Equations of Rotational Motion

Hey guys! When weÕre doing linear motion, you may remember that you had a set of four equations that you would use to solve a whole bunch of different types of problems. In rotational motion, it's exactly the same thing except they're going to take different letters. Let's check it out. As it says here, just like in linear motion, there are four equivalent motion equations for rotation. It's the same exact thing. They just have funny looking letters. As it says here, you often use these when youÕre given a lot of rotational quantities. It's usually a word problem and it starts throwing out things like the velocity, the acceleration and you would use these equations. The process is the same exact one. The equations just look a little bit different. We're going to rewrite these equations real quick. Instead of V IÕm going to have w or Omega, so it's the same thing. w final = w initial. Instead of a, IÕm going to write _t. Same thing here, w final^2 = w initial^2 + 2 ___. Then __ = w initial t + _ _t^2. Then this one is __ = _ (w initial + w final)*t. You can think of this as translating from linear to rotational, same exact stuff, the letters just look different, different variables. I have a star here, an asterisk, because remember same here in some cases your professor may only give you these three equations and want you to stick to three of them. This is the extra 4th equation. You should know by now whether your professor is cool with you using it or not. Remember also that when you're solving motion problems, you need to know three out of five variables. Remember that one variable will be your target and one variable will be your ignored variable. This is the one that will determine the equation to use. This is very straightforward. Let's do some examples. Here a wheel initially at rest. Initially at rest used to be that the initial velocity is zero. It still means that but now its initial angular velocity because this wheel is going to rotate around its central axis. You can think of it as a big disc, something like this. Imagine that's a disc and it has a central axis, meaning like some sort of stick and they can spin around it like that. It starts from rest so the initial w is 0 and it's going to accelerate with the constant 4. Radians per second is acceleration so _ = 4, until it reaches 80 radians per second squared. You can think of this as meters per second, but in rotation. That is your final velocity. It's not actually m/s. You can just think of it that way. w final = 80 radians per second. All the units here are correct. As I mentioned, you can tell you're supposed to use this because you start getting a lot of rotational quantity. In this case, I already know three of them so I know that I can already solve whatever IÕm about to be asked. It says by the time it reaches 80, how many degrees will it have rotated through? How many degrees it's going to have rotated through? ItÕs asking for __, but it wants the answer in degrees which means I'm going to get it in radians because the equations always spit out __ in radians and then you have to convert to degrees. I'm going to do what I always do which is list my five variables here. __ is what we're looking for. The variable out of the five that didn't get mentioned was _t. I'm going to put a little sad face here and IÕm going to pick the only equation out of the four that is missing a _t which is this one. ThereÕs no _t on this one. Same thing as before, w final = w initial and the squares, the squares, ___. __ is what I'm looking for. I'm going to move everything out of the way so __, the target variable, is by itself. wfinal^2 Ð winitial^2, this stuff comes to the other side dividing it. Now we're ready to plug in some numbers. IÕm going to set it up like this, now we're ready to stick the numbers inside of the parentheses. Final velocity was 80, the initial was zero and the acceleration is 4. If you do all of this, you end up with 800 radians. Remember, these equations always spit out radians. Then we're going to convert so I'm going to do ¹ radians at the bottom and then 180 degrees up top. Let me cancel radians with radians. We're left with degrees. 800*180 is 45800 degrees and that's a crap load of degrees. It spins a whole bunch. For Part B, Part B is asking how long in seconds does it take. In other words what is our _t. _t was originally my ignored variable but now we're looking for _t. We can use, since it's the same situation, I can use __. I actually have I know four out of five variables. I only needed three but I know four. When I know more than what I need, it means that I'm going to have more flexibility with the equations. Instead of having to use one specific equation, I can use any equations that have _t which in this case there's three of them. The simplest equation to use would be the first one so I'm going to use that one. We're looking for t, let me circle it. If I move everything out of the way so that t is by itself, it looks like this. t equals, let's plug it in. The final is 80, initial is zero. The acceleration _ is 4, so the answer is 20 seconds. That's it. Very straightforward just like it was before. You just have to basically make the adjustment for the letters and you see different units. It's going to say things like central axis and rotation. It's the same thing just in the rotational world. That's it for this one. Let's keep going.

Here we have a heavy disc or a very heavy disc. The word very obviously doesn't do anything because it's not a number. A very heavy disc 20 meters in radius, so a disc I wanna draw it like that, radius of 20 meters takes 1 hour to make a complete revolution. The time to make a complete revolution is called period and it's big T. T is 1 hour which is 60*60 seconds or 3600 seconds. Remember, we always convert to the standard units which in this case seconds. It says accelerating from rest at a constant rate. Presumably the disc is rotating around itself because it doesn't say otherwise. It starts with zero, it accelerates at a constant rate. So IÕm gonna write _ = constant but it doesn't tell us what it is so we don't know. We want to know what rotational velocity will the disc have 1 hour after it starts accelerating. After 1 hour or in other words after 3600 seconds, what rotational velocity will be disc have? I'm going to do my little bracket here with my motion variables. Remember, motion variables are the V initial, V final, acceleration, _t and the displacement which in this case is __. I'm missing w initial, I'm missing w final over here. That's what we want to know, what's my final angular velocity. T isn't really one of the five variables so I put it outside. Remember, we're supposed to know three of these things. We know this and this and we got a target. There's two variables here that I don't know. But to solve this problem, I'm supposed to know three. You have to figure out which one you do know here. The idea for this question is that you're supposed to figure out that if the period is 3600 seconds or an hour and I want to know the velocity after that same amount of time, well if it's been a full hour which is how long it takes to make a full revolution, then my __ isÉ Let's see if you can figure this out. What would your delta theta be if it takes an hour to make a full spin and you want to know your __ after that one hour? This would be 2¹ because it's been an hour. An hour is how long it takes to make a full revolution so __ is 2¹. Notice how this wasn't explicitly given to you. It was given to you in a tricky way. Now we know three things and I can solve. This _ here is my ignored variable. Therefore I could go straight into the fourth equation. The fourth equation would work here. Just in case your professor doesn't let you do it with the fourth equation, I'm going to show you how to do it without using the fourth equation. But again, if you could just plug it in and it's going to be really easy. What we're going to have to do is instead of using the fourth equation or use two equations. Why? Because you're going to have to find _ first, and then you're going to have to find w final. If I'm looking for _ first, that means that my ignored variable, while I'm looking for _ is w final. It flips. I was looking for this variable. This one is the ignored. Actually I got to find this first, so this is the ignored. Which equation doesn't have w final? The third equation doesn't have w final. I'm going to go with the equation number 3 and it's going to be __ = w initial t + _ _t^2. We're looking for _. The initial velocity is zero so this is gone and I'm going to move everything out of the way. 2 comes up, __ and the t comes back down over here, _. 2, __ is 2¹ and the time is 3600^2. If you do this, I have it here, you get a very small number. 9.7 x 10^-7 and the reason why the acceleration is so slow is because it took an hour for this thing to complete a full circle. That's the acceleration. Once I know the acceleration, we're now looking for w final. I have four out of five variables which means I'm going to be able to use more flexibility. I'm going to be able to use any equation that has w final in it. I can use the first equation, w final = w initial + _t. w initial is zero. This is just this tiny number, 9.7x10^-7 times time which is 3600 seconds. If you multiply all this, you get 3.5x10^-3 radians per second. That's it for this one. Let me know if you got any questions.