analyzing the roll of a bottle half filled with fluid

shoeman

This isn't really from my homework or anything like that, just something I was wondering about, but as my first post here I thought I might as well be on the cautious side and post it here.

What I was wondering about is the motion of a bottle of water rolling down an incline.
for the purpose of the exercise I decided to solve it for a non-viscous fluid (simply because i wouldn't know how to solve it otherwise).

so here are the known variables:

l, length of the cylinder (this will be our bottle), r- radius of the cylinder, Icm = mr^2, g-gravity, M-mass of the bottle, mf- mass of the fluid (although since it's exactly half filled, i guess it's the same as giving the density), θ (actually it's phi but i don't see it on the quick symbols), and last one - the friction between the bottle and the ground is big enough so the bottle would roll without slipping, and between the fluid and the bottle there's no friction.



Ok here is my attempted solution
http://i27.photobucket.com/albums/c1...leofwater3.png

Most of the Hebrew mambo-jumbo is the same data I wrote above.
then I explained what the water "feels" from inside the bottle (g and the bottles acceleration - De'alamber's force), and then explained that the mf(g+a) that they feel, the bottle would feel right back from the fluid (Newton's third).
inside the rectangle there's a short explanation of why the normal forces from the water wouldn't apply momentum.
and from there I wrote two equations, force on X coordinate, and moment equation from the center of the cylinder.


Also as a by product I can find (if solved correctly) the angle of the water β, and find a few interesting facts about it, although i'm not sure how true those results are...

basically i'm posting this here so that maybe you can find mistakes maybe in the calculations, assumptions, or simply something I missed.

shoeman

Ok, i've read in the rules that it's ok to bump once per thread, and 24 hours after the latest post.
I hope this one does the trick.

BUMP!