A very similar solution was proposed on the forum and rejected because he made use of the assumption of \omega_n^2 = sqrt(g/l). Yours doesn't make explicit use of that assumption, so perhaps they would find it acceptable. I personally would not.
The problem I have with your first assumption is that it doesn't align with what is observed. If you actually build one of these cart pendulum systems, or buy one [0], you will observe responses that don't follow x(t) = A*sin(w*t). Or you can numerically integrate the equations of motion (even the linear ones) and see that they will also predict responses that aren't pure sinusoids, even if you add some damping to be more realistic. But that slight inconvenience aside, how can you justify that response is a pure sinusoidal, and not some other periodic function? I realize you could do a Fourier series decomposition on any periodic function and get a good approximation of it with as a sum of sines an cosines, but this still doesn't seem to me a valid reason to assume the response is a pure sinusoid. The only way I can see that this assumption is valid is if you assume the response obeys a differential equation and you pick a sinusoidal solution and show that it satisfies the differential equation, or if you are able to integrate that differential equation directly and obtain a sinusoid. Otherwise, it seems to me, you are just saying -- "because the pivot motion is a pure sinusoidal, we assume the pendulum response is also pure sinusoid, just don't look behind the curtain -- there is a differential equation there that you aren't supposed to look at". >From a pedagogical point of view, what is the point of doing an exercise this way? It a) doesn't give a prediction that aligns with observation and b) doesn't teach a student how to systematically approach a mechanics problem, and c) doesn't (in my view) offer any insightful methods/techniques in the process of obtaining the "correct answer". After a bitter struggle to use this approach, I don't feel I have any deeper understanding of pendulums, mechanics, or geometry, than when I started. If I sound like I'm criticizing you here, I apologize, I am not. I am primarily dissatisfied with this question and others like it, and I really don't see the point of them. ~Luke [0] -- http://www.quanser.com/english/html/products/fs_product_challenge.asp?lang_code=english&pcat_code=exp-lin&prod_code=L2-invpen&tmpl=1 On Wed, Nov 16, 2011 at 12:52 PM, Aaron Meurer <[email protected]> wrote: >> Attached pdf of answer (see last section of attachment)! > > Your image on the last page is the geometrical solution that I had in mind. > > Aaron Meurer > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To post to this group, send email to [email protected]. > To unsubscribe from this group, send email to > [email protected]. > For more options, visit this group at > http://groups.google.com/group/sympy?hl=en. > > -- "Those who would give up essential liberty to purchase a little temporary safety deserve neither liberty nor safety." -- Benjamin Franklin, Historical Review of Pennsylvania, 1759 -- You received this message because you are subscribed to the Google Groups "sympy" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
