this perpendicular

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to our right-handed coordinate
system, direction k hat.

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Now let's differentiate that
to get our concept of angular

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acceleration.

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So alpha is the second
derivative d theta

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dt squared k hat.

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And this quantity is what we
call angular acceleration.

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Now we'll describe
the component alpha z

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as d squared theta dt squared.

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So it's the second
derivative of the angle.

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And also if we wrote
this as omega z k hat,

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we can write that as the
derivative of d omega z dt,

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as well.

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So this is the component.

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And now in circular motion, the
quantities of omega and alpha z

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are very much like
the linear quantities

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of the x component
of the velocity

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and the x component
of the acceleration.

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And again, when we've
chosen a reference frame,

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let's look at what
various components mean.

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Let's begin with the case 1
where omega z is positive.

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So when omega z
is positive, that

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tells us that the angle
d theta dt is increasing.

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And that corresponds to
counterclockwise motion.

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Now given that case,
let's look at what happens

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when alpha z is positive.

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Remember, that's the statement
that d omega dt is positive,

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that omega z is increasing.

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So if an object is moving with
a positive component of omega z

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and the angular acceleration
component is positive,

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that corresponds to increasing.

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The linear example, if you had
one dimensional motion, i hat,

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you had vx positive
and a x positive,

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corresponds to an
object increasing

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in its speed in the x direction.

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That's our first case.

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Now let's look at
the second example

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when alpha z is less than 0.

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So now the derivative of
d omega z dt is negative.

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What that corresponds
to-- remember,

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omega z is the z component
of the angular speed.

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And if that's slowing down,
then, with alpha z less than 0,

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the object is slowing down.

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So in our linear case, if
we had a x less than 0,

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this is the classic
example of breaking.

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The object is moving in the
x direction and slowing down.

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Now let's look at case 2.

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This is always a little bit
complicated for circular motion

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where omega z is less than 0.

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In that case, the
object is moving

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in the clockwise direction
because the angle

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theta is decreasing,
corresponding

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to clockwise motion.

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So in that case, once again,
let's consider the two

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examples.

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Well, the first example
is a positive component

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of angular acceleration.

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Now this is the one that can
be a little bit confusing.

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The object is moving
clockwise but it

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has a positive alpha z,
which will correspond

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to slowing the object down.

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And if the alpha z
remains positive,

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it will actually come to rest
and then reverse its motion

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and start to speed up.

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So this is the case where
d omega dz is increasing.

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And that's our first case.

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So something like that
could correspond to,

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if we plotted omega z
and we had an object that

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starts off with a negative
omega z and increases.

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Notice that the slope here,
which is alpha z positive,

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corresponds to a positive
angular acceleration component.

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And the object slows down
as omega gets closer to 0,

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stops, and now has
a positive omega

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z, corresponding to motion in
a counterclockwise direction.

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For our linear case,
this corresponds

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to, again, with i hat, our
object moving to the left,

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vx negative, and
if a x is positive,

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it breaks in this direction,
which means it's slowing down.

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And then eventually if
alpha x, ax, stays positive,

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it continues in that direction.

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Now our final case, and
I'll put it down here,

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b, this is again where omega z
negative and alpha z negative.

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It's always helpful to see this
immediately with the graph.

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Omega z is negative.

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Here, alpha z, which is the
slope, is also negative.

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This corresponds
to an object moving

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in the clockwise direction.

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And actually its
speed is increasing

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because alpha z is negative.

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So it's going faster and faster
in the clockwise direction,

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even though alpha z is negative.

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And for our linear
case, again, this

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corresponds to an object moving
in the negative x direction.

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And a x is negative,
it's moving faster

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in the negative x direction.

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And so these are
the cases of how

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we analyze the various cases
for angular acceleration

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and angular velocity.