# Archimedes’ Law

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Archimedes’ Law – a fundamental physical law discovered by the ancient Greek scientist Archimedes around the 3rd century BCE.

## Formulation of the Law

For any body submerged in a fluid (or gas) at rest, there acts upon this fluid (or gas) an upward buoyant force or Archimedean force equal to the product of the density of the fluid (or gas), the acceleration due to gravity, and the volume of the body submerged in the fluid (or gas).

Mathematically, Archimedes’ Law is expressed as:

F=ρ⋅V⋅g

Where:

F – buoyant force (in Newtons),

ρ – density of the fluid (or gas) in which the object is immersed (in mass per unit volume, e.g., kilograms per cubic meter),

V – volume of the fluid (or gas) displaced by the object (in cubic meters),

g – acceleration due to gravity (approximately 9.81 m/s2 on the Earth’s surface).

If the gravitational force on the body Fр​ is greater than the buoyant (Archimedean) force FA​, i.e., Fр>FA​, then the body sinks.

If Fр​=FA​, the body remains at rest at the depth to which it is submerged (floats).

If Fр<FA​, the body rises, and the ascent stops when the buoyant force equals the gravitational force.

The gravitational force of the fluid in a volume equal to the volume of the immersed part of the body is called the displacement, and the center of gravity of this volume is the center of buoyancy.

Archimedes’ Law is fundamental for understanding various phenomena, such as ship floating, the lift of hot air balloons, the operation of floats, and many others.

## Legend of Archimedes

One of the most famous myths about Archimedes is the story of the golden crown.

According to legend, King Hieron of Syracuse ordered a crown made of pure gold but suspected that the jeweler cheated him by adding some silver. The king asked Archimedes to determine whether the crown was made of pure gold.

Archimedes first established that a piece of pure gold is 19.3 times heavier than an equal volume of water. Thus, the density of gold is 19.3 times greater than the density of water. Archimedes needed to find the density of the material of the crown. If this density was greater than the density of water not by 19.3 times, but by a smaller factor, then the crown was not made of pure gold.

Weighing the crown was easy, but how to find its volume since the crown was of a complex shape.

Archimedes contemplated this problem for a long time until one day he entered a bath and noticed how the water rose when he got in. This discovery led to his famous exclamation “Eureka!” and allowed him to solve the problem.

By immersing the crown in water, he could determine its volume by measuring the volume of the displaced water.

By determining the crown’s volume, he could calculate its density. The density of the crown material turned out to be less than the density of pure gold.

From this, he concluded that the crown was not made of pure gold and perceived it as evidence of deceit.