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Find the period of a small oscillation

  1. Jan 20, 2016 #1
    1. The problem statement, all variables and given/known data
    A rod of lenght ##2L## is bent at point of its middle so that the rods now created are in a upside V shape and the angle between them is ##120°##. The system oscilates. Find the expression of the period of oscilation.

    2. Relevant equations
    3. The attempt at a solution

    I thought of one thing. Since the angle is ##120°## the system oscilates the same way as a normal pendulum rod of lenght ##L/2##. If that is true all that follows is the center of mass which for a rod would be ##L/2## i think. Therefore we have imagined a pendulum of lenght ##L/6##. Its period is ##T=2*\pi * \sqrt{\frac{L}{6g}}##. Is this correct?
     
  2. jcsd
  3. Jan 20, 2016 #2

    gneill

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    A pendulum that is not a simple bob on a massless string does not behave in the same way. You need to look at the moment of inertia around the pivot point.

    Look up: "Physical Pendulum". The Hyperphysics web site would be a good place to start.
     
  4. Jan 21, 2016 #3
    Well it seems i need to find the center of mass. But that means that the center of mass for a poll of lenght ##L## is ##L/2##. I can think of the center of mass being located at point ##L/4## below the point ##O##. Is that correct?
     
  5. Jan 21, 2016 #4

    gneill

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    Could be. Show how you worked it out.
     
  6. Jan 21, 2016 #5
    The poll of lenght ##L## has a center of mass located at ##L/2##. ##(L/2)*cos60=L/4## but that gives the period of ##\pi*\sqrt{\frac{L}{g}}##. The solution in the book read ##T=4*\pi* \sqrt{\frac{L}{3g}}## How can this be? That means that the center of mass is at ##4L/3## No idea how...
     
  7. Jan 21, 2016 #6

    gneill

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    What did you use for the moment of inertia of the pendulum about its pivot point?
     
  8. Jan 22, 2016 #7
    I didnt. I should have used ##I=mL^2/3## but i dont know how to use it to get the period.
     
  9. Jan 22, 2016 #8

    haruspex

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    Do you understand how to write down and solve the differential equation for a pendulum? It involves angular acceleration and moment of inertia.
    The formula for period is obtained from that.
     
  10. Jan 22, 2016 #9

    gneill

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    Did you investigate the term "physical pendulum" as I had suggested previously? The period of such a pendulum depends upon the moment of inertia about the pivot point and the distance of the pivot point from the center of mass.

    Hint: when you formulate your moment of inertia, remember that your pendulum was originally a single rod of mass m.
     
  11. Jan 23, 2016 #10
    I see now that the period of a physical pendulum is ##2*\pi*\sqrt{\frac{I}{mgL_{cm}}}##
    I know that the moment of inertia of a rod is ##mL^2/3## but that inputed doesnt give correct answer. I need the center of mass. Im not sure what to do
     
  12. Jan 23, 2016 #11
    I came up with something. The center of mass of these two rods is a rod of lenght ##L/2## staying vertical and oscilating. With above equation and inputing values of ##I=mL^2/3## ##L_{cm}= L/4## i get ##T= 4*\pi*\sqrt{\frac{L}{3g}}## correct?
     
  13. Jan 23, 2016 #12

    haruspex

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    Looks right to me.
     
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