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Falling pencil (inverted pendulum)

  1. Mar 8, 2014 #1
    1. The problem statement, all variables and given/known data
    A pencil of length l = 0.2 m is balanced on its point. How much time does it take to fall? Assume that the pencil is a massless rod and all of its mass is at the tip. To make the math easier, assume the small angle approximation.

    2. Relevant equations

    theta (double dot) = g/l sin theta

    T = 2 pi (l/g) ^ (1/2)

    3. The attempt at a solution

    Since the pencil seems to be an inverted pendulum and the fall would be only 1/4 of the period, I first tried T/4 = pi/2 (.2/9.8) ^ (1/2). But that is not a correct solution.

    Next I tried to rearrange and integrate

    1/sin theta d2theta = g/l dt2

    1/sin theta = sin theta/ sin2theta= sin theta/(1 - cos2theta)

    let u = cos theta so du=- sin theta d theta

    Integral( 1/ u2 - 1) du = 1/2 ln(u - 1) - 1/2 ln (u + 1)

    [ 1/2 ln (cos theta -1) - 1/2 ln (cos theta + 1)] d theta = gt/l dt

    I get lost here because the cos pi/2 is zero which makes the above meaningless.

    The way I interpret the problem, all the mass is at the balancing point. So how can you use energy to calculate potential difference or KE?

    If someone could just give a hint as to the approach, I would be greatful.

    Thank you.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 11, 2014 #2
    if the whole pencil can fall then θ will be π/2 how can the 'small angle approximation' hold?
  4. Mar 11, 2014 #3
    Sorry, I agree, but that was the problem that I was given.

    I think as in a lot of physics problems the assumption is made to make a good approximation.

    As in how to design an automatic chicken plucker: First you assume a perfectly symmetrical, spherical chicken...
  5. Mar 11, 2014 #4


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    The small angle approximation helps you to get started. ##\theta(0)=0## doesn't. :smile:
  6. Mar 14, 2014 #5
    Answer to pencil problem

    Attached is the answer for your falling pencil pleasure. Hope this helps someone.

    Attached Files:

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