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How Do I determine when a pendulum ceases to be SHM

  1. Dec 6, 2012 #1
    1. The problem statement, all variables and given/known data
    I have to find what angle a pendulum stops being simple harmonic motion experimentally and if possible theoretically.

    2. Relevant equations
    T=2pi√(L/g) [1]
    d2θ/dt2+(g/r)sinθ=0 [2]
    dθ/dt=√((2g(cosθ-cosψ))/L) where phi = theta initial [3]
    T = 4*∫oψdθ/[1] [4]


    3. The attempt at a solution
    My first thought was to measure the period at intervals and compare that to the small angle approximation formula T=2pi√(L/g). After all a pendulum would be SHM if the period followed that formula for all angles right? So what we did was get a rigid pendulum and construct an board marked with angles at intervals of 5 degrees to place behind the pendulum. Using a photogate we measured the period from 5 degrees to 60 degrees at 5 degree increments. I'm sure period is key to what my professor wants us to find, but he did tell me my approach was wrong. So comparing true period to the period equation [1] is a no go. So this is where I need help. What do I do with the period to show at a certain angle that pendulum motion is no longer SHM?


    The above question is what I want answered most of all, but also on the theoretical side of things I started with equation [2] and derived it to equation [3] which I solved for T giving me equation [4]. Now this appears to be right equation for T from looking at the graph. As theta deceases to 0, T is the same as T from equation [1]. And as theta increases, T increases at an increasing rate. At 90 degrees I calculated equation [1] to be 15.3% off assuming equation [4] is accurate.
    Now trying to find the equation for T on the internet leads me to believe that it involves an elliptical integral and that just goes right over my head. My calc knowledge stops at calc 2 for now. So I assume my equation is just completely wrong?
     
  2. jcsd
  3. Dec 6, 2012 #2

    haruspex

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    When you say you solved [3] to get [4], I assume you mean you approximated cos by expansion, then solved to get [4].
    From a theoretical perspective, there clearly is no specific angle at which it ceases to be SHM. It's never quite SHM, but the deviance gets worse as the amplitude increases.
    So the question becomes, at what amplitude can you detect the error in your lab set-up?
    The problem with comparing the measured period with a calculated period from theory is that it will be hard to know the radius of inertia and centre of gravity accurately enough. There will also be discrepancies caused by drag. So I would think it is a matter of simply trying different amplitudes, starting very small, and plotting the periods.
    You could back it up by also plotting the relationship based on theory. You can perform the integral numerically, so don't need advanced calculus. And it does not need to replicate the details of the lab model - just demonstrate a similar plot.
     
  4. Dec 7, 2012 #3

    rude man

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    I see no reason why your experimental method was faulty.

    As to your equations, I do have a big problem. Your second equation is correct but your next one lost me but good. This eq. is not solvable by elementary diff eq techniques. It's 2nd order nonlinear. That's why your textbook limits you to "small" angles where sin(θ) ~ θ. That makes the equation linear and easily solvable.

    Qualitatively, thinking about the relationship between force on the pendulum vs. angle, knowing the right force is ~ sinθ vs. the assumed force ~ θ, you should at least be able to see qualitatively that your statement of T increasing with increasing θ is correct.
     
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