Cruise ships and aircraft carriers are built from hundreds of thousands of tons of material, including a lot of steel, and they float. But throw a heavy metal anchor off the deck, and it will sink to the bottom of the ocean. Why?

Archimedes' principle describes how objects float or sink in fluids. In Newtonian physics, it is represented by the buoyant force.

## Who Was Archimedes of Syracuse?

Archimedes was a classical Greek thinker and tinkerer who lived from about 287 B.C. to 212 B.C. in Syracuse, an ancient Greek city-state on the island of Sicily. As a young man, Archimedes traveled to study in the world's largest library at the time, the Library of Alexandria in Egypt.

Known for his many mathematical formulations, including calculating pi to the most precise value until electronic calculators came along, he was also one of the first scientists to apply his math to physics and vice versa. Archimedes' discovery of a principle to describe buoyancy, or **how things float**, is at the center of one of the most famous stories in science history.

King Hiero II, a Sicilian tyrant of the era, reportedly received a new crown that he suspected was not made of pure gold. Fearing the crown maker had stolen some of the raw materials from him, substituting some of the material in the crown for silver instead, Hiero went to the island's resident genius Archimedes for help.

As legend has it, Archimedes was pondering the problem in the bathtub when he noticed that, as he got in and out of the water, the water level rose by a predictable amount. At this, he is said to have yelled "Eureka!" ("I've found it!"), a word that has now become indelibly stuck to discoveries and insights.

Presumably, the bathing scientist had put together two ideas: First, that for two objects of the same volume, the denser object has more mass. Second, the more space a submerged object takes up, the more fluid that is displaced when it's dropped in (an adult entering a bathtub sloshes more water around than a baby).

So, Archimedes reasoned, if he knew the weight of the crown he could gather an equal weight of pure gold, put both objects in water, and compare how much the water moved, or displaced. If they were equal, the crown was legitimate. If the gold moved more water by sinking deeper, the crown must be *less dense* than pure gold, meaning the crown maker was indeed tricking the king.

As it turned out, the crown was not pure: A win for Archimedes but likely catastrophic for the crown maker.

## Fluid Density

As Archimedes knew in the second century B.C., a fluid's density is a measure of its mass per unit volume. Mathematically, this is:

The more mass squeezed into the same volume, the denser the object. If the density of an object is more than the fluid in which it finds itself, it will sink.

Meanwhile, fluids that are more dense exert greater buoyant forces on objects placed in them.

These concepts together help explain why people can float almost effortlessly at the top of a very salty lake or sea, such as the Great Salt Lake or the Dead Sea, compared to in a less dense body of water.

## Fluid Pressure

Fluid pressure helps to describe the buoyant force in more detail.

Pressure in general is a **force per unit area**. All fluids have internal pressure, which pushes against any objects submerged in the fluid. This force per unit area exerted on the object by the water occurs from all sides, wherever water is pressing against it.

Additionally, fluid pressure depends on the density of the fluid and its depth. The deeper into the fluid an object is, the more fluid pressure the water exerts on it. This means for something like a boat in water, the bottom of the boat experiences more fluid pressure pushing it upward than the sides of the boat feel pushing inward.

## Archimedes' Principle

As Archimedes' bathtub anecdote illustrates, a convenient way to measure the force of fluid on an object, or the buoyant force, is to quantify the water displaced by that object when submerged.

This is true because the buoyant force equals the weight of the fluid the object displaces. In other words, for a canoe floating in a river, the amount of river water pushed away when it launches is equal to the amount of water that would fill the submerged portion of the canoe (however much of the inside of the boat is currently below the surface of the water).

The reason this happens is because pressure differences between the top and bottom of an object cause a net upward force equal to the difference between the object's weight the weight of the displaced fluid.

For example, consider a submerged cube in the water. The force vectors from the fluid pressure all around the cube are directed inward, but the vectors lower in the fluid are larger.

Hence, although the pressure at the top of the submerged object results in a downward force, and the pressure at the bottom results in an upward force, since the upward-directed vectors are larger, there will be a *net upward buoyant force* on the cube. So long as this force is at least equal to the additional downward force from gravity, or the cube's weight, it will float.

When the object is resting in the fluid, the weight of the object perfectly matches the weight of the displaced fluid. If the object weighs more than the displaced fluid though, the net force on it is downward, and it will sink; if it weighs less than the displaced water, it will accelerate upward.

Because in either case the volume of the object and the volume of fluid it displaces are set amounts, the only difference in their weights (the force of gravity acting on them) is from their respective masses. Since density is mass per unit volume, it follows that the density of the object is another way to determine whether it will sink or float: Objects denser than the fluid will sink and vice versa.

## Applications of Archimedes' Principle

Putting all these concepts together, a physicist can now explain how an incredibly heavy aircraft carrier, shipping vessel or cruise ship can float, even if it is made of materials like steel which have a density greater than the density of water. As long as the volume of water that is displaced by the boat is equal to the weight of the boat, the buoyant force on the boat will counteract gravity's downward pull.

Put another way, as long as there is enough space inside the ship beneath the water level, a really large hull, in seafaring terms, the ship can float. If however, the ship were a solid steel rectangle, or a giant solid steel anchor, it would *not float*. Such a shape would not displace as much water as something made from the equivalent mass but configured to have a large containment area inside, like a cruise ship with thousands of sleeping cabins.

While this article has focused on fluids and in particular ships floating in water, Archimedes' principle applies to gasses as well. Helium and hot air balloons are both floating objects in the same way as a ship. They displace a volume of air equivalent in mass to the mass of the balloon and its cargo. Eureka!