Laws Of Flotation

Laws Of Flotation

FLOTATION
When a body of density \(\rho_b\) and volume V is immersed in a liquid of density \(\rho_L\),
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) \( \rho _b > \rho _L \)
In this situation, 'W' is more than 'B' and hence the body will sink normal to the free surface
b) \( \rho _b = \rho _L \)
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) \( \rho _b < \rho _L \)
In this situation, W Then, the new upthurst force on the body is B¹ = W\(\to\) (i) (from translational equilibrium)
             

But from the expression for the thrust force, \( B^1 = V_{in} \rho _L g \to (ii) \)
    From (i) and (ii)  \( V\rho _b = V_{in} \rho _L \) \( \Rightarrow V_{in} = (\rho _b /\rho _l )V \)  

From these, it is clear that 
1)The body will float in a liquid only if \( \rho _b \leqslant \rho _L \)
2) When the body is floating, weight of the body is equal to the upthurst i.e,\( V\rho _B = V_{in} \rho _L \)
3)The apparant weight of an isolated floating body is zero.