Buoyant Force

RELATIVE DENSITY OF ARCHIMEDE’S PRINCIPLE

According to Archimedes principle, apparent loss of weight of a substance immersed in a liquid is equal to weight of displaced liquid . This principle can be utilised to determine the relative density of a solid substance (that does not dissolve in water).
 We know, relative density of a substance \( \text{ = }\frac{{\text{mass of any volume of substance in air}}} {{\text{mass of water of an equal volume}}} \)  Since weight of a body is proportional to mass, 
    Relative density of a substance \( \text{ = }\frac{{\text{weight of any volume of substance in air}}} {{\text{Apparant loss of weight of body in water}}} \)
The weight of the given substance is determined by a spring balance accurately (W₁). Then the body is completely immersed in water in a beaker without touching the sides and its weight in water W₂ is determined. 
    Then relative density of a substance \( = \frac{{W_1 }} {{W_1 - W_2 }} \)
    i.e., Relative density(R.D) of a solid \( = \frac{{Weight\text{ }of\text{ }solid\text{ }in\text{ }air}} {{loss\text{ }of\text{ }weight\text{ }of\text{ }solid\text{ }in\text{ }water}} = \frac{{W_{air} }} {{W_{air} - W_{water} }} \)   

b) Determine of Relative density (or R.D) or Specific gravity  of liquids using Archimedes’ principle :
    Take a body which sinks in both liquids and water. Determine its weight in air (W₁), in water (W₂) and in the given liquid (W₃) with a spring balance.
    Then the loss of weight of the body in water = W₁ – W₂ gram. weight loss of weight of the body in liquid (W₁– W₃) gram wt. The body displaces a liquid equal to its volume (V) 
    Relative density of the liquid = \( \frac{{\text{mass of 'V' volume of liquid}}} {{\text{mass of 'V' volume of water}}} \)  Since weight is proportional to mass, it can be replaced by weight
    Relative density of liquid =  \( \frac{{\text{weight of 'V' volume of liquid}}} {{\text{weight of 'V' volume of water}}} \)
                        = \( \frac{{\text{weight of displaced liquid by the body}}} {{\text{weight of displaced water by the body}}} \) 
                         \( = \,\,\frac{{\text{Loss of weight of the body in liquid}}} {{\text{Loss of weight of the body in water}}} \)    \( \frac{{W_{air} - W_{liquid} }} {{W_{air} - W_{water} }} \)

Applications of Archemides principle:
1. It is used in determining relative denisty of substances
2. Hydrometer and lactometers used for determining the denisty of liquid and purity of milk works based on Archimedes principle.
3. It is used in designing ships and submarines. 
4.  Hot air(or helium) balloon
5. Determination of purity of precious metals like gold, silver etc.