*The Archimedes Principle* was formulated by Archimedes, and it states that; an object immersed in a fluid experiences a buoyant (upthrust) force equal to the weight of the fluid that it displaces. Our earlier article on Engineering in Kenya has more information.

# Mathematical Definition of The Archimedes Principle

The buoyant force of a fluid on an object explained by The Archimedes Principle depends on the weight of the fluid displaced and thus on the density of the fluid and the volume of the fluid displaced (since M = D×V)

In case of a totally immersed object, the volume of the fluid displaced is just equal to the volume of the object and therefore if the buoyant force is measured, and the fluid density known, the density of the object can be readily calculated. From this known volume and its mass, the density of the object may be found. It is not easy to measure directly the volume of irregularly shaped objects with great accuracy, but The Archimedes Principle provides a way to find volume accurately, since only balance measurements are needed.

If two substances have densities D_{1} and D_{2}, the density of the second substance relative to the first is D_{1}/D_{2}. Since density = mass/volume, then D_{2} = m_{2}/v_{2} and D_{1 }= m_{1}/v_{1}. if one compares equal volumes of the two substances when applying The Archimedes’ Principle so that v_{2} = v_{1}, then the relative density D_{1}/D_{2} = M_{1}/M_{2}, showing that the relative density will equal the ratio of their masses and weight.

The density of a substance relative to that of water is called the specific gravity of the substance in The Archimedes Principle i.e. the specific gravity of a substance is equal to the density of the substance divided by the density of water, or the specific gravity of a substance is equal to the weight of a certain volume of the substance divided by an equal volume of water. In accurate work, it is necessary to specify the temperature at which the measurements were made.

E.g. a chunk of copper suspended from a balance weighs 156.8g in air. When it is completely surrounded by pure water at 20^{o}C, the reading on the balance is 139.2g. Calculate the specific gravity of copper.

Solution – the weight of copper in air = 156.8g

- the apparent weight of copper in water = 139.2g

- Therefore, the buoyant force of water = 156.8-139.3 = 17.6g.

- Applying The Archimedes Principle, the weight of water displaced = 17.6g.

Thus, the specific gravity of the copper = weight of copper/weight of water displaced.

Specific gravity of copper = 156.8g/17.6g = 8.91.

In The Archimedes Principle, the density of the copper in this sample can be said to be 8.91times the density of pure water at 20^{o}C. When the specific gravity of a substance is known, its density in any units can be calculated from the known density of water by applying The Archimedes Principle.

## Applications of The Archimedes Principle

Wherever the force of a fluid acts on a fluid, the fluid exerts a buoyant force as a result of the different pressures at different levels. Every fish and submarine in the sea is buoyed up by the force equal to the weight of the water displaced. To remain submerged, these objects must have a weight that is equal to, or greater than the buoyant force. If they wish to move from one level to another, the balance between the force and the gravity and the buoyant force must be disturbed.

Some fish can rise by expanding their bodies thereby displacing more water. Submarines are made to rise by decreasing their weight by forcing water out of their ballast banks. They apply **The Archimedes Principle**. The air of the earth, since it’s a fluid, also exerts a buoyant force on all objects immersed in it e.g. balloons utilize the buoyancy of air. If a gas such as hydrogen or helium is used to inflate the light weight plastic envelope, the buoyancy force of the air can be considerably greater than the weight of the balloon. Such balloons are used in making high altitude measurements of various properties of the atmosphere.

E.g. a weather balloon has a volume of 0.5m^{3} when inflated. The weight of the envelope is 350g. If the balloon is filled with helium (which is lighter than air), what weight of instrument can it carry aloft? The density of air is 1.29kgm^{-3} and the density of helium is 0.138 times the density of air.

Solution – by use of The Archimedes Principle, the buoyant force of air = the weight of air displaced = the weight of 0.5m^{3} of air. I.e. 0.5m^{3}×1.29kgm^{-3} = 0.65kg.

- The weight of helium in the balloon is therefore 0.138×650g = 90g.

- Therefore the weight of the balloon and helium = 350 + 90 = 440g.

- The excess of the buoyant force over the force of gravity = 650 – 440 = 210g.

Therefore, if a balanced force of 10g is left available to produce upward acceleration, the weight of the instrument load can be 200g.

### Conclusion on The Archimedes Principle

The application of The Archimedes Principle has made possible sea travel and even improvement of security measures at sea by use of submarines. The Archimedes Principle provides the basis for the design of ships made of steel. For a steel vessel to float, it is only necessary to spread the steel around so that it can displace an amount of water having a weight that exceeds the weight of the steel. It therefore follows that The Archimedes Principle is part of our day to day life.