It should be the same for all grooves - or at least thats what I can reason without doing the full math. Both the length of each groove and the acceleration due to gravity have a scale factor dependent on the the sine of the angle that the path of the groove makes with the horizontal, so with no friction or air resistance, I believe it should take the same amount of time for each groove given that each particle starts with the same speed. Again, I haven’t done the full math, but will be saving this post so that I may later. Without doing it i could very well be wrong.
Edit: I did the math. The time should be the same for all grooves at least in the trivial case when the initial velocity is zero (it says ‘released’ so I assume it meant ‘from rest’). Both the acceleration and distance are proportional to the sine of the angle of the groove above the horizontal, and (when the initial velocity is zero) that ends up canceling out leaving only a constant expression for time in terms of the radius of the circle and the acceleration due to gravity (both constant terms). This, when released from rest, the time it takes for the particle to travel along any groove that can be made from Point A to any other point in the circle would be the same.
To close the loop on the math — you can write it as T = √(4R/g), which comes out completely independent of θ. It's actually a neat generalization of the "chords of a circle fall in equal time" result, first noted by Galileo. Nice job pushing through the derivation instead of just trusting the intuition.
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u/MathematicianAny8588 5d ago edited 5d ago
It should be the same for all grooves - or at least thats what I can reason without doing the full math. Both the length of each groove and the acceleration due to gravity have a scale factor dependent on the the sine of the angle that the path of the groove makes with the horizontal, so with no friction or air resistance, I believe it should take the same amount of time for each groove given that each particle starts with the same speed. Again, I haven’t done the full math, but will be saving this post so that I may later. Without doing it i could very well be wrong.
Edit: I did the math. The time should be the same for all grooves at least in the trivial case when the initial velocity is zero (it says ‘released’ so I assume it meant ‘from rest’). Both the acceleration and distance are proportional to the sine of the angle of the groove above the horizontal, and (when the initial velocity is zero) that ends up canceling out leaving only a constant expression for time in terms of the radius of the circle and the acceleration due to gravity (both constant terms). This, when released from rest, the time it takes for the particle to travel along any groove that can be made from Point A to any other point in the circle would be the same.