Question

What fraction of the volume of an iceberg (density $\mathbf{917}\mathbf{}\mathit{k}\mathit{g}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$) would be visible if the iceberg floats: (a) in the ocean (salt water, density) and (b) in a river (fresh water, density $\mathbf{1000}\mathbf{}\mathit{k}\mathit{g}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$)? (When salt water freezes to form ice, the salt is excluded. So, an iceberg could provide fresh water to a community.)

Short answer

a) The volume fraction of an iceberg, which is visible if the iceberg floats in ocean, is$0.11$

b) The volume fraction of an iceberg, which is visible if the iceberg floats in river, is$0.083$

Step-by-step solution

The given data

i) The density of iceberg,${\mathrm{\rho }}_{\mathrm{i}}=917\frac{\mathrm{kg}}{{\mathrm{m}}^3}$

ii) The density of saltwater,${\mathrm{\rho }}_{\mathrm{s}}=1024\frac{\mathrm{kg}}{{\mathrm{m}}^3}$

iii) The density of freshwater,${\rho }_f=1000\frac{\mathrm{kg}}{{\mathrm{m}}^3}$

Understanding the concept of Archimedes Principle

We can use Archimedes’ principle to find the volume fraction of an iceberg, which is visible if the iceberg floats in the ocean or river in terms of densities of iceberg and freshwater. Then putting values for densities in that expression, we can get its magnitudes.

Formulae:

Downward force due to gravity, ${\mathrm{F}}_{\mathrm{g}}={\mathrm{m}}_{\mathrm{f}}\mathrm{g}$(i)

Density of a substance, $\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{V}}$(ii)

a) Calculation of fractional volume, if iceberg floats in ocean

The total volume of the iceberg is ${\mathrm{V}}_1$, and when it floats in the fluid, it displaces the same volume of the fluid. According to Archimedes’ principle and using equation (i), we get

Then the expression of mass for iceberg and fluid using equation (ii), we can write as

${\mathrm{m}}_{\mathrm{f}}\mathrm{g}={\mathrm{m}}_{\mathrm{i}}\mathrm{g}$

localid="1657550115952" ${\mathrm{m}}_{\mathrm{f}}={\mathrm{v}}_{\mathrm{i}}$

${\mathrm{m}}_{\mathrm{f}}={\mathrm{m}}_{\mathrm{i}}$

${\mathrm{m}}_{\mathrm{f}}={\mathrm{p}}_{\mathrm{f}}{\mathrm{V}}_{\mathrm{f}}$

Substituting the values of equations (a) & (b) in equation (1), we get${\mathrm{\rho }}_{\mathrm{i}}{\mathrm{V}}_{\mathrm{i}}={\mathrm{\rho }}_{\mathrm{i}}{\mathrm{V}}_{\mathrm{t}}$

$\frac{{\mathrm{\rho }}_{\mathrm{i}}}{{\mathrm{\rho }}_{\mathrm{f}}}=\frac{{\mathrm{V}}_{\mathrm{f}}}{{\mathrm{V}}_{\mathrm{j}}}$

$1-\frac{{\mathrm{\rho }}_{\mathrm{i}}}{{\mathrm{\rho }}_{\mathrm{f}}}=1-\frac{{\mathrm{V}}_{\mathrm{p}}}{{\mathrm{V}}_{\mathrm{i}}}$

Hence, the volume fraction can be given as:

$1-\frac{{\mathrm{V}}_{\mathrm{f}}}{{\mathrm{V}}_{\mathrm{i}}}=1-\frac{{\mathrm{\rho }}_{\mathrm{i}}}{{\mathrm{\rho }}_{\mathrm{f}}}$

If the iceberg floats in ocean, then the volume fraction is

$1-\frac{{\mathrm{V}}_{\mathrm{f}}}{{\mathrm{V}}_{\mathrm{i}}}=1-\frac{917\frac{\mathrm{kg}}{{\mathrm{m}}^3}}{1024\frac{\mathrm{kg}}{{\mathrm{m}}^3}}$

$=0.11$

$=11\%$

Hence, the fractional volume in case of ocean is$0.11$

b) Calculation of fractional volume, if iceberg floats in river

If the iceberg floats in river, then the volume fraction is

$1-\frac{{\mathrm{V}}_{\mathrm{f}}}{{\mathrm{V}}_{\mathrm{i}}}=1-\frac{917\frac{\mathrm{kg}}{{\mathrm{m}}^3}}{1000\frac{\mathrm{kg}}{{\mathrm{m}}^3}}$

$=0.083$

$=8.3\%$

Hence, the fractional volume in case of river is $0.083$