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# A Delicate Balancing Act

You are a competitive swimmer, and your coach wants you to swim a lap of the pool backstroke with a soda can balanced on your forehead. He gives you an empty soda can and you can add some water to it if that will make your task easier.

How much water should you place in the can so that the center of gravity of the water-plus-can is as low as possible?

Assume the can is a perfect cylinder with a top and bottom made of the same material as the sides. The density of water is 1 gram/centimeter³. Let H be the height of the cylinder and r its radius, and let the mass of the can be C grams.

The problem can be solved in general, but let me also pose the specific realistic case:

H = 13 centimeters
r = 3.5 centimeters
C = 50 grams

In this particular case, what is the height of water that minimizes the center of gravity?

NOTE: There is a related question here — namely, how much water should be placed in the can so that the chance of the can falling over is as small as possible? As many have pointed out to me, this is not the same as minimizing the center of gravity. It would be the same if the water were frozen. But normal water will move as the can tilts, so one has a more difficult problem: How much water to place in the can so that the tipping angle is as large as possible?

This problem is dedicated to Heather Lendway, a former Macalester College All-American swimmer and recent winner, for the second year in a row, of the US Triathlon Olympic-Distance National Championship. This problem is a true story about one of Heather's practice sessions when she was a student in my applied mathematics course.