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Elastic pendulum (continued)

  1. Dec 5, 2008 #1
    1. The problem statement, all variables and given/known data

    Originally I was considering this question:
    "You have a pendulum with a ball of mass m at the end, and the pendulum is held parallel to the ground at equilibrium length so there is no initial spring force. This pendulum is also a spring, however, with spring constant k and equilibrium length L. You let go of the weight. What is L', the new length of the spring at the bottom of the swing (pi/2), and what is the velocity of the ball?"

    Initially I assumed that at the bottom of the swing the velocity vector would be perpendicular to the acceleration vector, which would be parallel with the spring. Therefore I tried using uniform circular motion equations to solve the problem and assumed that the length at the bottom of the swing was the maximum spring length. I wasn't too certain that this was a safe assumption, however, so I tested it experimentally.

    I found that the spring actually didn't reach maximum length until after pi/2 radians. I talked to my professor about this and he had another professor use a simulation program to use 2nd order DEs in polar coordinates to solve the parametric path of the mass. This supported my experimental results, and the image is below if it shows up (radius vs. time, the angle vs. time and the parametric path):

    [​IMG][/URL][/IMG]

    The thing that interests me now is what determines the angle at which the spring reaches its maximum length and what is the conceptual or mathematical reason for it. I'm very curious about this question and I'd greatly appreciate any insight.

    2. Relevant equations

    about as far as I can get is:

    velocity vector = radius dot * radius unit vector + radius * theta dot * theta unit vector
    AND
    acceleration vector = radius dot dot * radius unit vector + 2 * radius dot * theta dot * theta unit vector + radius * theta dot dot * theta unit vector - radius * theta dot squared * radius unit vector

    but those could be wrong. After this you'd need to do 2nd order DEs.

    3. The attempt at a solution

    My intuitive hypothesis would be that a larger spring constant or lighter weight would make the angle smaller and a weaker spring constant or heavier weight would make the angle larger. It also seems that this angle could not be any smaller than pi/2. I've thought about conceptual reasons for these conclusions and haven't been able to come up with any support for them. I also am not familiar enough with the math to mathmatically derive a solution and I don't have access to the simulation they used so I can't just run a bunch of different masses and spring constants.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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