Question

The maximum depth ${\mathbf{d}}_{\mathbf{max}}$that a diver can snorkel is set by the density of the water and the fact that human lungs can function against a maximum pressure difference (between inside and outside the chest cavity) of $\mathbf{0}\mathbf{.}\mathbf{050}\mathbf{}\mathbf{atm}\mathbf{.}$What is the difference in${\mathbf{d}}_{\mathbf{max}}$for fresh water and the water of the Dead Sea (the saltiest natural water in the world, with a density of$\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{}\mathbf{kg}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}\mathbf{)}\mathbf{}\mathbf{?}$

Short answer

Difference in depth of fresh water and water in Dead Sea is$0.17m$

Step-by-step solution

The given data

(i) Density of dead sea water,$\rho =1500kg/m^3$

(ii) Density of fresh water,$\rho =998kg/m^3$

(iii) Maximum pressure,$p=0.05atmor5056Pa$

Understanding the concept of hydrostatic pressure

Using the formula for hydrostatic pressure difference in terms of density and height, we can find the height d for fresh water and the Dead Sea water and find the difference. Hydrostatic pressure is that, which is exerted by a fluid at any point of time at equilibrium due to the force applied by gravity.

Formula:

Pressure applied on a fluid surface, $p=\rho gh$ (i)

Net pressure applied on a body, $∆p=p_2-p_1$ (ii)

Calculation of pressure difference

Using equation (i), the depth can be given as:

$d=\frac{p}{pg}$

Hence the difference in depth for fresh water and dead water sea is as follows:

$$\begin{gathered} d_{max}=\frac{p}{p_{w}g}-\frac{p}{p_{sea}g} \\ =\frac{5056}{9.8}\left(\frac{1}{998}-\frac{1}{1500}\right) \\ =0.17m \end{gathered}$$

Hence, the value of d_max is 0.17 m.