A barge is carrying a load of gravel across a lake. You can assume no water enters or leaves the lake (i.e., the volume is constant). When it is in the middle of the lake the barge rocks and loses its cargo overboard, and the gravel sinks to the bottom of the lake. Does this cause the level of the lake to increase, decrease, or remain constant?
Solution:
The level of the lake at a given moment is proportional to the amount
of water displaced by foreign objects. For the barge to float on the lake,
it must displace a weight of water equal to its total weight, including the
weight of any cargo it is carrying. When the gravel is spilled into the
lake, the barge becomes lighter and floats higher. It no longer needs to
displace as much water. However, the gravel itself now sinks to the bottom
of the lake and displaces a volume of water equal to its volume.
We ask, has the displacement of water increased or decreased? In
other words, is the amount of water that is no longer displaced by the barge
less than or greater than the amount that is now displaced by the gravel
itself? Since solid gravel is more dense than liquid water, a greater amount
of water is required to equal the gravel's weight than to equal its volume.
Therefore, less water is being displaced after the spill than before, so the
level of the lake decreases as a result of the spill.
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