Deriving formula for centripetal acceleration from angular velocity | AP Physics 1 | Khan Academy

• [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.

In previous videos, we saw that that acceleration is constantly going to be inward. If you have your uniform circular motion, 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 want to 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 an 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. 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.