I'm looking for an intuition on the relationship between time period and amplitude (for a small pertubation) of pendulums. Why does the period not depend on the amplitude? I know the math of the problem. I am looking for physical intuition.
Answer
The higher you are, the greater the maximum velocity and maximum potential energy.
Consider one pendulum lifted higher than a second both released at the same time. When the higher pendulum reaches the starting point of the second, it already has a velocity greater than 0.
This higher velocity allows the higher pendulum to complete its swing in the same amount of time as the lower, even though it has a longer path.
Since I'm at a computer now I will address a majority of what is said in the comments. For starters, this does not exactly apply to pendulum; only approximately, and the approximation gets worse as $\theta$ increases.
A good way to visualize this is through the Tautochrone curve which is a frictionless curve where for all heights, the time to fall is the same (this is the equivalent of a pendulum period if you ignore the backswing, or have 2 of these curves mirrored; which will be a perfect mirror of the front swing if energy is conserved in the system).
In this scenario, the accelerations work out perfectly (under the same gravity) so that they all arrive at the same time. This is unlike the circular motion, which is only approximately correct for small angles. The interesting thing to note is that looking at a small displacement of a tautochrone curve; it looks approximately circular if you only look at a small section near the bottom. This is an intuitive way to explain why a circular pendulum approximately has this behaviour with small angles.
(Henning mentioned a tautochrone curve in his answer as well. It seemed to be an appropriate way to add more intuition to this)
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