Pendulum - How to calculate the time it takes to go from A to B?

Discussion in 'Physics' started by atferrari, Mar 13, 2019.

  1. atferrari

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    Jan 6, 2004
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    There is an ideal pendulum. Length L and period T are known.

    Amplitude is 15°.

    20190313_191942.jpg

    How could I calculate the time it would take to reach B, starting at A?
     
  2. Papabravo

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    Is the problem in 2 dimensions or 3? In 3 dimensions, the Foucault pendulum is subject to the Earth's rotation and the Coriolis acceleration. The equation of motion is non-linear and IIRC it follows an elliptical path when it's motion is projected on a surface perpendicular to the radius vector from the center of the Earth to the pivot.
     
  3. atferrari

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    Two dimensions in a perfect plane. Yes, non-linear. That's why me asking for help.
     
  4. Papabravo

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    For small displacements you can linearize the equation with the substitution sin θ = θ. That doesn't wok for θ ≥ 7° so not much help on your problem. Since there is no closed form solution we are left with a numerical approach.
     
  5. atferrari

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    Could you elaborate briefly? At lost here.
     
  6. MrChips

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  7. Papabravo

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  8. shortbus

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    The pendulum expert should be along soon.
     
  9. Papabravo

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    The equation of motion for a simple pendulum is:

    \frac{d^2\theta}{dt^2}\;+\;\omega^{}_ 0^{2}sin\theta\;=\;0

    The introduction of the sin function makes the differential equation non linear.
     
    Last edited: Mar 14, 2019
  10. atferrari

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    I can see that all this is well over my head. No matter how much I try, I cannot go any further. Being honest I should give up.

    I expected to have someone showing the calculation I should do to actually obtain a value. Gracias anyway.
     
  11. Papabravo

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    I have one reference that says you can derive an expression for time as a function of angle using elliptic integrals of the first kind. AFAIK these are evaluated with tables or numerical methods. The expression is:

    t\;=\;\frac{1}{\omega^{}_0}\int\limits_{0}^{\theta}\frac{d\theta}{\sqrt{1\;-\;\left(\frac{2\omega^{}_n}{\omega^{}_0}\right)^{2}sin^2\left(\frac{\theta}{2}\right)}}

    https://en.wikipedia.org/wiki/Elliptic_integral

    I should add that \omega^{}_0\; is the velocity at \theta\;=\;0\;

    and that \omega^{}_n\;=\; \sqrt{\frac{g}{l}}
     
    Last edited: Mar 15, 2019 at 10:46 AM
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  12. shortbus

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    As it's a pendulum, I really expected bahn to show up in this thread.
     
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