(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 =
Vin σ g
⇒ v ρ g = Vin σ g,
V ρ = Vin σ
Where Vin
is the volume of body in the liquid |
Important Points:
- A body will float in liquid only and only if ρ ≤ σ
- In case of floating as weight of body = up thrustSo WApp = Actual weight – up thrust = 0
- In case of floating Vρg = Vin σ gSo 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:
- When a body floats then the weight of body = Up thrustVρg = Vin σ g ⇒ Vin = (ρ/ σ) Vi.e., Fraction of volume outside the liquid fout = Vout/ V = (1 – [ρ/ σ])
∴ Vout = V - Vin = (1 – [ρ/ σ]) V - For floatation V ρ = Vin σ ⇒ ρ = [Vin/ V] σ = fin σ
- If two different bodies A and B are floating in the same liquid
then ρA/ ρB = (fin)A / (fin)B
- If the same body is made to float in different liquids of densities σA
and σB respectively.V ρ = (Vin)A σA = (Vin)B σB
∴ σA/ σB = (Vin)B/ (Vin)A - If a platform of mass M and cross-section A is
floating in a liquid of density σ with its height h inside the liquidMg = 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.
