Archimedes’s Principle

FIELDS OF STUDY: Fluid Mechanics; Classical Mechanics

ABSTRACT: This article discuss the buoyancy principle put forth by the ancient Greek mathematician Archimedes. Archimedes discovered that when an object is placed in a fluid, it will displace a volume of that fluid equal to its own volume. He further determined that the object in the fluid will experience an upward force equal in magnitude to the weight of the displaced fluid. This force is known as buoyancy. Archimedes’s principle is a fundamental law of fluid mechanics.

principal terms

• buoyancy: the upward force exerted by a fluid on a body immersed in that fluid.
• density: a measure of the amount of matter in a substance per unit area.
• displacement: in fluid mechanics, the process by which a body immersed in a fluid pushes the fluid out of the way and occupies the space in its stead. The volume of the displaced fluid is equal to the volume of the displacing body.
• mass: the amount of matter contained in an object.
• specific gravity: the ratio of the density of a substance to that of a standard reference substance; also known as relative density.
• volume: the amount of three-dimensional space enclosed within a given area.

The Eureka Moment

It is said that while in the public bath one day, the ancient Greek philosopher Archimedes of Syracuse (ca. 287–212 BCE) realized the solution to a difficult problem. A king had given an artisan an amount of pure gold with which to make him a crown. Believing that the artisan had replaced some of the gold in the crown with an inferior metal and kept the extra for himself, the king asked Archimedes to determine if this was the case. Archimedes, however, could not think of a way to determine the purity of the gold without destroying the crown in the process. Simply weighing the crown and comparing it to the original weight of the pure gold would not work. Even if the gold in the crown were of the same weight, it might have a different density. The artisan could simply have added more of a less dense material, or vice versa, to achieve the same weight.

According to popular legend, while pondering this problem, Archimedes visited a public bath to think it over. Once there, he noticed that the water level rose as he immersed himself in the bath and fell again as he stood to leave. Archimedes realized that his body was displacing the water when he was submerged. It is said that upon this realization, Archimedes leapt from the bath and ran through the streets naked, exclaiming, "Eureka!"

Further testing proved that when an object is submerged in water, the volume of the displaced water is equal to the volume of the object causing the displacement. Previously, Greek mathematicians had developed equations to calculate the volume of regular geometric objects, such as spheres and pyramids. However, they had no way to determine the volume of an object that could not be broken down into such shapes. Archimedes’s discovery was significant because it allowed him to accurately measure the volume of an irregular object—in this case, the king’s crown. Using this method, Archimedes could calculate the crown’s density, because an object’s density is equal to its weight divided by its volume. (Strictly speaking, density is in fact equal to mass divided by volume, while weight is equal to mass times the acceleration due to gravity. However, on Earth, an object’s mass is more or less equivalent to its weight, because the average magnitude of acceleration due to gravity near Earth’s surface—a quantity known as standard gravity—is defined as 1.) The density of the crown could then be compared to the density of pure gold. When Archimedes did this, he proved that the artisan had in fact cheated the king.

The discovery proved to be significant in another way as well: it allowed Archimedes to develop the principle that would bear his name. Archimedes’s principle states that an object submerged in a fluid experiences an upward force equal to the weight of the fluid displaced by the object. This upward force is called buoyancy. If the weight of the submerged object is less than the weight of the displaced fluid, and thus less than the buoyant force, the object will rise to the top of the fluid; if its weight is greater, the object will sink; and if the weight is the same, the object will neither rise nor sink.

Because the volume of the object and the volume of the displaced fluid are the same, Archimedes’s principle can be stated in another way: an object that is less dense than the surrounding fluid will float, while an object that is denser than the fluid will sink. This is related to the concept of specific gravity, also called relative density. An object’s specific gravity is equal to the ratio of its density to the density of a standard reference substance, typically water. If the object’s specific gravity is less than 1, it is less dense than the reference substance; if its specific gravity is greater than 1, its density is likewise greater.

The Buoyancy of Ships

While it is true that Archimedes formalized the concept of buoyancy—a concept that shipbuilders had understood for many years—he did not expand on the idea or undertake additional research in that area. However, the concept later spread throughout the region when the Romans used it to build coin-operated water dispensers. Archimedes’s method was used to check if the coins were genuine.

While ancient peoples might not have understood why wood floated, they knew that it did. They also knew that they had to be careful not to put too much cargo on board a ship and cause it to sink. Similarly, the ancient Chinese philosopher Zhuangzi (ca. 369–ca. 286 BCE) was aware that large ships need a great deal of water to float. He also understood that crumbs would float in a small bowl. He used such phenomena as a metaphor for knowledge and understanding. This understanding allowed people to use rocks for ballast in their ships, thus making them more stable by more closely approximating the density of water.

Additional experimentation in a formal setting took place in the model basins of Europe and the United States in the late 1800s and early 1900s. The findings from these early studies inform the computer models that eventually entered into common use. This enhanced understanding of Archimedes’s buoyancy principle remains crucial to the design of ships in the twenty-first century.

Sample Problem

A large model ship weighs 5 kilograms (kg) and displaces 0.004 cubic meters (m³) of water. The density of water is approximately 1,000 kilograms per cubic meter (kg/m³). Will the model ship float or sink?

Answer:

Given the density of water, calculate the weight of 0.004 m³ of water. Recall that the density (ρ) of a substance is equal to its mass (m) divided by its volume (V).

The mass of the displaced water is 4 kg. Mass on Earth is roughly equivalent to weight, so the weight of the water is also 4 kg. Because this is less than the weight of the model ship (5 kg), the model will not float in water, or at least not in freshwater. However, it might float in water that contains some other substance that increases its density, such as the highly saline water of an alkaline lake.

Archimedes's Principle in Action

Archimedes’s principle is one of the foundational ideas in fluid mechanics. It allows engineers to design ships that take into account the motion of waves, enabling them to move faster and more efficiently even in rough waters. Archimedes’s principle is also at the root of how submarines can ascend and descend in water. A submarine has floodable tanks that allow the operator to increase the vessel’s mass by taking in water. This increases the submarine’s density and makes it heavier than the water it displaces, causing it to sink to the desired depth.

Fish use a similar mechanism to alter their buoyancy. Many fish have an internal organ known as a swim bladder. They can expand or contract these bladders by taking in or expelling gas. When a fish’s swim bladder expands, its body likewise expands to accommodate it, causing its overall volume to increase. As a result, the fish’s density decreases, making it more buoyant. Contracting the bladder makes the fish smaller again, increasing its density and decreasing its buoyancy, so that it can swim in deeper water.

Bibliography

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