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Anderson Video - Rotational Kinematic Equations

Professor Anderson
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Hello class. Professor Anderson here. Let's talk about rotational motion. We're a little bit familiar with this idea of rotational motion already, but let's formalize it a little bit more. And let's convince you that it's maybe not quite as bad as it seems. Okay, when we talk about rotational motion, what do we really mean? Well, if we draw an xy coordinate system then rotational motion is just an object moving around in a circle like so. And we can define a few things to characterize that circle; the radius, r, of the circle; the angle, theta, between the line to the object and the x axis. We can also talk about the angular speed of the object and we can also talk about alpha; the angular acceleration. These are the important variables; theta, omega, and alpha. And in fact the equations that describe rotational motion are nearly identical to the ones that describe linear motion. We're just changing the variables. So for instance when we are thinking about linear motion we said; we're going to have x and we're going to have v. And we know that v is dx/dt. And we're going to have a. We know that a is dv/dt. And now in rotational we're going to do the same stuff, just changing the variables. So instead of an x we're going to use theta. What's omega? Omega is d theta/dt. What is alpha? Alpha is d omega/dt. Omega is called the angular speed. Alpha is called the angular acceleration. Okay? Theta is of course the angle. And typically we use theta in radians. Now, if we're making these substitutions we also realize that we have a whole set of equations -- the kinematic equations -- which govern motion, for instance this one; x final equals x initial plus vx initial times t plus one half a sub x t squared. That's one of those equations. And we can write the corresponding rotational equation like this; theta f equals theta i plus omega initial times t plus one half alpha t squared. All the math is exactly the same; we're just changing the variables. Now, there is a caveat with the kinematic equations. The kinematic equations apply when you have constant acceleration; a. If your acceleration is not constant, then you cannot use the kinematic equations. And so we have to be careful when we write these equations down over here, the caveat is constant, angular acceleration alpha. But if you have constant alpha you can use all of those kinematic equations that we developed before. Okay? Now, we also learned a little bit about some other terms in linear motion, like work. Let's see how that applies to rotational motion. Work -- w -- we said was the integral of f sub xdx where we're going to integrate from some initial position to some final position. We wrote it with a dot product before, but this is the same information here. f is the x component of the force, okay? Which is what the dots -- the dot product does, it picks out the x component. In rotational motion now, we're going to integrate from theta i to theta f. We're not dealing with a force anymore. We are in fact dealing with a new term -- torque -- and we have to integrate over angle. This thing -- Tao -- is torque. Okay? And torque is to rotational motion what force is to linear motion. So if I take a box and I put a force on it it will accelerate. If I take a wheel and I put a torque on it it will angularly accelerate. Okay? We also know that when things are moving they have kinetic energy. And the kinetic energy that we write with a capital K is just one half mv squared. If things are rotating they also have energy and we call that rotational kinetic energy; one half i omega squared. And now we've introduced yet another term; i, which is the moment of inertia. So in the linear case, objects have mass and if they're moving at speed v then they have kinetic energy. In the rotational case, objects have moments of inertia and if they are rotating at omega then they have rotational kinetic energy. Okay? And so you can actually store a lot of energy in rotating things, right? You have a big flywheel in a bus, when you come to a stop, instead of applying brakes, which is really just friction, right? Two brake pads hitting a piece of metal and it heats up the metal, okay? And you lose all that energy as heat. You can take a different approach, which is; let's spin up a flywheel and now you have a big rotating disk inside the bus and then when you start the bus up again you can tap into that energy and suck it back out. Okay? So they actually do use these giant rotating flywheels in busses and trains. Okay? Not in smaller cars because they're typically big, heavy wheels, but it's a way to transfer energy between systems. We also talked a little bit about power. And one of the things we said about power was; power, in the linear case, is equal to force times velocity. Okay? You apply a certain force to get your car moving and v, the velocity, is how fast you're going of course. So the power use goes like f times v. In the rotational case now, we don't use f anymore, we use torque. And we don't use v anymore, we use angular velocity omega. All right? So that's how much power is in a rotational system. So the whole point here is; everything that we've learned so far in linear kinematics applies to rotational kinematics if you just change variables. And so this is the trick is sort of convincing yourself that there's nothing that complicated here, I just have to start using some different letters. All right? And these -- in rotational -- are apparently usually Greek; Tao and omega, alpha, theta. And these over here are typically English letters.