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David,

(1) - Yes, your 1st result is right : divide 3 ft by 2*Pi and not by just Pi (C = 2*Pi * R) ... and get almost half a foot, therefore just under 6".

Thanks for your correction.

(2) - Thanks to your "extreme" case - i.e. the one assuming a zero radius - the second question has become clear to me now. I did not interpret it correctly at first.

2.1 - If the radius shrinks to zero, yes the height of "loop" cord - i.e. its apex - becomes 1/2 * 3 ft, i.e. 18". Your 2nd result is right too.

2.2 - I would not be otherwise surprised if such apex height stays independent of the radius too ... and always equal to half the extra rope length.

... Simply because assuming a "zero radius" is also equivalent to considering a "huge" extra length to the rope relatively to the radius. There are values - at least the huge ones - for which your statement holds true. Why - and when ??? - would this situation change if such extra length decreases then ? Let me have a look first.

There could be an utterly simple demonstration here and maybe / hopefully somebody else will publish it anytime.

Thanks again for your quick feedback and correction.

Kermit