kc1013,
You got me sucked in on this one. (Attracted to the gravity of the
question!). It's been a while since I studied gravitational mechanics,
so my hopes of answering your question directly were pretty slim.
However one of the main reasons Google has researchers is not because
of their immense knowledge (otherwise I would be in trouble) but
because of their ability to leverage the web and find the answer you
are looking for. And on that approach I was able to find a nice PDF
document that walks you through all of the steps to answer the
question.
So, short answer - go to the following link and read :
quiz.thphy.uni-duesseldorf.de/02/A0202_KMD.pdf
(Looks to me like the answer is "an egg" whose equation is given by
the rotation of the graph z/(x^2 + z^2)^(3/2) = 2*5^(1/3)*a)
From what I can tell, the analysis seems correct (again it's been a
couple of months since I did calculus of variations, but that would
make sense as the best approach to take on the question). It's not
surprising that the solution is cylindrically symmetrical, since any
mass would be better moved closer to the axis of action than applied
in a lopsided manner.
My main goal now should be to explain how I went about the search so
that you can find this kind of thing yourself.
My first approach (which turned out to be too direct) was do do a
search for what I thought the answer would be :
Google Search : "gravitational pull" "hollow sphere"
Unfortunately, the results produced by this search told me that I was
in fact 180 degrees off the correct answer (A hollow sphere has a zero
force inside of it - a fact that I would have known had I paid more
attention to Mr. I. Newton)
After realizing that this was not the correct answer, I did several
other fruitless searches before taking the more direct approach of
finding a general discussion of the problem :
Google Search : maximize "gravitational force" shape
As luck would have it, the above answer showed up as the first item
(so I could have clicked the "I'm feeling lucky" button if I wanted :)
So, make good use of the key terms in the problem; use quotation marks
around the key terms; and be prepared to learn from your erroneous
searches (Now if you get asked what shape Shmoo should be to minimize
the force, the answer would be to envelop the astronaut in a hollow
sphere (Like the astronaut wouldn't get the hint at that point :))
Anyway, I hope this helps (It's quite a technical answer, my apologies
- but to "prove" something like this (rather than give a reasonable
guess) is often quite messy).
Regards and happy shapeshifting!
calebu2-ga |
