# Relationship between centripetal force and linear velocity

### Episode Quantitative circular motion

When an object is traveling in a circular path, centripetal force is what How to Convert Angular Velocity to Linear Velocity You might not have known it at the time, but you were creating a balance between two forces - one real and one Its value is based on three factors: 1) the velocity of the object as it. Note that the centripetal force is proportional to the square of the velocity, implying that a doubling of speed will require four times the centripetal force to keep. Investigate the relationship between centripetal force and velocity in circular Next we are going to calculate the average linear speed, v, of the stopper for each.

It holds the pumpkin right over here. That's the pumpkin that's about to be shot away. And what it does is once it releases, I guess someone presses a button or pulls a lever, the pumpkin catapult is going to release, and so this arm is going to turn a certain amount and then just stop immediately right over there.

And then that pumpkin, that pumpkin is going to be released with some type of initial velocity.

### centripetal force - The relationship between velocity and radius - Physics Stack Exchange

So your pumpkin is just gonna go like that. So this is our small pumpkin catapult. This is our small one. But let's say we also have a large pumpkin catapult. Let me draw that. So our large pumpkin catapult right over here, similar mechanism. But let's say its arm is four times as long. So, that looks about four times as long right over there. So this is the larger one. The large pumpkin catapult. Let me make sure to draw the pumpkin to remember what we are catapulting.

And then it will go through actually the exact same angle, you go to the exact same angle, and then let go of its pumpkin. And this is gonna be a very useful video because you will find yourself making many pumpkin catapults in your life.

Now we know a few things about these pumpkin catapults. Let's say the small pumpkin catapult, the radius between the center of the pumpkin, the radius between the center of the pumpkin and the center of rotation right over there, let's say that this is r. While for the large one, this one is, this distance right over here, is four r.

We also know the angular velocity while this thing is moving. So we know that the angular velocity here, let's say the magnitude of the angular velocity is omega.

**Centripetal Acceleration, Basic Introduction, Physics Problems, Period, Frequency, Linear Speed**

It would actually be negative if we were to write it as a vector 'cause we're going in the clockwise direction because that's the convention.

But this right over here is the magnitude of the angular velocity and just to make that tangible for you, let's say that this is two pi radians per second. And let's say this thing, while it's in motion, it also has the same magnitude of its angular velocity, so this thing right over here is also. The magnitude of angular velocity is once again two pi radians per second. 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.

## Linear velocity comparison from radius and angular velocity: Worked example

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.