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# Equations of Rotational Motion

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Sections
Rotational Position & Displacement
Rotational Velocity & Acceleration
Types of Acceleration in Rotation
Rolling Motion (Free Wheels)
Equations of Rotational Motion
More Connect Wheels (Bicycles)
Intro to Connected Wheels
Converting Between Linear & Rotational

Concept #1: Equations of Rotational Motion

Transcript

Example #1: Rotational velocity of disc

Transcript

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.

Practice: 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.

Practice: 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?