Well, we've done a lot of work with how fast something moves, let's see if we can work with how fast something spins. Let's see what we can do. Since we're going to be working with things spinning, let me draw a circle. Since things that spin go in circles. And let me just draw the positive x-axis because it'll come in handy in a second. That's the positive x-axis. And let's say that I have an object, and the circle is the object's path. So let's say this is the object. And it's going around in a circle in a counterclockwise direction. Not squiggly counterclockwise, it's just going around this way. 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're probably familiar with is 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, 1 revolution per second. So after 1 second it goes back, then another second. So that's how fast it's spinning, 1 revolution per second. 1 revolution-- I'll just put rev-- per second. So let's see if we can quantify that in angles, and we'll do it in radians, but you could always convert it back to degrees, if you want. I don't know if you can see that line. Let's just say that theta is the angle between the radius from the center to that object, and the positive x-axis. So if this object is travelling at 1 revolution per second, how many radians per second is it traveling? Well, how many radians are there in a revolution? Well there's 2 pi radians in a revolution, right? 1 go-around in a circle is 2 pi radians. So we could say, so this equals 1 rev per second, times 2 pi radians per rev, right? And then the revolutions will cancel out. And you have 1 times 2 pi, so you have 2 pi radians per second. So this equals 2 pi radians per second. So that's interesting, we now know exactly after 5 seconds how many radians it has gone. Or after half a second, how many radians has it gone. But that might be vaguely useful. 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, because the direction is always changing. But the magnitude of the velocity is staying the same, so its speed is staying the same. But we'll say v for speed, because that's what they tend to do in most formulas that you'll see. So let's think about it this way, in 1 revolution-- so there's a couple ways you can think about this, but as we go 1 revolution, how far has this object traveled? Well, it's traveled the circumference of this circle. And in order to know the 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 meters will I travel in 1 second, then? Well, you could do the same thing up here. 1 revolution per second, times 2 pi r, where r is the radius-- whoops, 2 pi r, you can ignore that line-- meters per revolution, that's just the circumference of the thing, of the circle. And that equals-- the revolutions cancel out-- 2 pi r meters per second. So it's interesting, given the radius and how many revolutions per second, we can now figure out its velocity. So this right here is how fast it's spinning, and this is the object's actual speed, right? And this term of how fast something's spinning, that's called angular velocity. And of course you know that the term for how fast something is actually moving is velocity. And just so you know, the term for angular velocity is this curvy w, I think that's lower case omega, that's angular velocity. So in this case, angular velocity is equal to 2 pi radians per second. And what's the velocity equal to, or at least the magnitude of the velocity-- I know the direction's always changing. Well, we know that the velocity is equal to 2 pi r meters per second. So if we just ignore the units for a second, where do you see the difference between the angular velocity and the velocity? The angular velocity in this case is 2 pi, and the velocity is 2 pi 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 velocity. Or you can divide both sides of that by r, and you 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 know where it came from. I guess I did it this way to maybe give you an intuition, because I always have to work with numbers. Especially when I'm new to a concept-- so that's why I said 1 revolution per second, instead of just putting everything as a variable-- but another way to think about it is, what is the definition of a radian? By definition, a radian-- if this angle is x radians, it's an angle, and it also tells us that the arc that is kind of projected by this angle, is equal to x radiuses. So if each radius is 2 meters, it would be x times 2 meters. So if this is x radians, then this is going to be x times r meters. And that actually comes from the definition of the radian. And that might be more intuitive to you, than the original explanation, or less, so hopefully one of those two works. But as you can see, if this angle is x, and this distance is x times r, and if omega is change in that angle, over change in time. Then we know this is true too, that velocity is just change in this, over change in time, right. Velocity is change in-- the radius doesn't change-- change in x times r, divided by change in time. And we know once again that this is omega. So another way we just showed again, that omega times the radius is equal to the velocity. Or the angular velocity times the radius is equal to the velocity. And this is a useful thing to learn, we'll see it in a couple of things, when I do the proof for centripetal acceleration in calculus, I'm going to use this fact. And when I-- and actually I'm probably going to record that video now-- I'm actually going to show you 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 notion of angular velocity is going to come in useful. So this is the important takeaway, that w equals v over r. And hopefully my video has not confused you, and has shown you that w, the rate at which the angle is changing, is equal to the velocity of the object, or the magnitude of the velocity, divided by the radius of the circle that it's spinning. I'll see you in the next video.
Video Description: Angular Velocity and its relation with Linear Velocity for JEE 2023 is part of Physics For JEE preparation.
The notes and questions for Angular Velocity and its relation with Linear Velocity have been prepared according to the JEE exam syllabus.
Information about Angular Velocity and its relation with Linear Velocity covers all important topics for JEE 2023 Exam.
Find important definitions, questions, notes, meanings, examples, exercises and tests below for Angular Velocity and its relation with Linear Velocity.
Introduction of Angular Velocity and its relation with Linear Velocity in English is available as part of our Physics For JEE for JEE & Angular Velocity and its relation with Linear Velocity in
Hindi for Physics For JEE course. Download more important topics related with notes, lectures and mock test series for
JEE Exam by signing up for free.
Video Lecture & Questions for Angular Velocity and its relation with Linear Velocity Video Lecture | Study Physics For JEE - JEE | Best Video for JEE - JEE full syllabus preparation | Free video for JEE exam to prepare for Physics For JEE.
Information about Angular Velocity and its relation with Linear Velocity
Here you can find the meaning of Angular Velocity and its relation with Linear Velocity defined & explained in the simplest way possible.
Besides explaining types of Angular Velocity and its relation with Linear Velocity theory, EduRev gives you an ample number of questions to practice Angular Velocity and its relation with Linear Velocity tests,
examples and also practice JEE tests.