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The underlying fact on which the principle is based was discovered by Archimedes in about 300 B.C. The story goes that the King of Sicily suspected that the goldsmith has mixed some silver in his crown and cheated him. Without destroying the crown he wanted to know the truth. Archimedes was asked to find out whether this was so, without destroying the crown.
It seems that Archimedes got the solution while visiting the public baths. He leapt out of the bath and rushed naked through the streets shouting 'Eureka' (I have found it).
He obtained a lump of pure gold and a lump of pure silver, each with a weight equal to that of the crown. By immersing each in a vessel full of water he collected the volume of water which overflowed. The volumes were all found to be different, and he was able to calculate the percentage of silver in the crown which the goldsmith had made.
Principle of Archimedes
It states that when a body is totally or partially immersed in a fluid it experiences an upthrust equal to the weight of the fluid displaced.
Experimental Verification
Place a eureka can (over-flow vessel) on a table and place a beaker under its spout as shown in figure below.
The stone weighed 0.67 N in air and 0.40 N when immersed in water. The displaced water weighed 0.27 N (= 0.67 - 0.40).
Pour water into the can till the water starts overflowing through the spout. When the water stops dripping replace the beaker by another one of known weight.
Suspend a stone with the help of a string from the hook of a spring balance and record the weight of the stone.
Now, gradually lower the body into the eureka can containing water and record its new weight in water when it is fully immersed in water.
When no more water drips from the spout, weigh the beaker containing water.
Write down the results of the experiment as follows:
Weight of the stone in air = x gf
Weight of the stone in water = y gf
Weight of the empty beaker = a gf
Weight of beaker + water displaced = b gf
Apparent loss of weight of the stone = (x - y) gf
Weight of water displaced = (b - a) gf
You will notice that
x - y = b - a
Thus, the apparent loss of weight of the body, or the upthrust on the body equals the weight of the water displaced.
Numericals :
1. A body weighs 450 gf in air and 310 gf when completely immersed in water. Find
(i) the loss in weight of the body
(ii) the upthrust on the body
(iii) the volume of the body state the assumption made in calculating the volume.
Suggested answer :
(i) loss in weight = weight in air - weight in water
= 450 - 310
= 140 gf.
(ii) upthrust = loss of weight
= 140 gf
(iii) weight of displaced water = upthrust = 140gf
mass of displaced water = 140g
Assumption: density of water is 1 gcm-3
Volume of displaced water =
Volume of the body = volume of displaced water
= 140 cm3
2. The mass of a block made of certain material is 13.5 kg and its volume is 15 x 10-3 m3. Will the block float or sink in water? Give a reason for your answer.
Suggested answer :
Density of the block
= 0.9 x 103 kg m-3 (density of water is 1 x 103 kg m-3)
The block floats in water because its density is less than that of water.
