Solving a Falling Oscillator with the Lagrangian Method

[tom777]

Hello guys. This is not really a homework exercise, but I'm currently preparing for an
exam and this is a question I got from a textbook. I'm currently sort of stuck,
but here are the details.

Homework Statement

Link to a scan of the problem:
http://img443.imageshack.us/img443/4593/question1y.png

Homework Equations

The problem is supposed to be solved by using the Lagrangian Method.
(Since it's from a "theoretical-mechanics"-book)

The Attempt at a Solution

Okay. What I thought was that i have to choose a suitable accelerating coordinate system
and relate this coordinate system with an intertial frame of reference.
First one sees in the picture that:
l * q(t) = s(t) * l_1(t)
So if one finds an equation of motion for l_1(t) one is done.
Now the problem is I'm not sure which moving coordinate system to choose.
I could choose one with origin at B, which would mean that there's gravity but
only the spring-force acting on the mass m.
However I'm not quite sure how to tackle this problem in general and I'm a bit confused.

I'd be more than happy if you could help me!
Thanks in advance!

 

[gabbagabbahey]

Okay. What I thought was that i have to choose a suitable accelerating coordinate system
and relate this coordinate system with an intertial frame of reference.

I'm not sure why you would think that. Instead, just look at a single inertial reference frame and compute the equations of motion from the Lagrangian. You should find that [tex]m\ddot{l_1}[/itex] not only has a spring term and a gravitational term, but also a $kA\sin(\omega t)$ term.

 

[tom777]

Hey. Thanks for the quick response.
Here are some thoughts:

I'm picking a cartesian coordiante axis y with origin at the initial position of the block B
pointing "downwords" - that is to say: in the direction the entire system is falling.
As a next step I'm picking l_1 as a generalized coordinate.
Then: y= l_1 + s . (*)
The zero-level of potential energy is at y=0.
Hence I obtain
V=-mgy+(1/2)*k*(l_1 - l)^2 (not 100% sure about the signs though ;-) )
Then I got T=(1/2)*m*(d/dt y)
Substituting (*) in the equation for T and V gives me a Lagrange-function that
only depends on l_1 . Then I set up the differential equation and
use the relation between l_1 and q given in my first posting to rewrite everything
in term of q.

What do you think of that approach? Any errors?
Again thanks in advance for your reply.
cu