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# Deriving formula for centripetal acceleration from angular velocity

## Video transcript

- [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.