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I've been reading about pendulums and oscillators recently and I've learnt that the time period the pendulum is independent of its amplitude and we can conclude by symmetry two pendulaums at different amplitudes will come to the bottom most point at the same time

This should be same in the case of two balls in a bowl since both the bob on the pendulam and the balls in the bowl follow a circular path

But in the case of the bowl the tention force is replaced by the normal force I think that would not affect anything .

So why don't the two balls in the bowl come down together?

In the midst of formulating this question I suppose it would be valid for small amplitudes as in the case of a pendulam.But what do I know .

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  • $\begingroup$ Why do you say that two balls don't come down together? What are you referring to? $\endgroup$
    – Karim Chahine
    Commented Feb 6, 2021 at 8:43
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    $\begingroup$ @Karim Chahine I've read that only happens in a curved surface called a tautochrone $\endgroup$
    – Glowingbluejuicebox
    Commented Feb 6, 2021 at 8:52
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    $\begingroup$ @Glowingbluejuicebox Sounds like you've answered the question yourself using a good Google. Now, using what you now know about the tautochrone, and about when the small angle approximation of another curve to it fails, can you answer your own question? $\endgroup$
    – prolyx
    Commented Feb 6, 2021 at 10:39

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Your statement that "the time period of pendulum is independent of its amplitude" in fact only holds for small amplitudes (see Wikipedia). Therefore, if you release two pendulums at different heights (angles), their periods will be different. In addition, the equation of motions for balls in the spherical bowl is the same as the pendulum. Thus "two balls in the bowl come down together" is valid only for small height, as in the case of pendulums.

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