- #1
skrat
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Homework Statement
A body with mass ##m## can move without any friction on ellipse that ##(x/a)^2+(y/b)^2=1## describe. In ##y## direction homogeneous gravity field ##g## is present. For generalized coordinate we take angle ##\alpha ## defined with ##x=acos\alpha ##, ##y=bsin\alpha ##. Find equilibrium position and frequency of oscillation around that position.
Homework Equations
The Attempt at a Solution
##L=T-V##
##T=\frac{1}{2}m(\dot{r}^2+r^2\dot{\alpha }^2)## and ##V=mgsin\alpha ##.
##L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\alpha }^2)-mgsin\alpha ##
##\frac{\partial L}{\partial \alpha }-\frac{\mathrm{d} }{\mathrm{d} t}\frac{\partial L}{\partial \dot{\alpha }}=0=-mgbcos\alpha -2mr\dot{r}\dot{\alpha }-mr^2\ddot{\alpha }=0##
So finally
##\ddot{\alpha }+\frac{2\dot{r}}{r}\dot{\alpha }+\frac{gb}{r^2}cos\alpha =0##.
For equilibrium position:
##cos\alpha =0## so ##\alpha =-\frac{\pi}{2}##
Now for frequency, I am guessing I can write:
##\ddot{\alpha }+\frac{2\dot{r}}{r}\dot{\alpha }+\frac{gb}{r^2}\alpha =0##
But this will be like horrible to calculate, since ##r=\sqrt{a^2cos^2\alpha +b^2sin^2\alpha }##...
Hmmm, is that ok? :/
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