Introduction to angular velocity

We’ve done a lot of work with how fast something moves; let’s see if we can work with how fast something spins. Well, let’s see, what we can do. Let me, since we’re going to be working with things spinning, let me draw a circle. Since things that spin go on circles. So there’s my circle and let me just draw the positive X axis because it will come handy in a second.

So that’s the positive X axis. And let’s say that I have an object, and this circle is the object’s path. So let’s say this is the object. And it’s going around in a circle in a counter clockwise direction, is not quickly clockwise just going around this way. And let’s say I wanted to figure out, or I wanted to quantify, how much or how fast this thing is spinning. So one thing that you’ve probably familiar with is you know revolutions per second or rotations per second.

So let’s write that down, let’s just say for the sake of argument that this was moving at, I don’t know, one revolution per second so, after one second it goes back and another second, so that’s how fast it’s spinning, one revolution per second. One revolution, I’ll put this in rev per second. So let’s see if we can quantify that in angles and we’ll do it in radiance but you could always convert it back to degrees if you want.

So that be, I don’t know if you can see that line. But let’s just say that theta is the angle between the radius from the center to that object and the positive X axis, so this is theta. So if these object just travelling at one revolution per second, how many radiance per second is it travelling? Well, how many radiance are there in a revolution? Well, there’s two pie radiance in a revolution right? Well one, go around in circle is two pie radiance. So we could say times, so this equals one rev per second times two pie radiance per rev, right?

And then neither revolutions will cancel out, and you have one times two pies if two pie radiance per second. So that this equals, two pie radiance per second. That’s interesting I could, we now know exactly, you know after five seconds how many radiance has it gone or after half a second how many radiance is gone. Well that’s, that might be vaguely use. So let’s see if we can somehow convert from this notion of how fast something is spinning to its actual, speed.

I was tempted to say velocity but its velocity is always changing right? Because the direction is always changing. But the magnitude of the velocity stays the same so its speed is staying the same, but we’ll, let say V for speed, because that’s what they tend do in most formulas that you, you’ll see. So let’s think about it this way. In—in one revolution, so there’s couple of ways you can think about this, but as we go and—and as we go one revolution, how far has this object travelled? Well it’s traveled the circumference of this —of this circle. And in order to circumference, we have to know the radius of the circle so let’s say that the radius is R. Let’s say it’s in meters R meters.

So how many —how many— how many, meters will I travel in one second then? Well, you could —you could do the same thing up here, one revolution per second times two pie R, where R is the radius, oops to pie R, can we clear that line, meters per revolution right? That’s just a circumference of the thing of the circle and that equals the revolutions cancel out. Two pie R—two pie R meters per second. So it’s interesting we give in the radius and how many revolutions per second we can now figure out its velocity.

So this right here is its how fast its spinning and this is the objects actual speed, right? And this term of how fast something is spinning that’s called angular velocity and of course you know the term for how fast something is actually moving is velocity. So, and —and just so you know the term for angular velocity is, this curvy w, I think this is lower case omega but that little curve, that’s angular velocity. right? So in this case, angular velocity is equal to two pie radiance per second and what’s the velocity equal to or at least the magnitude of velocity? I know the directions are always changing. Well we know that the velocity is equal to two pie R meters per second.

So if we just ignore the units for second, where do you see the difference between the angular velocity and velocity? The angular velocity in this case is two pie and the velocity is two pie R. So in general if you just multiply the angular velocity times R you get the velocity. So angular velocity times the radius is equal to the velocity or you can divide both sides of that by R and you’ll get the angular velocity is equal to the velocity divided by the radius.

And this is a formula that you should know by heart, although it’s good to kind of know where it came from. An another way I guess even a, I did this way to maybe give you an intuition because I was have to work with numbers and especially when I’m new to the concepts so that’s why I said one revolution per second instead of just putting everything as a variable. But another way too really, to think about it is, what is the definition of a radiant?

By definition of radiant, if this is, if this angle is X radiance,—is X radiance, it’s an angle and its also tells us that the arc that is, that is kind of projective by this angle is equal to X radiuses, —radiuses, so if we, if each radius is two meters, it would be X times two meters so if this is X radiance and this is going to be X times R meters. And that actually comes from the definition —the definition of the radiant, and that might— that might be more intuitive to you than the original equation or less. So hopefully one of those two works, but as you can see if this angle is X, and this is, and this is just a X times R, and if omega is changed in that angle over changed in time, that’s omega, that’s just omega, like this omega. Then we know this is true too, that velocity is just a changed in this over changed in time, right, velocity is changed in, the radius doesn’t change, changed in X times R divide by changed in time.

And we know once again that is omega. So kind of another way, we just show it again that omega times the radius is equal to the velocity. The angular velocity times the radius is equal to the velocity. And this is a useful thing to learn we’ll see in the couple things would you know, when I do the proof for centripetal acceleration and calculus. I’m going to use this fact and when I, and I’m actually proud to be recorded video now, I’m actually going to show you that the law of conservation of angular momentum which is very similar to the law of conservation of momentum but it deals with things spinning and this is going to, this notion of angular velocity is going to come in useful.

So this is the important take way, that w equals v over Rand hopefully my video has not confuse you and has shown you that w, the radiant which the angle is changing is equal to the velocity of the object or the magnitude of velocity divided by the radius around, of the circle that’s spinning. I’ll see you in the next video.