Ponder This Challenge - January 1999 - Water glass center of mass

Ponder This Challenge:

When adding a new tumbler to your collection, you wish to record its mass, but have only the following information available. Given a cylindrical drinking glass, of uniform thickness, 12 centimeters high and 4 centimeters in radius. The location of the center of mass of the glass and water together depends on the depth of the water. As you pour water into the glass, the center of mass moves lower, then moves higher. The center of mass is at its lowest when the water is 4.5 centimeters deep. What is the mass of the tumbler?

Hint: Ignore the edges. The glass has a bottom and a side but no top. Bonus question: Is it possible to construct a glass such that the center of mass never drops when you add water? For this part the glass can have any shape you wish; it need not be cylindrical. FYI: Although this problem may appear simple at first, the solution is actually fairly complex.

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Solution

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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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