Ever wondered how huge ships manage to stay afloat in water, while a small iron nail sinks? Puzzling as it may appear, you can easily explain this, and many other similar phenomena, with the help of the Archimedes’ Principle of Flotation. So, what is the Archimedes’ Principle all about?

**Principle of Flotation: Definition**

*The Archimedes’ principle states that any object, wholly or partially immersed in a fluid, experiences an upward force equal to the weight of the fluid displaced by it.*

Here the term ‘fluid’ refers to all liquids and gases. For an object that is completely submerged in a fluid, the weight of the fluid displaced by it is less than its own weight. On the other hand, for an object that floats on the surface of the fluid, the weight of the fluid displaced by it, is equal to the weight of the object. Now, the upward force experienced by the body is termed as the **buoyant force**. Thus,

**Buoyant force = weight of the fluid displaced by the body**

Now, the weight of the fluid displaced by the body is directly proportional to the volume of the displaced fluid, since the density of the fluid is constant. This can be illustrated by the following equations.

Weight = Mass x g (where g is the acceleration due to gravity and is a constant)

and

*Mass = Density x Volume*

Thus, we can say

*Weight = Density x Volume x g*

From the above equation, we can conclude that a body shall float in a fluid under any of the two conditions.

The density of the body is less than the density of the fluid.

The volume of the fluid displaced by the immersed part of the body is such that its weight is equal to the weight of the body.

## The following two examples will help you understand this:

*1. Firstly*, consider two cubes of the same dimensions, one made of cork and the other made of solid iron. If you place them on the surface of the water, what will happen? Well, the iron cube would sink while the cube made of cork would easily float on water.

2. Now, take the example of a nail and a ship, both made of iron. While the nail sinks, the ship floats on water carrying several passengers and cargo.

In the first example, the cork cube sinks while the iron cube does not, because the density of iron is higher but the density of cork is lower than that of water. It is the same reason we find it easier to swim in sea water than in river water, as the density of sea water is high due to the dissolved salts present in it.

Now, let’s consider the example of the nail and the ship. When you place a nail in water, the weight of the water displaced by the nail is less than the weight of the nail itself. In other words, the buoyant force experienced by the nail (which is equal to the weight of the water displaced by it) is less than its weight, and the nail sinks in water. However, when you observe a huge ship that floats on water, you’ll see that the ship is hollow, which means it is filled with air. This makes the average density of the ship lower than that of water. Thus, with only a small part of it submerged in water, the weight of the water displaced by the ship, becomes equal to the buoyant force, and the ship floats on water. From this, we can conclude that, for a body to float in water or any other fluid, the weight of the fluid displaced by the body should be equal to the weight of the body. In other words, more the weight of a body, more the volume of fluid it needs to displace, in order to float in the fluid.

Now, consider this. Suppose an iron ball weighs 20 kg. When a string is tied to the ball and it is submerged in water, the weight of water displaced by the ball is, say, 7 kg. Therefore, the ball would experience an upward force equal to 7 kg. This means that the net downward force experienced by the string would be equal to 13 kg (20 – 7 = 13). Thus, it can be concluded that the weight of the ball decreases when it is immersed in water. This reduced weight of the ball is termed as the apparent weight. Hence, the Archimedes’ principle can be restated as follows.

Reduced weight of the body in water (Apparent weight) = Weight of the body – weight of the fluid displaced

**The Story Behind the Principle**

Most of the inventions of Archimedes were made to help his country during the time of war. However, the story behind the discovery of the principle of flotation is an interesting one. Briefly put, it goes this way. The king of the land had got a golden crown made, to be offered to the deity of a temple. However, he doubted the honesty of the goldsmith, due to which he wanted to make sure that it was only pure gold that was used to make the crown. The great scientist that Archimedes was, he was called by the king and was asked to check if the crown was indeed made from pure gold, without causing any damage to it. Now, this was certainly not an easy job and it put him in a fix. However, one day as he stepped into the bathtub, he noticed the water spilling over. At that very moment, an idea occurred to him. He realised that by measuring the volume of water displaced by the crown, he could easily calculate its density. All he needed to do was divide the mass of the crown by the volume of displaced water. So much was he excited by this discovery that he took to the streets shouting, “Eureka, eureka!” (I found it!)

Archimedes was one of the greatest mathematicians of all times. He was a Greek mathematician who was also a physicist, scientist and a great inventor. Born in 287 B.C. in Sicily, Archimedes had many great inventions to his credit before his death in 212 B.C. His most famous inventions include the screw propeller and the principle of flotation, among others. His mathematical works include inventing infinitesimals and formulas on the measurement of a circle, parabolas, spheres, cylinders and cones. The one theorem that he considered to be his most prized and valuable achievement, is the one which states that if you have a sphere and a cylinder of the same height and diameter, then the volume and the surface area of the sphere will be 2/3 of that of the cylinder, with the surface area of the cylinder inclusive of the surface areas of its bases. However, the principle of flotation remains one of his most popular inventions.

Credits- M. Mukherjee

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