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# Adding up resistance in series and in parallel

## Video transcript

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.