Explore connections in mathematics and nature with this article on buoyancy.
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Legend has it that while taking a bath one day, the Greek mathematician Archimedes made a tremendous discovery. As he lowered himself into the tub, he saw that the water level rose. After thinking about this for a moment, he leapt from his bath and ran down the street, shouting "Eureka!" ("I have found it!").What Archimedes saw has become known as the Archimedes Principle, or the Buoyancy Principle. This principle describes how an object immersed in a fluid will be buoyed up by a force equal to the weight of the volume of the fluid displaced.

The Archimedes Principle can help us to understand why some things float, while others don't. For example, if you fill up the bathtub and then drop a rock into it, the rock will sink. But if you drop a wooden stick into the water, the stick will float. Why is this? After all, the rock displaces water, and according to the Archimedes Principle, the water it displaces should help buoy it up!

The explanation for this comes from an understanding of density. The mass of the rock per unit of volume is larger than that of water, making the rock more dense than water. The rock, therefore, will sink. While the volume of water displaced by the rock does create a force of buoyancy, this force is not strong enough to compensate for the difference in densities to float the rock. In the case of the stick, a different outcome occurs. The stick is made of wood, which is less dense than water, so the stick will float. Even the small volume of water displaced by the stick creates enough buoyant force to keep it afloat.

Divers, swimmers, boaters, birds, fish, insects--all of these creatures are dependent on the Buoyancy Principle. What is true in water also works in the air. Why does a helium balloon rise? Because helium gas is less dense than air, it is buoyed up, just as a stick is buoyed up in water. The same idea works in hot air balloons. The hot air in the balloon is less dense than the air around it, so if you fill a balloon with hot air, it will rise. If you climb up on a chair and place your hand near the ceiling, you will notice that the air up there is warmer; it has floated to the ceiling because it is less dense than the cooler air close to the floor. What other examples of buoyancy and density can you think of?

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