Isaac Physics Maths Question

Does anyone know how to do part D of this question? https://isaacphysics.org/questions/maths_ch7_7_q3
I've solved 7.7.1 anad 7.7.2, and part A-C of this question. However I have no idea what they are after in part D. Am I supposed to differentiate part C and substitute the value back in?
Thanks for your help

What did you get for the first 3 parts?

part b: f=sin(alpha) * cos(gamma) - (u + w*cos(alpha)) * (sin(gamma))/(w) part csqrt(u**(2) + w**(2) + 2*u*w*cos(alpha)))/(w)

From part b) you know the form of f(). Youre also told that F_u = 0 at max speed (no other resistive forces), so f() = 0 ...

Thank you for your hints, looking back now it is quite straightfoward. What about part E? I differentiated my expression for u/w wrt alpha. I then got alpha_max=arctan(1/tan(gamma)) which is incorrect

Thank you for your hints, looking back now it is quite straightfoward. What about part E? I differentiated my expression for u/w wrt alpha. I then got alpha_max=arctan(1/tan(gamma)) which is incorrect

Its weird, Eii) seems to be based on the trig angle addition which Im guessing Ei) is based on and gives the correct answer. But Ei) doesnt.
Edit - I dropped isaac a line as like D) and Eii) it seems to be a simple follow on from B) so f() = 0. But this is marked incorrect.

Has Isaac got back yet?

Its the weekend, they usually get back in a few days.

Meanwhile, would you also mind taking a look at part B of this question? https://isaacphysics.org/questions/maths_ch7_6_q8 I have solved all the other questions regarding log horms and and part A of this as well. I'm just not quite sure what is the correct dw, I've got dw=b*w_0*e^(b*theta)*dtheta but that doesn't seem to be right, and I don't know how to dot product dA and dw as well

What did you get for the part A parts? As they have the dA in terms of the x,y,z components, theyd want you to get dw as a similar x,y,z vector and then dot.
As a small divesion, you can also interpret finding a 3*3 matrix determinant in a similar way, so cross two vectors (columns, rows) then dot with the third.

x= (r_0 + w*cos(alpha)) * cos(alpha) y=- b*(r_0*cos(alpha) + w) z=sin(alpha) * (r_0 + cos(alpha) * w) How am I supposed to get dw?

WIthout working it through, as its split over several parts, diff w in part A
https://isaacphysics.org/questions/maths_ch7_6_q6
with respect to theta, though that horn has the spiral defined by z=0 and Ive not checked to see if the current one is that as well.

Still don't quite get it. After differentiating w and setting theta=0 (that's what they did in 7.6.7), I'm left with only a y component in dw, is that correct?

Ill need to go through it properly tomorrow. dw should form the tangent to the sprial which will be perpendicular to the surface area slice. So will form dV when you do the dot product dw and dA and you integrate over that. But from a quick read, you may not set theta=0 for dw, but need to properly go through it.

Its weird, Eii) seems to be based on the trig angle addition which Im guessing Ei) is based on and gives the correct answer. But Ei) doesnt.
Edit - I dropped isaac a line as like D) and Eii) it seems to be a simple follow on from B) so f() = 0. But this is marked incorrect.

Hi, I actually solved part E)i) now. It's just pi/2+...

Not worked through the volume calculation fully for this part, but worked through most of the previous parts and have a reasonable understanding of it now. So the dA vector represents a small surface area section and is obviously perpendicular to the area section. dw is a differential displacement along a sectional radius which, like the 2d spiral in
https://isaacphysics.org/questions/maths_ch7_6_q5
isnt parallel to the previous perpendicular. So to calculate the volume, you dot the vectors dA and dw to get the volume of a small parallellepiped and integrate over that. It would probably have been better to rename dw as r_w (for the vector r as in the previous area calcs) so youd diff the x,y,z (= r, part A) in
https://isaacphysics.org/questions/maths_ch7_6_q6
with respect to w to get something like
r_w = (cos(alpha)cos(theta)dw, cos(alpha))sin(theta)dw, sin(alpha)dw)
The question is basically doing a horn in the same way that
https://math.stackexchange.com/questions/1761763/the-volume-of-a-torus
(answer) does a torus and in that s = w. Its a reasonble thing to refer back to and I guess partially what isaac was trying to do in
https://isaacphysics.org/pages/maths_ch7_6_text
As before though, Ive not worked it through to integrate and get the volume. THere are a few things with the question which make me uneasy, such as when it says "Consider horns with a sectional radius w=w_0​ such that the horn turns don't overlap -- at most they touch" Saying w=w_0 means the horn radius is constant and they must overlap, but I suspect its a typo and simply trying to saying w_0 is small enough so that the turns dont intersect so you dont double count volumes.

Still not getting there, I suspect I might be integrating with wrong limits. Have you made any progress?