1

concept

## Equations of Rotational Motion

7m

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Hey, guys. So when we're 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. Well, in rotational motion, it's exactly the same thing. Except they're gonna take different letters. Let's check it out. So as it says, you're just like linear motion. Therefore, equivalent motion equations for rotation. It's the same exact thing. They just have funny looking letters, All right. So as it says here, you often use these. When you're 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 thing. Equations just look a little bit different. So we're gonna rewrite these equations real quick. Um, instead of v, I'm gonna have W or Omega. So it's the same thing. Will make a final equals Omega initial instead of a I'm gonna write Alfa t. Same thing here. Omega Final squared, Omega Initial squared plus two Alfa Delta Fada. Hi. And then Delta fatum equals omega initial T plus half of Alfa T squared, and then this one is Delta Theta equals half or mayor Initial plus omega final times. T you can think of this as translating from linear to rotational. Same exact stuff. The letters just look different. Um, different variables. So I have a start here. NASA risk, because, remember, Same here. Um, in some cases, your professor may only give you these three equations and want you to stay three of them. That says the extra fourth equation. You should know by now whether your professors cool with you, using it or not. Remember also that when you're solving motion problems, you need to know you need two no. Three out of five variables. Remember that one variable will be your target, and one variable will be your ignored variable. And this is the one that will determined the equation to use equation to use. Okay, this is very straightforward. Let's do some examples. Right. So here, a wheel, initially at rest so 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 so you can think of it as a big disk. Something like this, right. Imagine that's disk. And it has a central axis, meaning, like some sort of stick, and they can spin around it like that. Okay, so it starts from rest. So the initial omega is zero, and it's going to accelerate with a constant four radiance per second is acceleration. So Alfa equals four until it reaches 80 radiance per second square. You can think of this as meters per second, but in rotation. Okay, so that is your final velocity. It's not actually meters per second. You could just think of it that way. Omega final equals 80 ratings per second. All the units you are correct. Eso as I mentioned, you can tell that you're supposed to use this because you start getting a lot of rotational quantity, right? In this case, I already know three of them. So I I know that I can already solve whatever I'm going to. I'm about to be asked. Okay, Cool. So it says by the time it reaches 80 how many degrees will it's have rotated through how many degrees it's gonna have rotated through? It's asking for Delta theta but it wants the answer in degrees, which means I'm going to get it in radiance because the equations always spit out Delta failing radiance. Then you have to convert two degrees. Cool. So I'm gonna do what I always do, which is list my five variables here. Delta Theta is what we're looking for. And the variable out of the five that didn't get mentioned was Delta T. So I'm gonna put a little sad face here, and I'm going to pick the Onley equation out of the four that is missing a delta T, which is this one. There's no delta t on this one. Okay, so same thing is before will make a final when you're sure that the squares to Alfa Delta theta Delta things what I'm looking for, I'm gonna move everything out of the way. So Delta Theta target varies by variables by itself, will make a final squared minus will make initial squared. This stuff comes to the other side dividing. Um, Now we're ready to plug in some numbers and set it up like this to now. We're ready to stick the numbers inside of the parentheses kind of lawsuit was 80 the initial zero, and the acceleration is four. So if you do all of this, you end up with 800 radiance. Remember, these equations always spit out radiance, and then we're gonna convert. So I'm gonna do pi ratings at the bottom and then 180 degrees of top when it canceled, ratings with radiance were left with degrees. So 800 times 1 80 is 1008 100 degrees. That's a crap load of degrees. Spends a whole bunch for part B Part B is asking how long in seconds does it take? In other words, what is our Delta T Delta? T was originally my ignored variable, but now we're looking for Delta T we can use since it's the same situation I can use Delta Fada. So I actually have I know four out of five variables. I only needed three, but I know four. And when I know more than what I need, um, it means that I'm gonna have more flexibility with the equations instead of having having to use one specific equation. I can use any equations that have Delta T which in this case there's three of them Okay, so the simplest equation to use would be the first one. So I'm gonna use that one. All right? We're looking for teeth. Let me circle it. So if I move everything out of the way so that teased by itself it looks like this and t equals, let's plug it in. The final is 80 initial zero acceleration. Alfa is four. So the answer is 20 seconds. Alright, that's it. Very straightforward. Just like it was before. You just have Thio basically make the adjustment for the letters on. Do you see different units? And it's gonna say things like Central axis and rotation eso. It's the same thing just in the rotational world. Alright, well, that's it for this one. Let's keep going.

2

example

## Rotational velocity of disc

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All right. So here we have a heavy disk or a very heavy disc. Uh, the word, very obviously doesn't do anything because it's not a number. A very heavy disc, 20 m in radius. So, disc, I'm gonna draw it like that. Um, radius of 20 m takes one hour to complete to make a complete revolution. The time to make a complete revolution is called period, and it's big T so t is one hour, which is 60 times 60 seconds or seconds room. Remember, we always convert to the standard units, which, in this case, seconds on git says accelerating from rest at a constant rates. Okay, so presumably the disk is rotating around itself because it doesn't say otherwise. Eso It starts with zero. It accelerates at a constant rate somewhere. Right? Alfa equals constant, but it doesn't tell us what it is, so we don't know. And we want to know what rotation of the lawsuit will the disk have? One hour after starts accelerating. Okay, So after one hour, Or in other words, after 3600 seconds, um, what rotational velocity will the disk have? Okay, so I'm gonna do my little bracket here with my motion variables. Remember, motion variables are the initial the final acceleration Delta t on the displacement, which in this case, is Delta Theta. Okay, so I'm missing Omega initial. I'm missing Omega Fine over here. Okay. And that's what we wanna know. What's my final angular velocity? Um, t isn't really one of the five variables, so I put it outside. Okay, remember, we're supposed to know three of these things. Um, we know this and this and we've got a target. There's two variables here that I don't know, but to solve this problem, I'm supposed to know three. So you have to figure out which one you do know here, All right. And 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, if it's been a full hour, which is how long it takes to make a full revolution, then my delta theta is Let's see if you can figure this out. What would your delta theta beat if it takes an hour to make a full spin and you wanna know how long? And if you want to know your delta theta after that one hour, this would be to pie, right? Because it's been an hour, an hours long it takes to make a full revolution. So Delta Thing is to pot Notice how this wasn't explicitly given to you. It was given to you in a tricky way. All right, so now we know three things, and I can solve this. Alfa here is my unknown. My my ignored Very okay. Therefore, I could go straight into the fourth equation. The fourth equation would work here. Now, just in case you have a professor who doesn't actually 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 gonna be really easy, so we're gonna have to do is instead of using the fourth equation or use two equations. Why? Because you're gonna have to find Alfa first, Okay. And then you're gonna have to find Omega final. All right, so if we're looking for Alfa. If I'm looking for Alfa first, that means that might ignored variable while I'm looking for Alfa is will make a final write. It flips. I was looking for this variable. This one was ignored. Well, actually gonna find this first. So this is the ignored. Okay, So which equation doesn't have to make a final? The third equation doesn't have to make a final. So I'm gonna go with equation number three and it's gonna be Delta theta equals Omega initial T plus half of 80 square Alfa T Square. Okay? And we're looking for Alfa the initial velocity zero. So this is gone, and I'm gonna move everything out of the way. So, too, comes up the theater and the tea comes back down over here. How far to delta fate is Two pi, and the time is 3600 squared. And if you do this, um, I have it here. You get a very small number 9.7 times 10 to the negative seven on the reason why the acceleration so slow is because it took a now, er for this thing to complete a full circle. Okay, so that's the acceleration. Once I know the acceleration I'm now looking for. I was first looking for acceleration. We're now looking for W final. Okay, I have four out of five variables, which means I'm gonna be able to use have more flexibility. I'm gonna be able to use any equation. Um, that has w final in it. I can use the first equation. W final w Nissho plus off T wh zero. So this is just this tiny number 9.7 times 10 to the negative seven times time, which is 3600 seconds. And if you multiply all this, you get 3.5 times 10 to the negative. Three radiance per second. Okay? And that's it for this one. All right. Let me know if you have any questions.

3

Problem

A tiny object spins with 5 rad/s around a circular path of radius 10 m. The object then accelerates at 3 rad/s^{2}. Calculate its angular speed 8 s after starting to accelerate.

BONUS:Calculate its linear displacement in the 8 s.

A

19 rad/s

B

29 rad/s

C

43 rad/s

D

53 rad/s

4

Problem

The turntable of a DJ set is spinning at a constant rate just before it is turned off. If the turntable decelerates at 3 rad/s^{2} and goes through an additional 30 rotations before stopping, how fast (in RPM) was the turntable initially spinning?

BONUS:How long (in seconds) does the turntable take to stop?

A

33.6 RPM

B

321 RPM

C

2,016 RPM

D

3,167 RPM

Additional resources for Equations of Rotational Motion

PRACTICE PROBLEMS AND ACTIVITIES (6)

- A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has...
- An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in...
- An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in...
- A wheel is rotating about an axis that is in the z-direction. The angular velocity ω_z is -6.00 rad/s at t = 0...
- A wheel is rotating about an axis that is in the z-direction. The angular velocity ω_z is -6.00 rad/s at t = 0...
- An electric turntable 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of 0...