Finding Equation of Motion for Oscillations Using Lagrangian Methods

[James1991]

1. A rigid straight uniform bar of mass m and length l is attached by a frictionless hinge
at one end to a fixed wall so that it can move in a vertical plane. At a distance a from
the hinge it is supported by a spring of stiffness constant k, as shown in the figure

Ignoring gravitational effects, make use of Lagrangian methods to find the equation of
motion for small oscillations about the position of equilibrium (in which the bar is
horizontal) and show that the angular frequency of the motion is

\omega = a/L * \sqrt{}3k/m

Where a is the length from the wall to the spring along the rod and L is the length of the rod

Homework Equations

F=-kx
U=1/2kx^2
I = 1/3ML^2

The Attempt at a Solution

Right so i think the inertia of the rod is 1/3ML^2
and i tried to set up the lagrangian
as L = K - V
where K = M\omega^2{}theta*x^2
and V = 1/2kx^2

and i did d/dt(dL/dthetaDOT) = dL/dtheta
but it didnt work at all tbh, maybe i don't need to do it via this method, either way I'm abit lost as to what to do next and I've spent ages on it now.
Any help would be greatly appreciated
:)

 

[zorro]

You forgot to attach the figure.
Where did you get this question from?
Do you mind telling what course you are pursuing?

 

[James1991]

It's from a Lagrangian Mechanics module I'm doing. I'm studying Physics.
I managed to do the question in the end :)