Archimedes' Principle

Have you ever had a swim in a pool? Don’t you feel lighter? Have you ever drawn water from a well and felt that the bucket of water feels heavier when it is out of the water? Have you ever wondered why a ship made of iron and steel does not sink, but if the same amount of iron and steel in the form of a sheet would sink? Well, all these phenomena occur in all fluids including water and are due to exerted pressure.

What we will study in this chapter :

1. Archimedes Principle
2. Theoretical proof of Archimedes' Principle
3. Application of Archimedes' Principle to determine densities of liquids

1. Archimedes Principle Take a spring balance, a piece of stone, a measuring cylinder and water. Measure the weight of stone in air by tying the string around in a loop, and hanging it from the spring balance. Take water in a measuring cylinder and note its volume level. Then dip the stone in the water while it is still hanging from the spring balance. You will see that the stone is weighing less!! If you see the water level now, you will see it has risen. Now from the volume of the water displaced, calculate the weight of water from the following equation for density :                                                 Mass of water (in gm) Density of water         =                                             Volume of water (in cubic cm)    Density of water is 1 gm/cm3. You will see that the mass of water displaced is exactly equal to the reduction in weight of the stone in water.

Archimedes was the first person to understand this phenomenon more than about 2,200 years ago  and hence the phenomenon is named after him. Click here for an interesting anecdote on Archimedes. 

Archimedes’ Principle states that a body immersed in a liquid, wholly or partly, loses its weight. The loss of weight is equal to the weight of the liquid displaced by the body.    

2. Theoretical proof of Archimedes’ Principle   

Consider the figure alongside, here a square piece of iron is immersed in liquid. The piece of iron is experiencing forces from all sides and they are: The down ward force due to its weight = W Downward force acting on the upper surface of the iron piece, due to water pressing on it = F1   Upward force due to the tension of the string = T Upward force acting on the lower surface of the iron piece due to water pressing on it = F2   Horizontal forces acting on the other surfaces due to water pressure = H

Since the piece of iron is stationary and is not moving either up or down or side ways, we can safely say that
H=0 and  

Total upward force = Total Downward force 

T+ F₂  = W + F₁

Pressure is defined as force per unit area. 

F₁ = P₁ (on the upper surface of the iron piece)  x area  

and  

F₂ = P₂ (on the lower surface of the iron piece )  x area.

Pressure at a point inside a liquid is proportional to the height at which the point is from the surface, multiplied by the density of the liquid () and the gravitational force.  In the above figure the pressure at the top surface of the iron piece is h₁ g and at the bottom surface is h₂ g.

Therefore F₁ = (h₁ g) x area    and    F₂ = (h₂ g) x area 

W - T  =   ( g ) x volume of the iron piece 

W - T  =  loss of the weight of the iron piece when immersed in liquid. 

( g ) x volume of the iron piece = ( g)  x  volume of the liquid displaced by the iron piece 

                                                = g x V = (mass of liquid displaced) x g

                                                =  weight of liquid displaced by the body 

Hence we can conclude that the loss of weight of a body in a liquid is equal to the weight of the liquid displace by the body. 

The Archimedes principle holds good for irregular as well as regular bodies and any liquids. 

The upward force experienced by the immersed body is also known as upthrust or buoyancy [1].

3. Application of Archimedes’ Principle to determine densities of liquids 

Density of a substance is given as the mass per unit volume. Quite often, it is easier to quote the relative density of the substance with respect to the density of water. Hence the relative density (R.D.) of a substance is defined as the ratio of the density of the substance with respect to that of water.

Density of substance R.D =                  Density of water Density of water is 1 gm/cm3. (Density  changes with temperature; density of water is 1 gm/cm3 at 4oC. It is taken as the same at all  temperatures unless the temperatures are close to 0oC or 100oC , where water changes to ice or steam respectively)     To determine the density of an unknown liquid by Archimedes’ method, please do the following :   Weigh a given object in air = W1 Weigh the same object in water = W2 Weigh the same object in the unknown liquid = W3

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