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Chapter 7: Problem 1028

By sucking through a straw, a student can reduce the pressure in his lungs to \(750 \mathrm{~mm}\) of \(\mathrm{Hg}\) (density $\left.=13.6\left(\mathrm{gm} / \mathrm{cm}^{2}\right)\right)$ using the straw, he can drink water from \(\mathrm{a}\) glass up to a maximum depth of (A) \(10 \mathrm{~cm}\) (B) \(75 \mathrm{~cm}\) (C) \(13.6 \mathrm{~cm}\) (D) \(1.36 \mathrm{~cm}\)

Short Answer

Expert verified
In this problem, we need to find the maximum depth of water a student can drink using a straw after reducing the pressure in his lungs. The pressure reduction is equal to \(1 cm\) of Hg. Using the relation \(h'= h \times \frac{{\rho_{Hg}}}{{\rho_{H2O}}}\), where \(h'\) is the height of water column, \(h\) is the height of mercury column, \(\rho_{Hg}\) is the density of mercury, and \(\rho_{H2O}\) is the density of water, we find that the student can drink water up to a maximum depth of \(13.6 cm\). Thus, the correct answer is (C) \(13.6 cm\).

Step by step solution

01

Understand the Problem

In this problem, a student reduces the pressure in his lungs by sucking through a straw, and as a result, he can drink water from a glass up to a certain maximum depth. We are asked to find this depth. The problem involves the concept of atmospheric pressure and its relationship with a column of liquid. The atmospheric pressure is equal to the pressure exerted by a column of mercury of height 760 mm, and the student can reduce this to 750 mm of Hg.
02

Choose the Correct Concept

To solve this problem, we need to use the concept that the amount of pressure reduction corresponds to the height of water column that can be sucked up. In essence, we know the change in pressure in mercury units (760mm-750mm=10mm), our task is to convert this change in pressure to the height in water column units.
03

Convert Units

Before we make the conversion, we need to change the pressure change measure to cm (the same height units as water) since our options are also in cm units. Given \(1 mm = 0.1 cm\), hence, the pressure change is \(10 mm = 1 cm\) of Hg.
04

Use Pressure and Density to Find Height

We will use the relation \(h'= h \times \frac{{\rho_{Hg}}}{{\rho_{H2O}}}\), where \(h'\) is the height of water column, \(h\) is the height of mercury column, \(\rho_{Hg}\) is the density of mercury and \(\rho_{H2O}\) is the density of water. Using the given densities, we convert pressure reduction in Hg to equivalent height in water. Simply substitute the values we have into the equation: \(h' = 1 \times \frac{{13.6 gm/cm^3}}{{1 gm/cm^3}}\), which simplifies to \(h' = 13.6 cm\). Thus, the student can drink water from a glass up to a maximum depth of 13.6 cm. Therefore, the correct answer is (C) \(13.6 cm\).

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