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‫Welcome back.

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‫And now we are going to make an assumption, if you look at this, you vector, right, then you know

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‫that we could rewrite it like this.

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‫In terms of Omega, ah, which was one of the components, and then V Sub V, which was another component,

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‫the reality is that the velocity VSA V is a lot smaller than Omega times are.

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‫And because of that fact, we can neglect our visa velocity for our purposes, meaning that it will

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‫play very little role in our thrust force calculations.

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‫If you do that, then our view equals the square root of Omega eight times are, and then all that squared.

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‫In other words, the squared and square root will cancel each other out.

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‫And you're just left with Omega R.

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‫In other words, you're assuming that this new velocity vector is pretty much parallel with this Omega.

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‫Times are velocity vector.

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‫In other words, what you're doing, you are assuming a small angle approximation for this fi angle.

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‫And so if you assume that your fi angle is very small, then that means that you can write your cosine

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‫phi like this one in your sine phi.

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‫You can write it down like this Phi Y.

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‫Well, let's look at the cosine function.

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‫This is your finances and this is your Y equals cosine phi axis.

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‫And so this is your cosine function, and when you have a small angle approximation for your fi angle,

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‫then what does it mean?

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‫It means that your fi angle hangs out very close to this zero point all the time, meaning that it never

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‫goes far away from this origin point, from this final zero point.

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‫Right.

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‫So this is your zero and a small angle approximation means that you just chill around this region all

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‫the time.

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‫And if you chill here all the time, then if you go up and you project this interval up, then you know

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‫that when you have a cosine function, then here you have one.

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‫So you see that if you go back and forth here a little bit, then you're pretty much always at one,

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‫right?

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‫So you could say that this red line here is almost a straight line at one on the Y axis.

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‫However, if you have a sign function, so this is your five axis and this would be your Y equals sign

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‫fi, then in case of a sign function which looks like this, then in this case you also chill close

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‫to your zero point, right.

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‫You don't go far away from your origin.

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‫You might wiggle a little bit to the left or to the right.

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‫But you stay very close to zero radiance because this axis here is in radiance, and so if you projected

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‫onto this sine function, then now your behavior will be something like this.

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‫You can't say like in this case when you had a cosmic function.

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‫Now, you cannot say that you have a horizontal straight line in the cosine function case.

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‫You had a horizontal straight line at Y equals one more or less.

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‫But here you don't have horizontal straight line, but you have a line that is something like a straight

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‫line function, because when you are very close to a region, then you have approximately forty five

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‫degrees here and forty five degrees here.

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‫But that's only if you're very close to your region.

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‫Obviously if you are somewhere here, then this assumption is not true anymore.

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‫But when you're very close to your origin then you have that thing that your angle very close to.

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‫The original is 45 degrees here and here.

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‫And in that case, you can approximate this thing as to be a straight line function Y equals five,

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‫the same thing.

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‫Like if you have your X here and then Y here and then your typical straight line function at forty five

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‫degrees would be this, then you would say that this is your Y equals X..

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‫And here you would just replace X by five.

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‫But again, it only is valid when you're very close to this zero degree radians, you cannot go far

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‫away.

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‫And that's what we call a small angle assumption.

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‫And that's why you can rewrite them like this.

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‫And we could have that small angle approximation because we said that our Visa V is very small compared

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‫to omegle times are, which essentially makes this angle very small.

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‫And so because of this assumption, you can get rid of this term and you can get rid of this term here,

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‫so your DL and the equations will become simpler.

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‫That's what they will become.

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‫And that also means that when you look at your differential thrust function, then this cosine PHI will

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‫become one and then this sine PHI will become simplify.

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‫So your DETI can be rewritten like this DL minus D, D times five, and then you can simply substitute

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‫the DL and equations with these ones here.

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‫So you can rewrite it like this, so this portion here, that would be your DL and then this portion

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‫here would be your D times fi and your Phi's here.

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‫And now I'm going to factor out some of the stuff.

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‫I'm going to factor out this and this.

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‫Then also the air density is one halves, then are squares and then the courts and the R's like this.

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‫So I'm going to put here Omega squared first, then one 1/2 times air density times the court length,

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‫then I'm going to put a bracket here and here.

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‫I'm going to write those things that I did not factor out.

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‫So C sub, L minus and then see some D times phi.

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‫And then on the other side of the bracket I'm going to put R-squared and D are so I can rewrite this

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‫equation like this.

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‫And now if I take this D.R and I'm going to put it here, then I have something very good.

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‫I have the change of my thrust force with respect to R.

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‫So if this is the top few of our blade, it rotates at Omega radians per second and we'll say that here

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‫you have a zero distance from the center of rotation, then you are would be in this direction.

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‫And let's just take random Air Force, let's say that this airfoil is at.

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‫Are one meters.

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‫And this airfoil here is that our two meters.

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‫And of course, the total blade length was our meters, but the capital are there now, we know how

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‫our thrust force varies with respect to the distance from the center of rotation.

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‫You might have one differential thrust here, DETI one, and then you might have another differential

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‫thrust here.

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‫Deti to.

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‫And in fact, you have some kind of differential thrust at every point on this blade when you go along

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‫this hour dimension.

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‫And so now you have an expression that tells you how your thrust force changes with respect to your

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‫R dimension.

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‫And to get the total thrust of the blade, you're going to have to sum up all the differential thrust

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‫on this blade, meaning that you have to integrate it over this entire length of the blade.