It is said that archimedes discovered the buoyancy laws when asked by king hiero of syracuse to determine whether his new crown was pure gold (sg = 19.3). archimedes measured the weight of the crown in air to be 11.8 n and its weight in water to be 10.9 n. was it pure gold?

The crown will be pure if it's specific gravity is 19.3

Now by definition of specific gravity it is the ratio between the weight of an object to the weight of water of equal volume

Since it is given that the weight of the crown is 11.8 N we need to find it's volume

Now According to Archimedes principle when the crown is immersed into water the water shall exert a force in upwards direction on the crown with a magnitude equaling to weight of the water displaced by the crown

Mathematically this is the difference between the weight of the crown in air and weight when immersed in water

Thus Buoyant force is

Now by Archimedes principle This force equals in magnitude to the weight of water of same volume as of the crown

Thus the specific gravity of the crown equals

As we see that the specific gravity of the crown material is less than that of pure gold hence we conclude that it is impure.  

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