Alright. So 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'm going to draw it like that. Meters takes 1 hour to complete, to make a complete revolution. The time to make a complete revolution is called the period, and it's big 'T'. So 'T' is 1 hour, which is 60 times 60 seconds or 3600 seconds. Remember, we always convert to the standard units, which in this case, seconds. And it says accelerating from rest at a constant rate. Okay. So presumably, the disc is rotating around itself because it doesn't say otherwise. So it starts with 0. It accelerates at a constant rate. So I'm gonna write α = constant, but it doesn't tell us what it is. So we don't know. And we want to know what rotational velocity will the disc have 1 hour after it starts accelerating. Okay? So after 1 hour or in other words, after 3600 seconds, what rotational velocity will the disc have? Okay. So I'm going to do my little bracket here with my motion variables. Remember, motion variables are the initial v, final acceleration delta 't', and the displacement, which in this case is delta θ. Okay. So I'm missing ωinitial, I'm missing ωfinal over here. Okay. And that's what we want to know. What's my final angular velocity? 'T' isn't really one of the 5 variables, so I put it outside. Okay? Remember, we're supposed to know 3 of these things. We know this and this and we got a target. There are 2 variables here that I don't know. But to solve this problem, I'm supposed to know 3. So you have to figure out which one you do know here. Alright? 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 wanna know the velocity after that same amount of time, Well, if it's been a whole hour, if it's been a full hour, which is how long it takes to make a full revolution, then my delta θ is let's see if you can figure this out. What would your delta θ be if it takes an hour to make a full spin and you wanna know your delta θ after that 1 hour, this would be 2π. Right? Because it's been an hour. An hour is how long it takes to make a full revolution, so delta θ is 2π. Notice how this wasn't explicitly given to you. It was given to you in a tricky way. Alright? So now, we know 3 things and I can solve. This α here is my unknown, my ignored variable. Okay? Therefore, I could go straight into the 4th equation. The 4th equation would work here. Now, just in case you're a professor who doesn't let you do it with the 4th equation, I'm going to show you how to do it without using the 4th equation. But again, if you could, just plug it in, and it's going to be really easy. So what we're going to have to do is instead of using the 4th equation or use 2 equations. Why? Because you're going to have to find α first. K. And then you're going to have to find ωfinal. Alright. So if we're looking for α, if I'm looking for α first, that means that my ignored variable while I'm looking for α is ωfinal, right? It flips. I was looking for this variable. This one is ignored. Well, actually, I got to find this first so this is ignored. K? So which equation doesn't have ωfinal? The third equation doesn't have ωfinal. So I'm gonna go with equation number 3, and it's going to be delta θ equals ωinitial * t + half of ατ2. K? And we're looking for α. The initial velocity is 0, so this is gone. And I'm gonna move everything out of the way. So 2 comes up, delta θ, and the 't' comes back down over here, α. Delta θ is 2π and the time is 3600 squared. And if you do this, I have it here, you get a very small number, 9.7 × 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. 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 ωfinal. Okay? I have 4 out of 5 variables, which means I'm gonna be able to use any equation that has ωfinal in it. I can use the first equation, ωfinal = ωinitial + ατ, ωinitial = 0. So this is just this tiny number, 9.7 × 10-7 times time, which is 3600 seconds. And if you multiply all this, you get 3.5 × 10-3 radians per second. Okay. And that's it for this one. Alright, let me know if you guys have any questions.
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12. Rotational Kinematics
Equations of Rotational Motion
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