# 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 Thread Starter AAC Fanatic!

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

Amplitude is 15°.

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

2. ### Papabravo Expert

Feb 24, 2006
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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 Thread Starter AAC Fanatic!

Jan 6, 2004
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Two dimensions in a perfect plane. Yes, non-linear. That's why me asking for help.

4. ### Papabravo Expert

Feb 24, 2006
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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 Thread Starter AAC Fanatic!

Jan 6, 2004
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Could you elaborate briefly? At lost here.

Oct 2, 2009
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Feb 24, 2006
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8. ### shortbus AAC Fanatic!

Sep 30, 2009
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The pendulum expert should be along soon.

9. ### Papabravo Expert

Feb 24, 2006
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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 Thread Starter AAC Fanatic!

Jan 6, 2004
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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 Expert

Feb 24, 2006
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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
atferrari likes this.
12. ### shortbus AAC Fanatic!

Sep 30, 2009
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As it's a pendulum, I really expected bahn to show up in this thread.