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Treebeard's Stumper Answer
11 Sep 98

My Cup Runneth Over?

We'll start the year in science studying water, so here's a water stumper. Take a glass of your favorite cold beverage, add a few ice cubes, and then fill it completely to the brim. The ice cubes of course will float above the top. Now set the glass in a warm place and wait. What will happen to the water-level as the ice melts? This is a fine experiment for an (almost) summer day, but make a guess or hypothesis first, and think about your reasons before and after. Would it make a difference if the ice cubes were hollow?

Fill a glass with water and floating ice cubes, and the water level will stay the same as the ice melts. Several kids connected this with the fact that water expands as it freezes. James explains: "When the ice melts, the ice on top of the water will shrink to keep the water level the same." The ice cube displaces its weight of water, which is the exact volume of liquid it contains. A hollow ice cube won't make a difference (unless it contains helium) since it will displace its weight of water. But what if you freeze a marble into an ice cube?

Note: I gave this stumper to my students at Dunn Middle School for their first weekend project. Most kids guessed wrong (except the 8th graders who have figured me out and know it can't be the obvious answer if I asked the question!), and then discovered that the level doesn't change.

Graybear sent the following:

The liquid level stays the same as long as the following assumptions are met:
  1. The Specific Gravity of the liquid produced by the ice is the same as the S.G. of the original liquid, which, if you want to get technical, means that even though we are dealing with regular tapwater, the initial and final temperature of the liquid must be the same.
  2. There is sufficient liquid, and the glass is shaped such that the ice is floating freely - not resting on the bottom, no friction on the sides, etc. - nothing holding the ice above or below its natural floating point.
The ice being hollow will not make a difference, since the ice cube displaces its weight in water, and after it melts, its volume of water. Unless the hollow space were filled with helium, hydrogen, carbon monoxide, or some other gas with a different density than regular air.....

Several kids reported that the level went down a bit when they used ice in 7-up or coke. I figure the drink would lose some volume just sitting as it went flat. Would dissolved gasses also effect specific gravity?

Freezing a marble into an ice cube is like the follow-up stumper I gave the kids at school. You're in a boat on a lake with a load of rocks. What happens to the water level of the lake if you toss the rocks overboard? It's easy to model this with a graduated cylinder for the lake and a test tube with a few nails for the boat and rocks. Pretty surprising results!

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