- #1
Plutoniummatt
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Homework Statement
A block of mass M can move along a smooth horizontal track. Hanging from the
block is a mass m on a light rod of length l that is free to move in a vertical plane
that includes the line of motion of the block. Find the frequency and displacement
patterns of the normal modes of oscillation of the system firstly by `spotting' the
normal modes of the system and
then secondly by writing the Lagrangian L = T - U
for the system and solving the Euler-Lagrange equations.
Homework Equations
[tex]\frac{d}{dt}\frac{\delta L}{\delta\dot{q}_i} = \frac{\delta L}{\delta\dot{q}}[/tex]
The Attempt at a Solution
Mode 1- Translation
Mode 2- Pendulum swings, at the same time the Block also oscillates from side to side?
Kinetic Energy:
[tex]\frac{1}{2} M\dot{x}^2 + \frac{1}{2}ml^2\dot{\theta}^2[/tex]
Potential Energy:
[tex]mg(l - l cos\theta)[/tex]
Then write down the Lagrangian as L = T-U
applying the Euler-Lagrange equations for variables [tex]x[/tex] and [tex]\theta[/tex]
I get [tex]M\dot{x} = 0[/tex]
and for the [tex]\theta[/tex] coordinate, I just get the trivial pendulum equation...with [tex]\omega = \sqrt{g/l}[/tex]
Is that it? because this doesn't seem to have yielded me the answer they're looking for I guess...