Chapter 10, Problem 33P

To determine

To Find: The fraction of an iceberg is above the surface of the water

Answer to Problem 33P

The fraction of iceberg above the water surface is $10.5\%$ .

Explanation of Solution

Given information:

For ice, $SG=0.917$

For sea water, $SG=1.025$

Density of water, ${\rho }_w=1000\mathrm{kg}/m^3$

Formula used:

• When an object is submerged into the fluid, then a force is provided by the fluid to the object, this force is known as Buoyant force.

This force is written as,

  $F_B={\rho }_{F}Vg$

• Specific gravity is the density of the substance divided by the water’s density, so,

  $\begin{array}{l}S{G}_a=\frac{{\rho }_a}{{\rho }_w}\\ {\rho }_a=S{G}_a{\rho }_w\end{array}$

Calculation:

Now, substitute the given values in the above formula for calculating the density for ice and sea water.

  $\begin{array}{l}S{G}_a=\frac{{\rho }_a}{{\rho }_w}\\ {\rho }_a=S{G}_a{\rho }_w\end{array}$

  $\begin{array}{l}{\rho }_{ice}=S{G}_{ice}{\rho }_w\\ {\rho }_{ice}=\left(1000\right)\times \left(0.917\right)=917\raisebox{1ex}{$kg$}\!\left/ \!\raisebox{-1ex}{$m^3$}\right.\end{array}$

And,

  ${\rho }_{seawater}=S{G}_{sw}{\rho }_w=\left(1000\right)\left(1.025\right)=1025\raisebox{1ex}{$kg$}\!\left/ \!\raisebox{-1ex}{$m^3$}\right.$

The net force is

  $\begin{array}{l}{F}_B-F_{ib}=0\\ F_B=F_{ig}\end{array}$

Now, the buoyant force for iceberg is

  $F_B={\rho }_{iceberg}\times V_{ib}\times g$

From the above two equation,

  $\begin{array}{l}{F}_B=F_{\begin{array}{l}ig\end{array}}\\ {\rho }_{sw}\times V_{sw}\times g={\rho }_{iceberg}\times V_{ib}\times g\\ {\rho }_{sw}\times V_{sw}={\rho }_{iceberg}\times V_{ib}\\ V_{sw}=\frac{{\rho }_{iceberg}}{{\rho }_{sw}}\times V_{ib}\end{array}$

Now, percentage of iceberg above water $=\frac{V_{above}}{V_{iceberg}}\times 100\%$

But, $V_{above}=V_{sw}-V_{ib}$

So,

  $\begin{array}{l}{V}_{above}=V_{ib}-V_{sw}\\ \%=\frac{V_{ib}-V_{sw}}{V_{iceberg}}\times 100\%\\ =\frac{V_{ib}-\frac{{\rho }_{iceberg}}{{\rho }_{sw}}\times V_{ib}}{V_{iceberg}}\times 100\%\\ =\left(1-\frac{{\rho }_{iceberg}}{{\rho }_{sw}}\right)\times 100\%\\ =\left(1-\frac{917}{1025}\right)\times 100\%=10.5\%\end{array}$

Conclusion:

The fraction of iceberg above the water surface is $10.5\%$ .