What Are the Normal Mode Frequencies of a Rod on a String?

[Plutoniummatt]

Homework Statement

A uniform rod of length a hangs vertically on the end of an inelastic string of
length a, the string being attached to the upper end of the rod. What are the
frequencies of the normal modes of oscillation in a vertical plane?

Answer: $$\omega^2 = (5 \pm \sqrt{19})g/a$$

Homework Equations

N/A

The Attempt at a Solution

I have tried a few attempts, all failed, like trying to take 2 pivots, one at the top of the string, one at the string-rod interface, or the rod COM, and trying to use torque and such like...no successes, so if someone could push me in the right direction, I would be grateful.

 

[vela]

I wrote down a Lagrangian in terms of two coordinates, the angle the string makes with the vertical and the angle the rod makes with the vertical. Using small angle approximations, I was able to derive those normal mode frequencies.

What level course is this for?

 

[Plutoniummatt]

2nd year undergrad, we haven't done Lagragian mechanics yet though...

I will try it anyway...How can I express the translational Kinetic energy of the rod?

 

[Plutoniummatt]

ok, is it:

$$(1/2)I\dot{\phi}^2 + (m/2) (d/dt((a/2)sin\phi + asin\theta))^2$$

I is the moment of inertia of rod about 1 end, (1/3)ma^2

phi is the angle the rod is to vertical

theta is angle of string to vertical

and once i haven't the lagragian in terms of the angles, do I minimize the action or?

 

[vela]

You've only included its motion in the horizontal direction; you need to account for its motion vertically too. Also, you have to subtract the potential energy to get the Lagrangian.

But as you haven't learned about the Lagrangian formulation of mechanics, I don't think this is going to be a very wise use of your time. (Besides, I just kind of hand-waved my way to the answer from piecing together stuff from my old classical mechanics homework. I wouldn't be able to explain why my guess worked.)