Gravitation

ARCHEMEDE’S PRINCIPLE:

It states that when a body is immersed completely or partly in a liquid, it loses it’s weight which is equal to weight of liquid displaced
 Let W₁ be weight of a body in air and W₂ be the weight, when completely immersed in a liquid.
    Loss in weight of the body inside the liquid =W₁ – W₂
When the body is immersed in the liquid, it experiences upthrust. The upthrust is equal to the weight of the liquid displaced by the body. Since the upthrust acts upwards, the weight of the body inside the liquid decreases apparently by an amount equal to the weight of the displaced liquid. Hence, apparent loss in weight of the body (W₁-W₂)
        = weight of the liquid displaced by the body= F_B 

Condition of floatation of a body. When a body is thrown into a liquid,one of the following may happen
When a body of density  and volume V is immersed in a liquid of density ,the force is acting on the body are:
(1) Weight of the body W = mg = \( V\rho _b g \)  acting vertically downwards through the centre of gravity of the body.
(2) the upthrust \( B = V\rho _L g \) acting vertically upwards through the centre of gravity of the displaced liquid which is also known as the centre of buoyancy.
So, the following then situations are possible when  the body is released.
a)(\( \rho _b > \rho _L \)) 

1.The body may sink in the liquid:It happens,when the weight of the liquid displaced (upthrust of the liquid)is less than the weight of the body .In case,the body is not hollow from inside,then density of the material of the body must be greater than the density of the liquid 
 In this situation, 'W' is more than 'B' and hence the body will sink normal to the free surface
b) \( \rho _b = \rho _L \)  
2.The body may float just completely submerged in the liquid:It happens,when when the weight of the liquid displaced is just equal to the weight of the body .Again if the body is not hollow from inside,then density of the material of the body must be equal to  the density of the liquidc) .
In this situation, W = B so, the body will float while it is fully submerged or just floats
in liquid or just immersed in liquid. 
c) \( \rho _b < \rho _L \)  
3.The body may float with a part submerged in the liquid:It happens ,when the weight of the of the liquid displaced by the submerged  part in the liquid is equal to the weight of the body .Again if the body is not hollow from inside,then density of the material of the body must be less than  the density of the liquid 
In this situation, W<B. So, the body will move upwards normal to free surface. After some oscillation, it will finally settle in an equilibrium position while partially submerged. 

Buoyancy:
Buoyancy is the upward force exerted by a fluid (liquid or gas) on an object placed in it. This force is equal to the weight of the fluid that the object displaces. The concept of buoyancy is most commonly observed in the context of objects floating or sinking in water.
The key principle behind buoyancy is Archimedes' principle, named after the ancient Greek mathematician and scientist Archimedes. According to this principle, when an object is immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid it displaces. This buoyant force acts in the opposite direction to the force of gravity, and if it is greater than the object's weight, the object will float.

The mathematical expression for buoyant force (F_B) is given by: F_B = \( \rho _f .g.v_{fluiddisplaced} \)  
Where:
\( \rho _f \) is the density of the fluid,
 g is the acceleration due to gravity,
  \( v_{fluiddisplaced} \) is the volume of the fluid displaced by the object.
If the object's weight is less than the buoyant force, it will float; if it is equal, it will remain suspended at a certain depth; and if it is greater, the object will sink.