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Falling Oscillator

  1. Jun 26, 2010 #1
    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.

    1. The problem statement, all variables and given/known data
    Link to a scan of the problem:
    http://img443.imageshack.us/img443/4593/question1y.png [Broken]

    2. Relevant equations

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

    3. 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!
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jun 26, 2010 #2


    User Avatar
    Homework Helper
    Gold Member

    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 [itex]kA\sin(\omega t)[/itex] term.
  4. Jun 26, 2010 #3
    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.
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