A popular earring design features a circular piece of gold of diameter D with a

Question

A popular earring design features a circular piece of gold of diameter D with a circular cutout of diameter d. If this earring is to balance at the point P, show that the diameters must satisfy the condition: D = d,where = (1 + 5 ) /2= 1.61803 … is the famous “golden ratio.”

Explanation / Answer

1) It is clear that d>D/2 or the earring would not balance at P. 2) This problem is equivalent to not cutting out the small circle and instead placing a disk of diameter d so that its center lies on the vertical diameter thru the center of D and it is externally tangent to the location of the original small circle at P. That is it "kisses" the location of the original small circle. 3) This is true because the additional circle is symmetric about P with the added small circle and this added circle counterbalanced its weight having the same effect as cutting out the small circle. 4) The distance from the center of the big circle to P is then (d-D/2) and the distance from the center of the added small circle to P is just d/2. 5) Since the large circle is now treated as not missing a piece these circles should balance as if all their weight were at their centers (that's at their centers of gravity). 6). Therefore the following equation must hold: (pi)(D/2)^2(d-D/2)=(pi)(d/2)^2(d/2) 7) This simplifies to: (D/d)^3-2(D/d)^2+1=0 8) Solving this cubic equation or better testing the proposed answer by substitution we obtain the solution: D = d (1+sqrt5)/2.

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