Angular Velocity and its relation with Linear Velocity Video Lecture -

Well, we've done a lot of work
with how fast something moves,
let's see if we can work with
how fast something spins.
Let's see what we can do.
Since we're going to be working
with things spinning,
let me draw a circle.
Since things that spin
go in circles.
And let me just draw the
positive x-axis because it'll
come in handy in a second.
That's the positive x-axis.
And let's say that I have an
object, and the circle is the
object's path.
So let's say this
is the object.
And it's going around
in a circle in a
counterclockwise direction.
Not squiggly counterclockwise,
it's just
going around this way.
Let's say I wanted to figure
out, or I wanted to quantify
how much, or how fast this
thing is spinning.
So one thing that you're
probably familiar with is
revolutions per second, or
rotations per second.
So let's write that down, let's
just say for the sake of
argument that this was moving
at, I don't know, 1 revolution
per second.
So after 1 second it goes back,
then another second.
So that's how fast
it's spinning, 1
revolution per second.
1 revolution-- I'll just
put rev-- per second.
So let's see if we can quantify
that in angles, and
we'll do it in radians, but you
could always convert it
back to degrees, if you want.
I don't know if you
can see that line.
Let's just say that theta is the
angle between the radius
from the center to
that object, and
the positive x-axis.
So if this object is travelling
at 1 revolution per
second, how many radians per
second is it traveling?
Well, how many radians are
there in a revolution?
Well there's 2 pi radians
in a revolution, right?
1 go-around in a circle
is 2 pi radians.
So we could say, so this equals
1 rev per second, times
2 pi radians per rev, right?
And then the revolutions
will cancel out.
And you have 1 times
2 pi, so you have 2
pi radians per second.
So this equals 2 pi radians
per second.
So that's interesting, we now
know exactly after 5 seconds
how many radians it has gone.
Or after half a second, how
many radians has it gone.
But that might be
vaguely useful.
Let's see if we can somehow
convert from this notion of
how fast something is spinning
to its actual speed.
I was tempted to say velocity,
but its velocity is always
changing, because the direction
is always changing.
But the magnitude of the
velocity is staying the same,
so its speed is staying
the same.
But we'll say v for speed,
because that's what they tend
to do in most formulas
that you'll see.
So let's think about it this
way, in 1 revolution-- so
there's a couple ways you can
think about this, but as we go
1 revolution, how far has
this object traveled?
Well, it's traveled the
circumference of this circle.
And in order to know the
circumference, we have to know
the radius of the circle.
So let's say that the radius
is r-- let's say it's in
meters, r meters.
So how many meters will I travel
in 1 second, then?
Well, you could do the
same thing up here.
1 revolution per second, times
2 pi r, where r is the
radius-- whoops, 2 pi r, you can
ignore that line-- meters
per revolution, that's just
the circumference of the
thing, of the circle.
And that equals-- the
revolutions cancel out-- 2 pi
r meters per second.
So it's interesting, given
the radius and how many
revolutions per second, we can
now figure out its velocity.
So this right here is how fast
it's spinning, and this is the
object's actual speed, right?
And this term of how fast
something's spinning, that's
called angular velocity.
And of course you know that
the term for how fast
something is actually
moving is velocity.
And just so you know, the term
for angular velocity is this
curvy w, I think that's
lower case omega,
that's angular velocity.
So in this case, angular
velocity is equal to 2 pi
radians per second.
And what's the velocity equal
to, or at least the magnitude
of the velocity-- I know the
direction's always changing.
Well, we know that the velocity
is equal to 2 pi r
meters per second.
So if we just ignore the units
for a second, where do you see
the difference between
the angular
velocity and the velocity?
The angular velocity in this
case is 2 pi, and the
velocity is 2 pi r.
So in general, if you just
multiply the angular velocity
times r, you get the velocity.
So angular velocity times the
radius is equal to velocity.
Or you can divide both sides of
that by r, and you get the
angular velocity is equal
to the velocity
divided by the radius.
And this is a formula that you
should know by heart, although
it's good to know where
it came from.
I guess I did it this way to
maybe give you an intuition,
because I always have to
work with numbers.
Especially when I'm new to a
concept-- so that's why I said
1 revolution per second,
instead of just putting
everything as a variable-- but
another way to think about it
is, what is the definition
of a radian?
By definition, a radian-- if
this angle is x radians, it's
an angle, and it also tells us
that the arc that is kind of
projected by this angle,
is equal to x radiuses.
So if each radius is 2 meters,
it would be x times 2 meters.
So if this is x radians,
then this is going to
be x times r meters.
And that actually comes from the
definition of the radian.
And that might be more intuitive
to you, than the
original explanation, or
less, so hopefully one
of those two works.
But as you can see, if this
angle is x, and this distance
is x times r, and if omega is
change in that angle, over
change in time.
Then we know this is true too,
that velocity is just change
in this, over change
in time, right.
Velocity is change in-- the
radius doesn't change-- change
in x times r, divided
by change in time.
And we know once again
that this is omega.
So another way we just showed
again, that omega times the
radius is equal to
the velocity.
Or the angular velocity
times the radius is
equal to the velocity.
And this is a useful thing to
learn, we'll see it in a
couple of things, when I do
the proof for centripetal
acceleration in calculus, I'm
going to use this fact.
And when I-- and actually I'm
probably going to record that
video now-- I'm actually going
to show you the law of
conservation of angular
momentum, which is very
similar to the law of
conservation of momentum, but
it deals with things spinning.
And this notion of angular
velocity is
going to come in useful.
So this is the important
takeaway, that w
equals v over r.
And hopefully my video has not
confused you, and has shown
you that w, the rate at which
the angle is changing, is
equal to the velocity of the
object, or the magnitude of
the velocity, divided
by the radius of the
circle that it's spinning.
I'll see you in the
next video.