Chapter 10, Problem 71GP

To determine

To calculate: The height of the water.

Answer to Problem 71GP

The height of the water is found to be $14.2\text{cm}$ .

Explanation of Solution

Given:

The height of the alcohol = $18\text{cm}$

The density of alcohol = $790\text{kg}/{\text{m}}^{\text{3}}$

The density of water = $1000\text{kg}/{\text{m}}^3$ .

Formula used:

The pressure at a given depth for both liquid but starting at each liquid top with the atmospheric pressure is given as,

  $P_{alcohol}+{\rho }_{alcohol}g\Delta h_{alcohol}=P_{water}+{\rho }_{water}g\Delta h_{water}$

Where, $\Delta h_{water}$ is the water level, $\Delta h_{alcohol}$ is the alcohol level, and ${\rho }_{alcohol}and{\rho }_{water}$ are the density for the alcohol and water respectively.

Calculation:

At the top of each liquid, the pressure will be atmospheric and at the place where the liquid meets, the pressure must be the same. The pressure at a given depth for both liquid but starting at each liquid top with the atmospheric pressure is given as,

  $\begin{array}{l}{P}_{alcohol}+{\rho }_{alcohol}g\Delta h_{alcohol}=P_{water}+{\rho }_{water}g\Delta h_{water}\\ \Rightarrow \Delta h_{water}=\Delta h_{alcohol}\frac{{\rho }_{alcohol}}{{\rho }_{water}}\end{array}$

By substituting the values in this formula, the water level is given as,

  $\Delta h_{water}=18\text{cm}\times \left(0.79\right)=14.2\text{cm}$

Conclusion:

Thus, the height of the water is found to be $14.2\text{cm}$ .