**(1)** **Translatory equilibrium: **When a body of density ρ and volume V is immersed in a liquid of density σ, the forces acting on the body are

Weight of body W = mg = Vρg, acting vertically downwards through centre of gravity of the body.

Up thrust force = Vσg acting vertically upwards through the centre of gravity of the displaced liquid i.e., center of buoyancy.

If density of body is greater than that of liquid ρ > σ
Weight will be more than up thrust so the body will sink. |
If density of body is equal to that of liquid ρ = σ
Weight will be equal to up thrust so the body will float fully submerged in neutral equilibrium anywhere in the liquid. |
If density of body is lesser than that of liquid ρ < σ
Weight will be less than up thrust so the body will move upwards and in equilibrium will float partially immersed in the liquid Such that, W = V => v ρ g = V |

**Important Points:**

**(i)** A body will float in liquid only and only if ρ ≤ σ

**(ii)** In case of floating as weight of body = up thrust

So W_{App} = Actual weight – up thrust = 0

**(iii)** In case of floating Vρg = V_{in} σ g

So the equilibrium of floating bodies is unaffected by variations in g though both thrust and weight depend on g.

**(2)** **Rotatory Equilibrium: **When a floating body is slightly tilted from equilibrium position, the centre of buoyancy B shifts. The vertical line passing through the new centre of buoyancy B¢ and initial vertical line meet at a point M called meta-centre. If the meta-centre M is above the centre of gravity the couple due to forces at G (weight of body W) and at (up thrust) tends to bring the body back to its original position. So for rotational equilibrium of floating body the meta-centre must always be higher than the centre of gravity of the body.However, if meta-center goes below CG, the couple due to forces at G and tends to topple the floating body.

That is why a wooden log cannot be made to float vertical in water or a boat is likely to capsize if the sitting passengers stand on it. In these situations CG becomes higher than MC and so the body will topple if slightly tilted.

**(3) Application of floatation:**

**(i)** When a body floats then the weight of body = Up thrust

Vρg = V_{in} σ g Þ V_{in} = (ρ/ σ) V

∴ V_{out} = V – V_{in} = (1 – [ρ/ σ]) V

i.e., Fraction of volume outside the liquid f_{out} = V_{out}/ V = (1 – [ρ/ σ])

**(ii)** For floatation V ρ = V_{in} σ Þ ρ = [V_{in}/ V] σ = f_{in} σ

If two different bodies A and B are floating in the same liquid then ρ_{A}/ ρ_{B} = (f_{in})_{A} / (f_{in})_{B}

**(iii)** If the same body is made to float in different liquids of densities σ_{A} and σ_{B} respectively.

V ρ = (V_{in})_{A} σ_{A} = (V_{in})_{B} σ_{B}

∴ σ_{A}/ σ_{B} = (V_{in})_{B}/ (V_{in})_{A}

**(iv)** If a platform of mass M and cross-section A is floating in a liquid of density σ with its height h inside the liquid

Mg = hA σg …… (i)

Now if a body of mass m is placed on it and the platform sinks by y then

(M + m)g = (y + h)A σg …… (ii)

Subtracting equation (i) and (ii),

mg = Aσyg, i.e., W α y …… (iii)

So we can determine the weight of a body by placing it on a floating platform and noting the depression of the platform in the liquid by it.