Solution Found!
The golden earring A disk of radius r is removed from a
Chapter 14, Problem 65AE(choose chapter or problem)
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 \((1+\sqrt{5}) / 2 \approx 1.618\).
(Source: R Glaister, "Golden Earrings," Mathematical Gazette 80 ( 1996): 224-225.)
Questions & Answers
QUESTION:
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 \((1+\sqrt{5}) / 2 \approx 1.618\).
(Source: R Glaister, "Golden Earrings," Mathematical Gazette 80 ( 1996): 224-225.)
ANSWER:
Solution 65AE1. The center of mass of the earring when the origin of the co