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Homework Help: SHM of Mass Oscillating between Two Identical Springs

  1. Mar 5, 2010 #1
    1. The problem statement, all variables and given/known
    A mass m is connected between two identical springs (along a y-axis) with identical spring constants k. The equilibrium length of each spring is L, but they are stretched to twice this length when m is in equilibrium. By analyzing the force acting on the mass when it is displaced by a small distance x, find the equation of motion of the mass and thus find the angular frequency of oscillation. Ignore gravity.

    Hint: You will have to make an approximation, but you only need to keep the term that is linear in x, higher powers of x should be ignored.


    2. Relevant equations
    F_sp = -kx = ma


    3. The attempt at a solution
    When you displace the mass a small x from equilibrium, the y components of the spring forces cancel, leaving twice the spring force in the x direction.

    If the angle made from the equilibrium (horizontal) to the stretched state is [tex]\theta[/tex] and the displacement is positive x, then

    - 2Fsp sin[tex]\theta[/tex] = max

    I'm really confused about what exactly the spring force is: do I need to consider that the spring is stretched or is that irrelevant? And how does the approximation apply to x (aren't we approximating [tex]\theta[/tex]?). Can someone lead me in the right direction?

    EDIT:
    I found the magnitude of the spring force to be k(2L-L) = kL.
    I approximated sin[tex]\theta[/tex] to be [tex]\theta[/tex] = x/2L
    So then the net force in the x direction on the mass = -2Fspsin[tex]\theta[/tex] = -kx = max
    So then the equation of motion is just the differential equation for SHM for a mass on a spring?

    Is this correct?
     
    Last edited: Mar 5, 2010
  2. jcsd
  3. Mar 5, 2010 #2

    ehild

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    Sketch the arrangement, please.

    ehild
     
  4. Mar 5, 2010 #3
    http://webserv.kmitl.ac.th/~physics/mb/images/stories/publisher/witoon/Problem_oscillation.pdf [Broken]

    Essentially question #53, except with springs.
     
    Last edited by a moderator: May 4, 2017
  5. Mar 5, 2010 #4

    ehild

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    Your solution is all right.

    ehild
     
  6. Mar 5, 2010 #5
    Great, thanks!
     
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