Normal frequencies and normal modes of a multi-part system

[the_kid]

Homework Statement

***This is problem 11.29 in Taylor's Classical Mechanics***
A thin rod of length 2b and mass m is suspended by its two ends with two identical vertical springs (force constant k) that are attached to the horizontal ceiling. Assuming that the whole system is constrained to move in just the one vertical plane, find the normal frequencies and normal modes of small oscillations. [Hint: It is crucial to make a wise choice of generalized coordinates. One possibility would be r, φ, and $\alpha$, where r and φ specify the position of the rod's CM relative to an origin half way between the springs on the ceiling, and $\alpha$ is the angle of tilt of the rod. Be careful when writing down the potential energy.)

Homework Equations

The Attempt at a Solution

Right now, I'm just trying to set up the Lagrangian for this system, but the potential is giving me some problems. I recognize that there is a gravitational potential and a spring potential. I'm attempting to find the positions of the ends of the rod relative to some fixed point; however, I'm not sure what fixed point I should choose. Ultimately, I'm trying to find the lengths of the springs in terms of r, φ, and $\alpha$. I'm getting a little frustrated with the trig and trying things out. Could someone point me in the right direction?

 

[vela]

Using the suggested coordinate system, write down the (x_R, y_R) coordinates of the right end of the rod. The top of the right spring is located at (b, 0). The distance between those two points is the length of the right spring. You can do the same thing for the left spring with the top end connected to the ceiling at (-b, 0).