Fluid Statics: Archimedes Principle Made Simple

The Story of Archimedes' Principle

Thousands of years ago in 250 B.C a man named Archimedes in Greece ran naked into the street shouting at the top his lungs eureka!. Such enthusiasm is justified when you just discovered an experimental method to find the gold content in the crown of king of Syracuse (or else get Zapped!). The story goes such that the king suspected some blacksmith had short changed him by making him a crown of alloyed gold (Blasphemy!). So, he consulted Archimedes a wise and a great scientific mind of that time (remember it is Greece) to check if such is the case.

The Approach to the problem

Most people would approach this problem by taking into the account the weight of crown and try to compare it to the weight of a crown made by pure gold. But lets say the forger was very clever and made sure that this crown of alloyed gold was to weigh the same in air compared to a crown made with pure gold (say the king already has). This now complicates the problem (for that year at least). But there is another very simple solution he could just measure the density of the crown as density of the material (19,300 kg/m^3 for Gold) remains the same (extremely minutely affected by the temperature). However, measuring the volume of the crown by conventional means is a daunting task ( remember density = mass/volume) as crowns come in very complex shapes.

The moment of Epiphany

He realized that when he got into his bath, the water level rose around him because his body was displacing the water and that this was linked to the feeling of weight loss; that uplifting feeling that everyone experiences in the bath. As the baths were public places, he probably noticed as well that smaller people displaced less water. This was the key to solving this problem. It is to be noted that displaced water has nothing to do with the weight but rather its the volume that is the culprit. When Archimedes placed both crowns in the water he noted both of those crowns had different weights or experienced different loss in weight. Understand this in figure 1.1

                                                Figure 1.1: Crown Displacing volume of water

In figure 1.1 the crown displaces volume of water in another container by means of an overflow pipe. The interesting part is that the volume of water collected by overflow pipe in the container is exactly equal to the volume of the crown. Moreover, the weight of the displaced volume is equal to the loss in weight of the crown (Due to natural upward push of water) . Which in turn means Archimedes measured the weight of the displaced volume i.e weight of an equal volume of water to that of the crown. 

By measuring the loss in weight, Archimedes was in fact measuring the weight of displaced water, that is, the weight of an equal volume of water. As the weight density of water is a fixed value (9810 N/m^3) it is a simple matter to convert this weight of water into a volume and so determine the density of the crown.

We get the answer to the previous problem of finding the volume the complex shape of the crown and now density of the crown can be compared to density of gold to determine if it is alloyed or not.

Formal Definitions

Archimedes' principle states that:

An object, fully or partially submerged in a fluid (liquid or gas), experiences an upward buoyant force. This force is equal in magnitude to the weight of the fluid displaced by the object.

where,

• Buoyant force: The upward push exerted by the fluid on the object.
• Displaced fluid: The amount of fluid pushed out of the way by the submerged object.

Examples:

• Ships: Designed to displace a large volume of water, creating a significant buoyant force to stay afloat.
• Submarines: Control their buoyancy by adjusting internal water tanks (ballast) to rise or sink.
• Hot air balloons: Heated air inside the balloon is less dense than surrounding air, creating a large buoyant force for lift.

Conclusion: 

Archimedes principle is now observed in our daily lives to design and maintain large water floating objects. However, the next time you sit in the bathtub do remember this principle.