‫Welcome back.
‫And now we are going to make an assumption, if you look at this, you vector, right, then you know
‫that we could rewrite it like this.
‫In terms of Omega, ah, which was one of the components, and then V Sub V, which was another component,
‫the reality is that the velocity VSA V is a lot smaller than Omega times are.
‫And because of that fact, we can neglect our visa velocity for our purposes, meaning that it will
‫play very little role in our thrust force calculations.
‫If you do that, then our view equals the square root of Omega eight times are, and then all that squared.
‫In other words, the squared and square root will cancel each other out.
‫And you're just left with Omega R.
‫In other words, you're assuming that this new velocity vector is pretty much parallel with this Omega.
‫Times are velocity vector.
‫In other words, what you're doing, you are assuming a small angle approximation for this fi angle.
‫And so if you assume that your fi angle is very small, then that means that you can write your cosine
‫phi like this one in your sine phi.
‫You can write it down like this Phi Y.
‫Well, let's look at the cosine function.
‫This is your finances and this is your Y equals cosine phi axis.
‫And so this is your cosine function, and when you have a small angle approximation for your fi angle,
‫then what does it mean?
‫It means that your fi angle hangs out very close to this zero point all the time, meaning that it never
‫goes far away from this origin point, from this final zero point.
‫Right.
‫So this is your zero and a small angle approximation means that you just chill around this region all
‫the time.
‫And if you chill here all the time, then if you go up and you project this interval up, then you know
‫that when you have a cosine function, then here you have one.
‫So you see that if you go back and forth here a little bit, then you're pretty much always at one,
‫right?
‫So you could say that this red line here is almost a straight line at one on the Y axis.
‫However, if you have a sign function, so this is your five axis and this would be your Y equals sign
‫fi, then in case of a sign function which looks like this, then in this case you also chill close
‫to your zero point, right.
‫You don't go far away from your origin.
‫You might wiggle a little bit to the left or to the right.
‫But you stay very close to zero radiance because this axis here is in radiance, and so if you projected
‫onto this sine function, then now your behavior will be something like this.
‫You can't say like in this case when you had a cosmic function.
‫Now, you cannot say that you have a horizontal straight line in the cosine function case.
‫You had a horizontal straight line at Y equals one more or less.
‫But here you don't have horizontal straight line, but you have a line that is something like a straight
‫line function, because when you are very close to a region, then you have approximately forty five
‫degrees here and forty five degrees here.
‫But that's only if you're very close to your region.
‫Obviously if you are somewhere here, then this assumption is not true anymore.
‫But when you're very close to your origin then you have that thing that your angle very close to.
‫The original is 45 degrees here and here.
‫And in that case, you can approximate this thing as to be a straight line function Y equals five,
‫the same thing.
‫Like if you have your X here and then Y here and then your typical straight line function at forty five
‫degrees would be this, then you would say that this is your Y equals X..
‫And here you would just replace X by five.
‫But again, it only is valid when you're very close to this zero degree radians, you cannot go far
‫away.
‫And that's what we call a small angle assumption.
‫And that's why you can rewrite them like this.
‫And we could have that small angle approximation because we said that our Visa V is very small compared
‫to omegle times are, which essentially makes this angle very small.
‫And so because of this assumption, you can get rid of this term and you can get rid of this term here,
‫so your DL and the equations will become simpler.
‫That's what they will become.
‫And that also means that when you look at your differential thrust function, then this cosine PHI will
‫become one and then this sine PHI will become simplify.
‫So your DETI can be rewritten like this DL minus D, D times five, and then you can simply substitute
‫the DL and equations with these ones here.
‫So you can rewrite it like this, so this portion here, that would be your DL and then this portion
‫here would be your D times fi and your Phi's here.
‫And now I'm going to factor out some of the stuff.
‫I'm going to factor out this and this.
‫Then also the air density is one halves, then are squares and then the courts and the R's like this.
‫So I'm going to put here Omega squared first, then one 1/2 times air density times the court length,
‫then I'm going to put a bracket here and here.
‫I'm going to write those things that I did not factor out.
‫So C sub, L minus and then see some D times phi.
‫And then on the other side of the bracket I'm going to put R-squared and D are so I can rewrite this
‫equation like this.
‫And now if I take this D.R and I'm going to put it here, then I have something very good.
‫I have the change of my thrust force with respect to R.
‫So if this is the top few of our blade, it rotates at Omega radians per second and we'll say that here
‫you have a zero distance from the center of rotation, then you are would be in this direction.
‫And let's just take random Air Force, let's say that this airfoil is at.
‫Are one meters.
‫And this airfoil here is that our two meters.
‫And of course, the total blade length was our meters, but the capital are there now, we know how
‫our thrust force varies with respect to the distance from the center of rotation.
‫You might have one differential thrust here, DETI one, and then you might have another differential
‫thrust here.
‫Deti to.
‫And in fact, you have some kind of differential thrust at every point on this blade when you go along
‫this hour dimension.
‫And so now you have an expression that tells you how your thrust force changes with respect to your
‫R dimension.
‫And to get the total thrust of the blade, you're going to have to sum up all the differential thrust
‫on this blade, meaning that you have to integrate it over this entire length of the blade.