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Now we'd like to talk
about angular velocity.

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So for a particle
traveling in a circle,

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we've seen that the velocity
can be written as r d theta dt.

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In the theta hat direction.

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And we've also seen that d
theta dt can be positive,

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in which case,
theta is increasing.

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And so the particle is
traveling around the circle

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in this direction.

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You can call that the
counterclockwise direction.

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We've also seen that d theta dt
can be negative, in which case

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the angle is decreasing, so
the particle's traveling around

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the circle in this direction,
which we could call clockwise.

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Let's now look at rotation
in an arbitrary plane.

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So if I have a plane
like this and I

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have some particle traveling
in a circle like this.

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And I have some observer
that's above the plane

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looking down on
this plane, then it

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will see this
rotation as being in

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the counterclockwise direction.

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Whereas if I have another
observer down here

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and they're looking
up at this plane,

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they'll see the motion
as being clock wise.

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And so you can
see we have a need

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for the more formal definition
for the rotation of this.

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And so what we're going to
do is use the right hand rule

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to define a direction
that tells you

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both the direction
it defines the plane

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and it also tells you what the
positive direction of rotation

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is for that plane.

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So the way we'll
do this right hand

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rule is we'll take
our right hand,

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we'll curl our fingers in the
direction of the rotation.

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And our thumb will
point in the direction

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of the positive direction
that we're defining.

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So in this case, I'm going
to have an arrow like this.

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I'll call it n hat to indicate
that it's the normal unit

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vector to that plane.

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All right.

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Let's come back to
this example now,

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where the plane of the motion
is this plane of the board.

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And let's look at
our two cases again.

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So in the case of d theta
dt positive, our circle

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looks something like this.

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And you can see that if
I use my right hand rule,

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the plane, the vector that I've
defined, is out of the board.

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And I'm going to use this
symbol, a circle with a dot,

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to indicate this direction
of out of the board.

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In our other case, for d
theta dt less than zero,

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our particle is traveling
in this direction.

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And so you can see
by my right hand rule

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that now the direction that
I've defined is into the board.

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And for that, I'm going to draw
this as an x in the circle.

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So these are symbols
that you'll see

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throughout the rest of the
course, the out of the board

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and into the board symbols.

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And so now, let's look back
at our coordinate system

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that we've defined.

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We have r hat direction
and a theta hat direction.

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And in this coordinate
system, there's

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a third direction
that's defined,

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which is the k hat
direction, which you

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can see is out of the board.

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And so now, I'm going to
define what we actually call

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the angular velocity as omega.

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And I'm going to write
that as d theta dt.

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Now, in the k hat
direction, in this case,

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it could be in an
arbitrary n hat direction,

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depending on what you're
playing of motion is.

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In this case, I'm going to
call it in the k hat direction,

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and now you can see that these
signs are going to line up

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with these directions.

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So omega, as d theta dt k hat,
when d theta d t is positive,

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k hat is in the same
direction as this unit

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normal that we've
defined that defines

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both the plane of the motion
and the direction of the motion.

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And when d theta
dt is negative, I

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have that omega is in the
negative k hat direction, which

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is exactly this
direction here that we've

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defined as being the
motion for this plane.

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And so this is how we
define angular velocity.

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So we can define d theta dt to
be the component of the angular

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velocity omega z, because
it's in the k hat direction.

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And we can also define
the angular speed,

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which is just omega as being the
absolute value of this angular

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velocity omega.

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So that is, in other words,
it's just the absolute value

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of the d theta dt.

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So when a particle's
undergoing circular motion,

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it has a velocity
that you can describe,

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which is the tangential
motion around the circle.

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And it also has an
angular velocity,

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which we define as being
in a direction that's

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perpendicular to that
direction of rotation.

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And it's defined by
the right hand rule.

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And now that we have defined
this omega z, the component

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of the angular velocity,
we can rewrite the velocity

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as just being equal to
r times omega z still

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in that theta hat direction.

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And so this is another way
that we can write the velocity

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and connect it back to
the angular velocity.