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We all know that the pendulum is isochronic, i.e. that it takes the same time regardless of the amplitude is this is less than 20 degrees.

But how do we prove it mathematically? What happens when the amplitude grows that breaks this law?

I am not looking for the intuition, but the math behind it, and I would like to see explicitly how larger amplitudes break the isochronicity, so this is different from the other questions.

Qmechanic
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user
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1 Answers1

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The math behind it is in the restoring force:

$$ F = mg\sin{\theta} = mg(\theta - \frac 1 3 \theta^3 + \cdots )$$

For small displacement, you only keep $\theta$, which gives the isochronic simple harmonic oscillator. For larger $\theta$ the cubic term means there's not enough restoring force to get back in time to maintain the SHO frequency.

When you get to $\theta = \pi $, well, you're stuck. Sort of.

JEB
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