Deriving Moment of Inertia

  1. 1. The problem statement, all variables and given/known data
    A uniform rod of mass M and length L is free to rotate about a horizontal axis perpendicular to the rod and through one end. A) Find the period of oscillation for small angular displacements. B) Find the period if the axis is a distance x from the center of mass


    2. Relevant equations
    I = summation(Mx^2)
    T restoring = k*theta = mgtheta*x
    period = 2pi*root(I/k)

    3. The attempt at a solution
    The first part is no problem. I = 1/3ML^2 and the restoring constant is LMg/2. T = 2pi*root(2L/3G)

    For the second part, I know that mgtheta acts at the distance x, but how do I derive moment of inertia for an axis x distance from the com? I know it changes with the axis and it has to fall inbetween 1/3ML^2 and 1/12ML^2. But Im not familliar with the integration that goes into deriving I.

    Should I leave it as 2pi*root(newI/xMg) ?
     
  2. jcsd
  3. radou

    radou 3,217
    Homework Helper

    This should help: http://en.wikipedia.org/wiki/Parallel_axis_theorem.
     
  4. Thanks a lot. So the period is 2pi*root[(1/12L^2 + x^2)/(gx)]
     
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