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Adding up resistance in series and in parallel
We had talked a little bit about the resistance equation that we got from Dr. Poiseuille. And the equation looked a little bit like this. Actually, let me just replace this. We had 8 times eta, which was the viscosity of blood times the length of a vessel divided by pi times the radius of that vessel to the fourth power. And all this put together gives us the resistance in a vessel. So thinking about this a little bit more, let's assume for the moment that the blood viscosity is not going to change. It certainly won't change from moment to moment, but let's say that, in general, blood viscosity is pretty constant. Now, given that, if I want to change the resistance, then I have two variables left. I've got the length of my vessel and I've got the radius. So if I have a vessel-- like this-- and let's say it's got a certain radius and length. And let's say that radius is r, and the length is here. And I apply a number. Let's say the number is 2 for the resistance. Well, I have two options for changing that resistance. If I want to increase the resistance, I can do two things. So let's say I want to increase that resistance. And you can look at the equation and tell me what the answer would be. Two things. And I'll actually draw it out. So one thing would be to keep the radius basically the same, but make it much longer. Because if I make it longer, since the L is now, let's say, twice as long and r is the same, now my resistance is going to double. So now we go 2 times. And 2 times 2 is 4. So my resistance is 4. OK. Option 2. Let's say I don't want to change the length. I keep the length the same. Instead, I could actually maybe change the radius. And let's say I half the radius. I make it half of what it was. And I actually worked out the math in the last one. And it turned out that, if you half the radius-- in the last video, that is-- then the resistance actually is 16 times higher. And you can see that because the resistance equals r to the fourth power here. So because r is to the fourth power when you half it, it goes 16-fold. And so 16 times 2 is 32. So our resistance is 32. So these are the two strategies, if you think of it that way, that a blood vessel can use to increase resistance. And of the two, you can see that one of them is definitely more effective. I mean, I can see that because it's raised to the fourth power, this is going to work much more effectively to raise the resistance than changing the length. And additionally, if you think of it kind of from a practical standpoint, keep in mind that I have smooth muscle. So it's actually pretty easy to accomplish this-- or at least possible to accomplish this. Whereas trying to actually change the length-- which is option 1-- is not feasible. I mean, it's much, much more complicated to actually expect a vessel to simply double in its length because it wants to raise the resistance. So for multiple reasons, changing radius, again, becomes the name of the game. OK. So let's complicate this a little bit. Let's say instead of one vessel, I've got three vessels. I've got, let's say, one vessel here. And there's, let's say, 5. And then I've got, let's see, a longer vessel here. And this one happens to have a resistance of, let's say, 8 because it's longer. And let's do the same radius for all these, but shorter now. This one is 2. And I want blood to flow through all three of these. What is my total resistance? And here we're talking about the three vessels being in a series-- meaning that you actually expect the blood to go through all three of the vessels or tubes. So if they're going to go through all three tubes, what you have to do is simply add up the total. So resistance total-- so this is total resistance. And I just put a little t to remind me that that means total. So total resistance equals the resistance of one part plus the second part plus the third part. And if you had a fourth or fifth part, you just keep adding them up. So in this case, you have 5, 8, and 2. So Rt becomes 5 plus 8 plus 2, and that equals 15. So total resistance would be 15. And actually, I'm going to give you a general rule. Total resistance is always, always greater than any component. And you can see how this is very intuitive. I mean, how could you possibly have a situation where-- if you're just simply adding them up, because we don't expect any negative resistance in this situation. You're simply adding up all these positive resistances. Of course, the total will be always greater than any one component. Seems intuitive, but I just wanted to spell that out. So now, let's take a scenario where you have a human body, a vessel in the body. And let's say you have three parts to it, and these are equal parts. So let's say the resistance here is 2, 2, and 2. Obviously, I want to calculate-- as before-- my total. So my total will be 2 plus 2 plus 2, which is 6. And then, an interesting thing happens. So you have, let's say-- I'll draw the same vessel again. A really interesting thing happens. This is the same blood vessel, but now you have a blood clot. And this blood clot is floating through the blood vessels. And it kind of makes its way to this one that we're working with. And it goes and lodges right here. So right here you have a lodged blood vessel. Wow. That's pretty big, but it's right in that middle third of our vessel. So we have now a tiny little radius here. It's about, let's say, half of what we had before. The new radius equals half of what the old radius was. And you know from the last example that's going to increase the resistance in that part by 16. So the resistance here stays at 2. Here it stays at 2. But here in the middle, it goes from 2 to 32 because it's 16 times greater. So you end up increasing the resistance in the middle section by a lot. So let me just write that out for you. So 2 times 16 gets us to 32. So here the resistance is 32. And so if I wanted to calculate the total resistance, I'd get something like this-- 32 plus 2 plus 2 is 36. So I actually went from 6 to 36 when this blood clot came and clogged up part of that vessel. So just keep that in mind. We'll talk about that a little bit more, but I just wanted to use this example and also kind of cement the idea of what you do with resistance in a series. Let's contrast that to a different situation. And this is when you have resistance in parallel. So instead of asking blood to either kind of go through all of my vessels, I could also do something like this-- I could say, well, let's say, I have three vessels again. And this time, I'm going to change the length and the radius. And let's say this one's really big. And the resistance here, let's say, is 5, here is 10, and here is 6. So you've got three different resistances. And the blood now can choose to go through any one of these paths. It doesn't have to go through all three. So how do I figure out now what the total resistance is? So what is the total resistance? Well, the total resistance this time is going to be 1 over 1 R1, plus 1 over R2, plus 1 over R3. And you can go on and on just as before. But in this case, we only have three. So let's just put that there, that there, and that there. And I can figure this out pretty easily. So I can say 1 over 1 over 6 plus 1 over 10 plus 1/5. And the common denominator there is 30. So I could say 5/30. This is 3/30, and this would be 6/30. And adding that up together, I get 1 over 14/30 or 30 over 14, which is 2 and let's say 0.1. So 2.1. So the total resistance here is 2.1. Putting all three of these together is pretty interesting. And I want you to realize that the resistance in total is actually less than any component part. So unlike before where we said that the total resistance is greater than any component, here an interesting feature is that you have total resistance is always less than any component. So a pretty cool set of rules that we can kind of go forward with.