SNHS Physics Blog 1: Archimedes' Principle

William Shi

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Legend has it that Archimedes, in the midst of taking a bath, suddenly ran naked through the streets of Syracuse shouting “Eureka!” after discovering the eponymously named Archimedes’ principle. But what is Archimedes' principle, and why is it so important?

Archimedes’ principle states that a body partially or fully immersed in a fluid experiences an upward force (called a buoyant force) equal to the weight of fluid displaced by the body. In a partially submerged body we only take into account the volume of the body inside the fluid. While this principle may sound simple, it can be incredibly tricky to apply in practice, and has lengthy and significant implications.

The first immediate implication is that a dense body always sinks in a less dense fluid. We see that the weight of the body must always be greater than the buoyant force applied by the fluid, so the body will experience a net downward force, and will sink.

Another property is that the force on a submerged body is independent of the actual shape of the body, and depends only on the volume. This is something we might take for granted, but is actually not obvious without Archimedes’ principle. In addition, we see that a denser fluid applies a stronger buoyant force, and if the body is completely submerged, then the buoyant force is constant no matter the depth, since the volume submerged does not change!

We see also that for a floating body, we can also use the Archimedes principle to determine the fraction of a body submerged. For instance, the tip of an iceberg is much smaller than the rest of the iceberg. But using Archimedes’ principle, the percentage of an iceberg occupied by the tip is actually completely determined by the density of ice! If an iceberg is floating, it is stationary, which by Newton’s Second Law implies that the total (net) force on the iceberg is zero. The Ice has a density roughly 90% that of water, which means that roughly 90% of the iceberg is submerged underwater for the buoyant force from the water and gravitational force from the Earth to cancel.

As another example, consider if I am in a kayak in a lake. A heavy rock is in the kayak, but I throw it overboard. How does the water level in the lake change?

This problem is again a tricky application of Archimedes’ principle. When the rock is in the kayak, it displaces a volume of water equal to its weight. When the rock is thrown overboard, it displaces a volume of water equal to its own volume. Since the rock is presumably denser than water, it must hold that the displaced water decreases, which means the water level in the lake decreases.

As we’ve seen, Archimedes’ principle easily explains many common day phenomena. We’ll now consider another final scenario. Consider if a student has a glass of ice water. The student is not particularly thirsty, so he does not drink from the glass. When the ice melts, how does the water level change? Again, since the ice is floating, we see that the water the ice displaced has the same mass as the ice. This means that once the ice melts and becomes water, it will occupy the same volume as the initial submerged volume of ice, which means the water level remains constant.

Of course, Archimedes principle has many applications on an industrial scale as well. Nonetheless, these interesting but tricky everyday examples demonstrate on their own the deceptively complicated nature of Archimedes’ principle.