On 11/16/2011 04:57 PM, Luke wrote:
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<asmeu...@gmail.com> 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
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I agree. I would rather solve the problem from the fundamental equation
of motion point of view. I have heard from some that the Feynman lectures
are like a Chinese dinner. You think you have had a large dinner, but
half and hour later you are hungry again. I think that you need to get
the math you
need first and then do the physics correctly rather than coming up with
tricks that allows you to avoid doing the math.
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