# Relationship between centripetal acceleration and speed

### Visual understanding of centripetal acceleration formula (video) | Khan Academy

In the first expression v is the speed of a single point on the disk, actually any point that is a distance r from the center. In the second expression. Bottom line: You first of all must know the relationship between the . the same would reduce the centripetal acceleration, but increasing the speed and keeping . Looking at the equation: centripetal acceleration = velocity^2 / radius, is it find" push is relative to centripetal speeding (denoted alternating current or even ar).

So the radius of this circle is v. The radius of this circle, we already know, is equal to r. And just as the velocity vector is what's giving us the change in position over time, the change in position vector over time, what's the vector that's going to give us the change in our velocity vector over time?

### homework and exercises - radius of centripetal accleration - Physics Stack Exchange

Well, that's going to be our acceleration vectors. So you will have some acceleration. We'll call this a1. We'll call this a2.

And I'll call this a3. And I want to make sure that you get the analogy that's going on here.

As we go around in this circle, the position vectors first they point out to the left, then the upper, kind of in a maybe 11 o'clock position, or I guess the top left, and then to the top. So it's pointing in these different directions like a hand in a clock. And what's moving it along there is the change in position vector over time, which are these velocity vectors. Over here, the velocity vectors are moving around like the hands of a clock.

And what is doing the moving around are these acceleration vectors. And over here, the velocity vectors are tangential to the path, which is a circle. They're perpendicular to a radius. And you learned that in geometry-- that a line that is tangent to a circle is perpendicular to a radius.

And it's also going to be the same thing right over here. And just going back to what we learned when we learned about the intuition of centripetal acceleration, if you look at a1 right over here, and you translate this vector, it'll be going just like that. It is going towards the center. So all of these are actually center-seeking vectors.

And you see that right over here. These are actually centripetal acceleration vectors right over here. Here we're talking about just the magnitude of it.

And we're going to assume that all of these have the same magnitude. So we're going to assume that they all have a magnitude of what we'll call a sub c. So that's the magnitude. And it's equal to the magnitude of a1. That vector, it's equal to the magnitude of a2. And it's equal to the magnitude of a3. Now what I want to think about is how long is it going to take for this thing to get from this point on the circle to that point on that circle right over there?

So the way to think about it is, what's the length of the arc that it traveled? The length of this arc that it traveled right over there. The circumference is 2 pi r. So that is the length of the arc. And then how long will it take it to go that? Well, you would divide the length of your path divided by the actual speed, the actual thing that's nudging it along that path.

So you want to divide that by the magnitude of your velocity, or your speed. This is the magnitude of velocity, not velocity. This is not a vector right over here, this is a scalar. So this is going to be the time to travel along that path. Now the time to travel along this path is going to be the exact same amount of time it takes to travel along this path for the velocity vector. So this is for the position vector to travel like that. This is for the velocity vector to travel like that.

So it's going to be the exact same T. And what is the length of this path? And now think of it in the purely geometrical sense. We're looking at a circle here. The radius of the circle is v.

The circumference of this circle is 2 pi times the radius of the circle, which is v. Now what is nudging it along this circle? What is nudging it along this path? What is the analogy for speed right over here? Speed is what's nudging it along the path over here. It is the magnitude of the velocity vector. So what's nudging it along this arc right over here is the magnitude of the acceleration vector.

So it is going to be a sub c. And these times are going to be the exact same thing. The amount of time it takes for this vector to go like that, for the position vector, is the same amount of time it takes the velocity vector to go like that. So we can set these 2 things equal each other.

**Centripetal force and acceleration intuition - Physics - Khan Academy**

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.

## Visual understanding of centripetal acceleration formula

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.