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Simple Pendulum- Real formula

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data

    The motion of a simple pendulum is only approximately simple harmonic. How small should the amplitude should be for the approximation to hold good?. Obtain the general expression for the time period of a simple pendulum. How much does the actual time period differ from the approximate time period when the amplitude is 15 degree?

    2. Relevant equations


    3. The attempt at a solution
    On obtaining the period of oscillation using SHM, we approximate sin(a)=a in radians. I was thinking of keeping it as sin(a). Any help with the first question? The amplitude one.
     
  2. jcsd
  3. Nov 11, 2009 #2

    gabbagabbahey

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    The approximation [itex]\sin\theta\approx\theta[/itex] is derived via Taylor series expansion. To determine how accurate it is, you need to look at the Taylor series remainder.
     
  4. Nov 11, 2009 #3
    [tex]\ T=2pi \sqrt{L/g}(1\theta^2/16+11\theta^4/3072+173\theta^6/737280+22931\theta^8/1321205760+...)[/tex]

    That was the equation I got for time period. But I cant approximate the smallest value of amplitude.
     
    Last edited: Nov 11, 2009
  5. Nov 11, 2009 #4

    gabbagabbahey

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    Surely you mean

    [tex]T=2\pi\sqrt{\frac{L}{g}}\left(1+\frac{1}{16}\theta_0^2+\frac{11}{3072}\theta_0^4+\frac{173}{737280}\theta_0^6+\ldots\right)[/tex]

    where [itex]\theta_0[/itex] is the initial amplitude (angular displacement) of the pendulum....right?

    Read the question again. You aren't asked to approximate the smallest value of amplitude.
     
  6. Nov 12, 2009 #5
    "How small should the amplitude should be for the approximation to hold good?"
    It is given in the question. I dont know how to do it.
     
  7. Nov 12, 2009 #6

    gabbagabbahey

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    The approximation they are referring to is

    [tex]T=2\pi\sqrt{\frac{L}{g}}\left(1+\frac{1}{16}\theta _0^2+\frac{11}{3072}\theta_0^4+\frac{173}{737280}\theta_0^6+\ldots\right)\approx2\pi\sqrt{\frac{L}{g}}[/tex]

    ...how large can you make [itex]\theta_0[/itex] before this is no longer a good approximation?
     
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