Hey guys, So, I've got a great example here for you. We're gonna work this one out together. So we have this engine that's operating and it's extracting some heats and all this stuff, but we're asked to find out how many cycles it's gonna take to lift a 10 kg mass. So I want to draw out what's going on here. So imagine that I have some kind of a box that's on the ground like this, It's 10 kg and we want to lift it up by, you know, five m. So this height here is five. Well, we remember back from energy conservation, that if you want to lift a mass by some heights, you have to do some work. So, for example, the work that's required to lift this mass is equal to MG. H. And so therefore it's just 10 times 9.8 Times the five m. So in other words, it takes 490 jewels In order to lift this mass by five m. So, what does this have to do with the heat engine? What's going on here is I have a carnot engine that's operating between these temperatures and these heats, and that's the thing that's actually supplying the energy to lift this mass. So, I'm gonna draw out my sort of heat diagram or my, my energy flow diagram. So I've got my hot reservoir that's connected to my engine and this is connected to my cold reservoir, It's T. C. So what's happening here is this engine? This is my car, no engine here is actually the thing that is supplying the work in order to lift this mass. So, you can imagine that this carnot engine, like, you know, it's just like a box with a pulley or something, maybe it's like a little pulley and basically it's sort of cranking this this box To lift this mass by five m. So that's basically what's going on here. The thing that's supplying the work here is my car, no engine. And so remember, in a carnot cycle or in any heat engine cycle, you have a heat that flows in, you have some work that's done by the engine and then you have some heat that flows out to the hot the cold reservoir. So what happens here is we have to figure out how many cycles it takes for this engine. This work that's produced by the engine to lift this mass here. So, basically, what we're trying to find is the number of cycles times the work that's done in each cycle is going to equal the work that I need to lift this mass. So, basically, so this is going to be number of cycles, like this number of cycles times the work that I do per cycle Is going to equal the work that is required to lift this mass here. And ultimately, what I'm trying to find is what is in what is the number of cycles here. We know how much energy and how much work is required to lift, which is just the 490 that we calculated over here. So really all we have to do is figure out well how much work is produced by the engine each cycle. So basically that's really what we're trying to figure out in this part of the problem. So how do we calculate the work that's done by the engine? Well now we're just going to stick to our heat engine equations. We have a couple of them to deal with work and heats. So let's just try to use our sort of basic work equation. So W equals Q. H minus Q. C. So we do we have the Qh well and this probably we're told one of the heats these are the two temperatures 1 82 and zero. But we're told that it extracts 25 jewels of energy from the hot reservoir. So that's my cue. H this is gonna be 25 while I'm at it I'm just gonna fill in the rest of the variables here, my th this hot reservoir here is 1 82. So if you add to 73 to it, it's just going to be 455 Kelvin. We do the same thing, this is just gonna be T. C. Equals zero. So that's just gonna be 2 73 kelvin. Alright, so we don't have where we do have with the Q. H. Is unfortunately we don't have what Q. C. Is, what is the heat that gets discarded out to the cold reservoir. And we can't find it using this equation because we don't know W or Q. C. So it turns out we can't use this equation over here because we have multiple unknowns. And so we're gonna have to use a different equation. The only other equation that has efficient or the work in it is going to be the efficiency equation. So we can't use this. But it turns out we're gonna have to use this. So our efficiency equation, remember this is a Carnot engine is equal to the work done divided by the heat that's taken in from the hot reservoir, right? It's how much you get out versus how much you paid into this heat engine. However, because this is a Carnot engine, we also have one more equation. We have that. This is one minus TC over th if you look through the variables here, what's going on by the way? This w here is the work done by the engine. That's not to confuse you by w lift. Those are not the same thing. Okay, so what's going on here is that if you look through your variables, we have Q. H, we have T. C. And th here. So what I can do is I can rearrange this equation, I can say the work that's done by the engine. Each cycle is equal to Q. H. Times the efficiency. And because I don't have what the efficiency is, but I can solve it by using this part of the equation. In other words, it's Q. H times one minus TC over th this equation here gives me the work that's done each cycle. So basically all I have to do is just plug some numbers in. So I've got the work done by the engine is in equal to Q. H. Which is 25 times one minus TC over th T. C. here is to 73 And th is 455. So when you work this out, what you're gonna get here is that the work done? Each cycle is 10 jewels. So if you come up here and this kind of makes sense here because you have 25 jewels and you have a work that's done, it's 10, it should be a little bit less than the heat done uh than the heat transferred from the hot reservoir. So what happens here is now that we've figured out the work that's done by the engine. Each cycle, we can now just plug this into our last equation over here, that's the 10 jewels. So all we have to do is we just do n equals the work that's required to lift this mass here divided by the work that's done by the engine. In other words, it's 490 jewels, but each each, each cycle the engine does 10 jewels of work. So in other words, what you're going to need here is you're gonna need 49 cycles of this heat engine running in order to lift this mass by five by five m. Alright, so hopefully all that stuff made sense. Let me know if you guys have any questions in the comments. All right, that's it for this one.
