Change in angular velocity when velocity doubles

We're told that a car with wheel radius r moves at a linear velocity v, and this is a bolded v to show that it's a vector. Suddenly, the car accelerates to velocity 2v. How does the angular velocity of the wheels change? So pause this video and see if you can figure this out on your own.

All right, now let's work through this together. So just to visualize what's going on, let's just draw at least the front wheel of a car here. So let me draw the wheel first, actually. So this is the front wheel of the car, and I'll draw the part of the car. So this is our car right over there. It might look something like—noted, looks something like that. You've got the windshield right over there, so you can visualize this car. So our car looks something like—it’s got our headlights and all the rest.

All right, enough of drawing the car. So let's imagine this car right over here has a velocity v. Let's say it's going forward, so let me draw that. So it's velocity—I'll just draw it right over here—it is v. So this is a velocity of v. And since it's hard to hand-draw something bolded, I put a little arrow on it, but this is equivalent to this bolded vector v that you might see sometimes in your textbook. Bolded would be a vector, or you put an arrow on it to show it's a vector.

Now, I want to make notation clear. If we talk about the magnitude of this vector, the magnitude of velocity is just speed. I'm going to denote, I'm going to denote that as just v without an arrow on it and not bolded. So this is going to be the same thing as the magnitude of our velocity vector, which is the same thing as our speed.

Now, let's take a point on the outer end of our wheel. Any point on the outer end of the wheel, let's say we take this point right over here, and the center of our wheel is right over here. So if the car is moving forward to the right with velocity v, what's the velocity of this point right over here? Well, the velocity of this point is going to have the same magnitude but in the opposite direction, so it's going to look something like that. I don't know if those lines are exactly equal, but these are supposed to have the same magnitude. And so you could even imagine this would be— you could say that this would be a negative v, just like that. But for the sake of r, for the purposes of this question, I want to think about its speed.

So the speed here—the speed here is just going to be v again. So the speed of this point, which is going to be the same as the speed of this point—the direction here is different, but all of these points along the rim, or along the outside of the tire, I should say, are all going to have the same speed, speed v.

And we already have a relationship that we've explored in previous videos between speed and angular velocity. We have seen that speed—I'll do it in a new color here, just I'll say v for speed with no arrow, not bolded—is going to be equal to the magnitude of our angular velocity, so the magnitude of our angular velocity times our radius. Or another way of saying this, we could say that our speed is equal to— instead of writing it like this, let me just write just an omega without an arrow over it to say this is the magnitude of our angular velocity—so that times our radius. So here, I'm dealing with all scalar quantities. I'm saying speed is equal to the magnitude of my angular velocity times my radius.

So if I wanted angular velocity as a function of speed, I could just divide both sides by r. And so I could say angular velocity is going to be equal to my speed over r. So let's think about what happens, and let me be clear—this right over this distance right over here—that is r, that is r right over there. So if our velocity goes to 2v, so if this goes from v to—let me just a different color—if that goes from v to 2v, then this is going to go from negative v to negative 2v, and our speed is going to go from v to just 2v, twice the speed.

And if you have twice the speed, if this goes by 2, well then the angular velocity is going to increase by a factor of two. So this is going to increase by a factor of two. You're not squaring this or something else; you're just multiplying this by one over r. You're just dividing it by r, and r is not changing—just the speed is changing. And since omega is equal to v over r, if you double v, you're going to double omega.

So how does the angular velocity of the wheels change? Well, it's going to double. It will double, and we are done.