Expert: Steve Holleran - 5/14/2007

Question
I've been developing this model on my spare time and am trying to optimize t wrt the angle x.  I've written a few of my key steps below.  Unfortunately, I've reached a point that has stumped me.

t = (lw)/(mv) + ( w*sqrt(l^2+w^2)/(lcos(x)-wsin(x))*( 1/(vtan(x)) + 2*r*sin(x)/m)

Differentiating this, I've found:

dt/dx = w*sqrt(l^2+w^2)/(lcos(x)-wsin(x)) * [ (lsin(x) + wcos(x))/(lcos(x)-wsin(x)) * (1/(vtan(x)) + 2*r*sin(x)/m) + (2*r*cos(x)/m) - csc^2(x)/v ]

To find the critical points, I've started solving the roots of this equation:

0 = w*sqrt(l^2+w^2)/(lcos(x)-wsin(x))
Lead simply to w=0

but the trouble comes with
0= (lsin(x) + wcos(x))/(lcos(x)-wsin(x)) * (1/(vtan(x)) + 2*r*sin(x)/m) + (2*r*cos(x)/m) - csc^2(x)/v

Here, I've simplified the equation to
0=alsin^2(x) + lcos^3(x) + wsin(x) + wcos^2(x)sin^2(x),
where a=2*r*v/m

Now that the problem is framed, my main concern is with my own differentiation, and with finding the roots, and consequently critical points.

I would appreciate any help you can offer, and can provide more information/clarification upon request.


Answer
Hi Derek,

Well, I've done this out, and I cannot see how to get a solution for dt/dx = 0.  I came out with a slightly different derivative than you did, though.

I don't know if it matters, but this is what I came up with.  I am assuming that w, l, m, v and r are constants.

dt/dx = -w * sqrt(l^2 + w^2) * [(l* cos x - w * sin x)(1/v * -csc^2 x + 2r/m * cos x) + (1/v * cot x + 2r/m * sin x)(-l * sin x - w * cos x)] / (l * cos x - w * sin x)^2 * (1/v*tan x + 2r* sin x / m)^2


And this = 0 when the numerator = 0, so if you do that and then divide out the -w * sqrt(l^2 + w^2) factor, you have

(l * cos x - w * sin x)(2r/m * cos x - 1/v * csc^2 x) +

(cot x / v + 2r/m * sin x)(-l * sin x - w * cos x) = 0

but I can't get anywhere after that.


This is one winner of a problem.  You must have ample spare time!!

Let me know if you come up with anything.
Sorry I couldn't help more.

Steve