What fraction of the volume of an iceberg (density 917 kg/m3) would be visible if the iceberg floats in (a) the ocean (salt water, density 1024 kg/m3) and (b) in a river (fresh water, density 1000 kg/m3) ? (When salt water freezes to form ice, the salt is excluded. So, an iceberg could provide fresh water to a community.)

a) 10.4% b) 8.3%

Explanation:

According to Archimedes' principle, any object submerged in a liquid, receives an upward force (called buoyant force) numerically equal to the weight of the volume removed by the liquid.

When this force is equal to the force of gravity on the object (which we call weight, always downward) the object floats in the liquid.

Let's see what happens in the two situations we have:

a) salt water (density 1024 Kg/m³)

We can express the force of gravity on the iceberg, as follows:

Fg = mice*g = ρice*Vice*g

For the Fb (buoyant force), we have:

Fb = ρsw*Vsub*g, where Vsub, is the fraction of the iceberg volume that is submerged.

So we can replace Vsub as follows:

Vsub = k Vice, where k is the fraction submerged, i. e. 0
Therefore, if we need that the iceberg float, we need that Fg=Fb, as follows:

Fg = ρice*Vice*g = Fb = ρsw*k*Vice*g (1)

Simplifying common terms, we have:

ρice/ρsw = k ⇒917 kg/m³ / 1,024 kg/m³ = 0.896

This means that 89.6% of the volume is submerged, i. e., is not visible, so the visible percentage is just the difference between 100% and 89.6%:

Vvis = 100% - 91,7% = 10.4%

b) For fresh water, we can do the same procedure, repeating (1), with the value for density of fresh water, as follows:

Fg = ρice*Vice*g = Fb = ρsw*k*Vice*g

Simplifying common terms, we have:

ρice/ρsw = k ⇒917 kg/m³ / 1,000 kg/m³ = 0.917

This means that 91.7 % of the volume is submerged, i. e., is not visible, so the visible percentage is just the difference between 100% and 91.7%:

Vvis = 100% - 91.7 = 8.3%