This weekend I was out of town to attend a graduation. The night before the ceremony the family was gathered, visiting, and the graduate shared a recent event from her physics class. The professor described a scenario, asked the members of the class to make a prediction of the outcome, told them that they were all wrong, but was unable to explain why they were wrong.
Here is a scenario: a container of water has an ice cube floating in it, and a pebble sits on the ice cube. The ice cube melts. The pebble drops to the bottom of the container. Does the water level in the container rise, fall, or remain the same?
Along with most of the other family members, I predicted that the water level would rise. I had pictures of Archimedes running through the streets shouting “Eureka!” after realizing that the volume of a solid object could be measured by dropping it into a container of water and measuring the displacement of the water. Moreover, it seems that the water level should rise because of the melting of the ice. The graduate said all the members of the class had made the same prediction and it was wrong, but she still did not understand why.
One family member, an engineer, said that the professor was correct, and he explained why. The explanation puzzled most of the family members, although I caught on after a couple times through the scenario. The engineer wanted to produce a mathematical explanation with paper and pencil, but the rest of the family assured him that would not be necessary. We did try to experiment by creating the scenario with a measuring cup, an ice cube, and a pebble, but we could not find the right size ice cube or pebble to conduct the experiment.
The next day there was a party in the same house after the graduation ceremony. In addition to family members, several fellow graduates and other college students were present. To fill a lull in the conversation, I reintroduced the scenario from the physics class. One of the college students, a mathematician, insisted that the water level would rise. The engineer again countered that it would drop. This time the two of them did resort to pencil, paper, a laptop computer, and information from the internet, including the density of water and ice. The engineer was able to convince the mathematician that the water level would indeed drop.
It happens that the classic form of this scenario involves a boat and an anchor rather than an ice cube and a pebble. When the anchor is removed from the boat and dropped into the water, the water level drops, even though it seems that it should rise. The reason for the counterintuitive answer is that the boat with the anchor in it displaces some of the water in the pond. When the anchor is removed from the boat, the boat rises and the water level falls. When the anchor is dropped into the water, some water is displaced and the water level rises, but not to the height that it had been when the anchor was in the boat. The reason this happens is that the anchor sinks because it is denser than the water. (If the anchor floated and did not sink, it would not be an anchor, said the engineer.) Because of its density, the anchor displaces less water than its weight alone displaced when it was in the boat, being supported by the water.
By the same token, ice floats because it is less dense than water. As it floats, it displaces some of the water. When it melts, the volume of the water that was previously frozen is less than the volume of water displaced by the floating ice. Therefore, the pebble-ice cube combination displaced more water when the ice was frozen and floating, supporting the pebble, than the pebble displaced after the ice melted; even the melted ice did not add enough water to raise the water level to the height it had been when the ice was still frozen.
The rest of the weekend, including the graduation, was also nice. J.