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Book Pendulum Oscillation

  1. Sep 14, 2012 #1
    1. In the figure below, a book is suspended at one corner so that it can swing like a pendulum parallel to its plane. The edge lengths along the book face are 28 cm and 19 cm. If the angle through which it swings is only a few degrees, what is the period of the motion?

    W0358-N.jpg

    2. I=(ML^2)/12
    I=(ML^2)/3
    T=2∏√I/M*G*dcom


    3. I got dcom by using the distance formula and it's .1692 m
    Then I tried to using both inertia equations and using length, width. Then I plugged in all the numbers. (multiple attempts)

    All wrong answers (.788s, .394s, 1.062s)






    Appreciate your help! :)

    .
     
  2. jcsd
  3. Sep 15, 2012 #2

    Simon Bridge

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    You don't have the right moments of inertia.
     
  4. Sep 15, 2012 #3
    Oh.

    Then that leaves me with I= M(a2 + b2)/12.

    That gave me a period of .476 second which has been marked wrong.
     
  5. Sep 15, 2012 #4

    Simon Bridge

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    Yep - that's the wrong moment of inertia as well.
    That is for an oblong rotating about it's center.

    You book is not rotating about it's center - otherwise it could not act as a pendulum.

    Look up: parallel axis theorem.
     
  6. Sep 16, 2012 #5
    debad27da7daea89152bd3c7b5d34dd7.png

    I|| = [itex]\frac{1}{12}[/itex]Mdcom2 + Mdcom2

    This doesn't make sense though because it leaves me with an M on the top in the equation for T.


    EDIT: I asked somewhere else and they used T = 2π√[L/g] and got .83 seconds. This doesn't sound quite right. It's too easy this way!
     
    Last edited: Sep 16, 2012
  7. Sep 16, 2012 #6

    Simon Bridge

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    $$I_{CM}=\frac{M}{12}(a^2+b^2)$$
    ##r## is the distance from the corner to the center of the book; by pythagoras: $$r^2=\frac{a^2}{4}+\frac{b^2}{4}$$... therefore, by the parallel axis theorem: $$I=I_{CM}+Mr^2=\cdots$$... you finish up.
     
  8. Sep 16, 2012 #7

    Simon Bridge

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    That would be pretty normal ... he's modeled the book as a simple pendulum.

    For a simple pendulum ##I=ML^2## ... completing the calculations above will tell you how taking the mass distribution into account affects the period.
    http://en.wikipedia.org/wiki/Pendulum_(mathematics)#Compound_pendulum
     
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