Problem

$\bullet$ A thin, light string is wrapped around …

05:38

Problem 13 Medium Difficulty

\bullet A $22,500$ N elevator is to be accelerated upward by connecting it to a counterweight using a light (but strong!) cable passing over a solid uniform disk-shaped pulley. There is no appreciable friction at the axle of the pulley, but its mass is 875 $\mathrm{kg}$ and it is 1.50 $\mathrm{m}$ in diameter. (a) How heavy should the counterweight be so that it will accelerate the elevator upward through 6.75 $\mathrm{m}$ in the first 3.00 $\mathrm{s}$ , starting from rest? (b) Under these conditions, what is the tension in the cable on each side of the pulley?

Answer

a) $3.14 \times 10^{4} \mathrm{N}$
b) $2.66 \times 10^{4} \mathrm{N}$
$2.60 \times 10^{4} \mathrm{N}$

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Video Transcript

blue in problem number 13. We're looking at an elevator that will be accelerated upward by connecting it to a counterweight over a uniform cylindrical pulley. So let's take a look at the numbers we have right now. We know that this, um this is the elevator, and it is got a 22 1005 100 new weight. And we don't know the size of our counterweight or the holy has a diameter 1.5, which gives us then a radius here of 0.75 We know the mass of that pulley. You 875 kilograms, right? I ask. This is what is the size of this counter way? We also need to know, um, what's the tension in both sides on the robe? Um, besides, now we got us a little help here and that they tell us that the elevator will be accelerating. Award a distance of 6.7 to 5. That the distance that this elevator will be moving is 6.75 meters. When it does, um, in three seconds. We know also that we now that it's the velocity initial is zero. That helps us a little bit there and gives us a few things to help us. First, we can use this information to find our acceleration of our elevator. Do this of you. The page here. So we know using our kin O Matics formulas that we can find the acceleration we were given this displacement that this elevator is 6.75 we now it's started from rest, and that occurred in three seconds. Something for a get an acceleration of one point I meters per second squared. So that helps us out. Remember, it's going up, which means this counterweight would be coming down. Now we need to look it kind of the three separate things going on here. Courage. Look at the elevator itself. From a free body standpoint, we know. And as a downward weight of 22,500 Newtons, whereas it up for attention and I'm gonna call this tension The left tension. They are looking at the counter way. Similar free body. You just don't know what that way it is. But we know this is gonna call us the right attention. No, we do have some torque on our pulley system here that we will want to look at. We know that there is a torque that's gonna rotate this pulley clock Waas on. That is due to the tension in the right cable. Then we also have the tension in the left cable that's going to do a counter clock Waas roads ish calculate here is we will need to know our moment of inertia. Moment of inertia for a cylinder. Our moment of inertia of the cylinder is 1/2 huh? But how m R squared? In this case, it's 1/2 Our mass is 8 75 times Intel Limiter square. This actually is 246 kilogram meter square. Thanks. So how we're gonna look at the's three different diagrams, dignities loss statements for each one of them. Let's go ahead and start with our right side. So, looking at this, we know the net force is on this are equal to mass times acceleration. So we feel here for ah, that force is in the wind direction ends. This would show us that our downward gravity Linus, our tension in the right cable is the unknown mass times acceleration. We do know this ingenious. We don't know this So we got a bunch of unknowns here, but I'm sure we're going to get some help Looking at some of our other that takes care of this right side. It's so that are left side. Oops. Are left sign. We know a little bit war. We know that this time our attention in the left side is gonna be greater than the weight of the elevator on the way to the elevators. 22. My 100 in the mass of that elevator. Is this 22? I'm 100. It's about it by ah, acceleration due to gravity then. So here we have a little bit more since we know our acceleration already, actually Consol for the tension in the left side. Your call Our acceleration. We saw four words 1.5. It does actually give us the tension in the left side to be 26,000 Britain's. This is actually act they're asking for. I think part. See, guys taking it back here in looking at our talk situation here we know our networks our moment of inertia. Times are angular acceleration. You remember. We know relationship between angular acceleration in our tea and gentle acceleration. Our angle in our 10 general acceleration we already know is 1.5. Our radius is 0.75 which allows us to solve for angular acceleration of two radiance she could square sort. Your talks are provided because of the rights tension in the left tension. Now the positive attention, the one that is going to be on the right side because we know it's moving this jingle and then we're going to subtract are left side tension or talk. So we have tension on the right side times. It's LeBron minus tension on the left side times it's weather arm, and that's gonna be our moment of inertia. Times are angular acceleration. We don't know tension on the right side. What would you know? I love her arm is the radius of the pulley. We just finished solving for attention. Once she's 6000 Newtons, also same lever arm and we sold for our moment of inertia of the cylinder to be 246 times. An angular acceleration of two little algebra will help us with finding our 26,600 new intention and the rights. Now we can plug that in here you know, we know our acceleration to be 1.5 and using a little bit of algebra we can solve. I'm sorry. Down into a war, It's not any It is him. So combining like terms, we console for our mass and it turns out the masses 31 80 kilograms. We did technically ask us for the wades of the counterweight. So when you default by that masked by 9.8, you know, we get 3.16 times 10 to the fourth. It's Thank you. Sure, learning with me today.

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