IBM Research | Ponder This | January 1999 solutions
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January 1999

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The trick to the answer is to realize that when the center of mass is at its lowest point, the center of mass corresponds with the height of the water. This is because if the water were lower, then adding a bit more water would lower the center of mass; and if the water were higher, then removing a bit of water would lower the center of mass. Once we realize that, we only need to do some straightforward computations to relate the center of mass to the dimensions, the density of water (1 gram per cubic centimeter), and the AREAL density of the tumbler (T grams per SQUARE centimeter, assuming uniform thicknness). Let H=12 be the height of the glass, R=4 its radius, and Z=4.5 the optimal depth of the water. The water has volume ZR2 = 72 cubic centimeters and mass 72 grams. The bottom of the cylinder has area R2 = 16 square centimeters and mass 16 T grams. The side of the cylinder has area 2 RH = 96 square centimeters and mass 96T grams. The total mass is then 72 + 112 T. The moment is calculated as (72 * Z/2) + (16 T * 0) + (96 T * H/2) = 162 + 576 T. Because the center of mass is known to be at height Z=4.5 we have (72 + 112 T) * 4.5 = 162 + 576 T, so that T = 2.25 grams per square centimeter. Then the tumbler has mass 2.25 * (16 + 96 ) = 252 grams.

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