Chapter 13.6, Problem 65E

The golden earring A disk of radius r is removed from a larger disk of radius R to form an earring (see figure). Assume the earring is a thin plate of uniform density.

a.    Find the center of mass of the earring in terms of r and R. (Hint: Place the origin of a coordinate system either at the center of the large disk or at Q: either way, the earring is symmetric about the x-axis.)

b.    Show that the ratio R/r such that the center of mass lies at the point P (on the edge of the inner disk) is the golden mean $\left(1+\sqrt{5}\right)/2\approx 1.618$.

(Source: P. Glaister, Golden Earrings, Mathematical Gazette, 80, 1996)