Archimedes' principle

Archimedes' principle is a basic idea in science that explains why things float or sink in liquids and gases. It was discovered by Archimedes, a famous Greek scientist who lived in the 3rd century BCE. The principle says that an object put in a fluid (like water or air) feels an upward force called buoyancy. This force is equal to the weight of the fluid that the object pushes out of the way. If the upward force is greater than the object’s weight, it floats. If the force is less, it sinks. Archimedes' principle is important in many areas like physics, engineering, and designing boats.

There is a famous story about how Archimedes discovered this principle. The king asked him to check if a crown was made of pure gold, but without breaking it. Archimedes noticed the water rising when he got into a bathtub. He realized that the amount of water that spills out shows the volume of the object. From this, he could find out if the crown was made of real gold or a lighter metal like silver, because the gold would be heavier, and thus cause more water to spill out than the lighter silver. When he understood this, he is said to have shouted “Eureka!”, which means “I have found it!”

Archimedes’ principle works for all fluids, liquids and gases, and for all kinds of objects, no matter their shape or size. Scientists write the formula for the buoyant force like this:

$F_{b}=\rho \cdot V\cdot g$

Where:

• F_b is the buoyant force
• ρ (rho) is the density of the fluid
• V is the volume of fluid displaced
• g is the acceleration due to gravity

This principle is used today in many technologies. It helps in building ships and submarines, making hydrometers to measure liquid density, and in finding the volume of irregular objects. It is also important in weather science, ocean studies, and hot air balloon flight, because it helps explain how air and water move and why things rise or fall in them.

Principle

Archimedes may have used his principle of buoyancy to determine whether the golden crown was less dense than solid gold.

Archimedes' treatise, On floating bodies, proposition five states:

Any floating object displaces its own weight of fluid.

— Archimedes of Syracuse

For more general objects, floating and sunken, and in gases as well as liquids (i.e. a fluid), Archimedes' principle may be stated thus in terms of forces:

Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

— Archimedes of Syracuse

For a sunken object, the volume of displaced fluid is the volume of the object, and for a floating object on a liquid, the weight of the displaced liquid is the weight of the object.

Briefly: Buoyancy = weight of displaced fluid.

Gold

To use the principle to tell the difference between gold and another substance, the concept of mass (in practice, weight) must be added.

Apply this formula to a suitably pure piece of gold:

${\frac {\text{density of object}}{\text{density of fluid}}}={\frac {\text{weight}}{{\text{weight}}-{\text{apparent immersed weight}}}}\,$

That gives you the density of the gold sample. The only unknown is the density of the (gold) object; the density of water is 1.

Repeat for the experimental object (non-gold), when you get a different (and usually lesser) density.

Now you can tell what is gold and what is not, and collect your reward from the King of Syracuse. That is why Archimedes shouted "Eureka!"

Second thoughts

We do not actually know if Archimedes used exactly this method. An alternative is to use a scale. On one side put the object to be tested (e.g. the crown). On the other side put gold of equal weight. Immerse the scales. The gold will go down, and the crown up (if it is not gold). That is because, being less dense than gold, it occupies a larger volume and receives more buoyancy.

Other sites and pages

• Buoyancy
• Theory