 ## Physics

Learn the toughest concepts covered in Physics with step-by-step video tutorials and practice problems by world-class tutors

12. Rotational Kinematics

# Equations of Rotational Motion

1
concept

## Equations of Rotational Motion 7m
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2
example

## Rotational velocity of disc 5m
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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/s2. Calculate its angular speed 8 s after starting to accelerate.

BONUS:Calculate its linear displacement in the 8 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/s2 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? 