- [Instructor] In multiple videos we have already talked
about if something is moving in a circular motion
at a fixed speed, its velocity is constantly changing.
Why is that?
Because velocity is a vector
and a vector has not just a magnitude
which would be its speed
but also its direction,
so even if I have the same speed at this point
as I have at say this point right over here,
my velocity will be pointing in a different direction
so now the magnitude might be the same,
I'll try to draw it roughly the same,
the length of this arrow should be the same,
so the magnitude,
V without an arrow on top,
you could view that as the linear speed,
that will be the same
but now the direction has changed
and in order to change the direction,
you must have this ball
that's moving in this circular motion
must be accelerated.
That's the only way.
If you have a change in velocity,
then you must have acceleration
and it's a little counterintuitive at first
because you're saying well, my magnitude didn't change,
only my direction did
but any change in your velocity implies that acceleration
and in previous videos we saw
that that acceleration is constantly going to be inward
if you have your uniform circular motion
and we call that inward acceleration,
we call that centripetal acceleration
and though if I wrote this A sub C like this,
this means the magnitude of my centripetal acceleration.
If I'm talking about the magnitude and the direction
I would put an arrow on it just like that.
Now, we have also, in previous videos
have been able to connect what is the magnitude
of centripetal acceleration,
how can we figure that out
from our linear speed and the radius
and we had the formula,
the magnitude of centripetal acceleration
is equal to the magnitude of our velocity
or our linear speed squared
divided by our radius.
Now, what I wanna do in this video
is see if I can connect our centripetal acceleration
to angular velocity,
our nice variable omega right over here
and omega right over here
you could use angular speed.
It's the magnitude, I could say our magnitude
of our angular velocity,
so our angular speed here.
So, how can we make this connection?
Well, the key realization is to be able to connect
your linear speed with your angular speed.
So, in previous videos, I think it was the second
or third when we introduced ourselves
to angular velocity or the magnitude of it
which would be angular speed,
we saw that our linear speed is going to be equal
to our radius, the radius of our uniform circular motion
times the magnitude of our angular velocity
and I don't like to just memorize formulas.
It's always good to have an intuition
of why this makes sense.
Remember, angular velocity
or the magnitude of angular velocity
is measured in radians per second
and we typically view radians as an angle
but if you think of it as an arc length,
a radian you could view it
as how many radii in length am I completing per second?
And so, if I multiply that times the actual length
of the radii, then you can get a sense
of well, how much distance am I covering per second?
Hopefully that makes some sense
and we actually prove this formula,
we get an intuition for this formula in previous videos
but from this formula it's easy to make a substitution
back into our original one
to have en expression for centripetal acceleration,
the magnitude of centripetal acceleration
in terms of radius and the magnitude
of angular velocity
and I encourage you, pause this video
and see if you can drive that on your own.
All right, let's do this together.
So, if we start with this,
we have the magnitude of our centripetal acceleration
is going to be equal to,
instead of putting V squared here,
instead of V, I can write R omega,
so let me do that,
R and then omega.
There you go and all I did I said look,
our linear speed right over here
is equal to our radius
times the magnitude of our angular velocity
or angular speed,
so everywhere I saw a V here,
I'm just replacing it with an R omega
and so, I have R omega,
the entire quantity squared over R
and then we can simplify this.
This is going to be equal to,
I just use my exponent properties here,
R omega times R omega
is gonna be R squared
times omega squared,
all of that over R.
If I have it R squared over R,
well, that's just going to simplify to an R,
so there we have it,
we have our formula for the magnitude
of our centripetal acceleration
in terms of the magnitude of our angular velocity.
It is going to be R
times the magnitude of our angular velocity omega squared.
There you have it
and in future videos
we'll do worked examples
where we actually apply this formula.