# Analytic Solution For Equilibrium Point

1. Feb 18, 2007

I recently made an interesting problem involving a uniform rod. I would like to find an analytic solution to this problem however I do not know if this is possible. The equation yielding the equilibrium points is given by

omega^2*R*cos(phi) + omega^2*(L/2)*cos(phi)*sin(phi) + g*sin(phi) = 0

omega is a constant being a measure of angular velocity
R is also a predetermined radius
L is a constant but we are to consider the variations L < R, L = R, L > R

phi is the coordinate we are trying tofind equilibrium points for.

The problem is a uniform rod rotating freely in a vertical plane whos basepoint is a radial distance R away from the primary axis of rotation. The length of the rod is L, and its vertical orientation is given by phi measured clockwise from the vertical.

Note that there are 4 angles that yield a solution if L is long enough, (this is not to say that all 4 are stable). Only two will exist however (between 0 and 180 degrees if L is less than this k*R (k > 1)). We can easily find stability by either looking at the numerical solution and checking concavity (When L = k*R the concavity becomes zero at the critical points and it is tangent to the line y = 0). So this is all great for insight and such but is there an analytic solution? Pretty much this is becoming a math problem with independent coefficients. So if we normalize all the terms by omega^2*r the nwe have something like

cos(phi) + A*cos(phi)*sin(phi) + B*sin(phi) = 0

Anyone have any ideas? I'm completely stumped (tried s(2a) = 2cos(a)sin(a))
also sin(a) = sqrt(1-cos(a)^2) hasnt gotten me anywhere either. Thanks