Problem 34 - College Physics

Question

The pressure exerted by a person's lungs can be measured by having the person blow as hard as possible into one side of a manometer. If a person blowing into one side of an open tube manometer produces an \(80-\mathrm{cm}\) difference between the heights of the columns of water in the manometer arms, what is the gauge pressure of the lungs?

Step-by-Step Solution

Answer

The gauge pressure of the lungs is 7848 Pa or approximately 80 cmH2O.

Step 1: Understanding the Problem

We need to find the gauge pressure exerted by the person's lungs based on the given height difference of the water columns in a manometer. This is a typical manometer pressure problem where the height difference helps determine the pressure.

Step 2: Identify Known Values

The height difference between the columns of water is given as 80 cm. Since the liquid in the manometer is water, its density is approximately \(1000 \, \text{kg/m}^3\). The acceleration due to gravity \(g\) is \(9.81 \, \text{m/s}^2\).

Step 3: Convert Units

Convert the height difference from centimeters to meters for consistency in SI units: \( 80 \, \text{cm} = 0.8 \, \text{m} \).

Step 4: Apply the Manometer Formula

The gauge pressure \(P_g\) in a manometer is determined by the formula: \[ P_g = \rho \cdot g \cdot h \] where \(\rho\) is the water density, \(g\) is the gravitational acceleration, and \(h\) is the height difference.

Step 5: Calculate the Gauge Pressure

Substitute the known values into the formula: \[ P_g = 1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 0.8 \, \text{m} \]Solving this gives: \[ P_g = 7848 \, \text{Pa} \]

Step 6: Convert the Pressure to cmH2O

To express the pressure in cm of water (often used for respiratory measurements), recognize that 1 Pa equals 0.0102 cmH2O. Thus, \[ P_g = 7848 \, \text{Pa} \times 0.0102 \, \text{cmH2O/Pa} \approx 80 \, \text{cmH2O} \]

Key Concepts

Gauge Pressure

Gauge pressure is an essential concept in understanding how pressure measurements are made relative to the surrounding atmosphere, rather than an absolute reference point. When we talk about gauge pressure, we refer to the pressure of the system compared to atmospheric pressure.

Gauge pressure is often used because it conveniently tells us whether the inside of a container is under higher or lower pressure than the outside. For instance, when measuring tire pressure, we use gauge pressure to see how inflated or deflated it is compared to the air around it.

• It does not take atmospheric pressure into account, as opposed to absolute pressure, which includes it.
• Gauge pressure can have both positive and negative values.

In our exercise on manometer readings, the gauge pressure measured represents the pressure exerted by the person's lungs above normal atmospheric pressure. This measurement helps judge the effort personnel can exert through breathing.

Manometer

A manometer is a simple yet effective device used to measure pressure levels in fluids. In our exercise, we work with an open tube manometer, which features two columns of liquid whose heights can vary depending on pressure conditions applied at one end.

Manometers rely on the principle that the pressure exerted by a fluid is related to its depth, or height, hence height differences can indicate pressure differences.

• They can measure both positive and negative gauge pressure.
• The liquid often used in manometers is water, or sometimes mercury, due to their known properties.

When the person blows into one side of the manometer, the difference in height between the two columns directly indicates the gauge pressure of the lungs. This method is often employed in physiology for simple and quick measurement of respiratory pressures.

Fluid Mechanics

Fluid mechanics is the branch of physics that deals with the behavior of fluids (liquids and gases) at rest and in motion. Understanding it is essential for solving problems like those involving pressure in a manometer.

In the context of the manometer exercise, fluid mechanics principles such as the balance of forces due to gravity on the fluid columns help determine pressure differences.

• The pressure exerted by a column of fluid is proportional to its height, a principle central to understanding manometer function.
• Other forces, like atmospheric pressure, also affect fluid motion and pressure observations.

By applying these core principles, along with the known density of the fluid and the height difference in a manometer, we can accurately measure pressures, such as the gauge pressure that was calculated for the lung capacity in this exercise.

Unit Conversion

Unit conversion is a vital part of solving real-world physics problems, ensuring all measurements are consistent and comparable. In our manometer problem, converting units assists in using the standard SI units needed to calculate gauge pressure.

The exercise begins with a height difference given in centimeters, which is not an SI base unit. Converting this to meters aligns with the SI unit system to coherently calculate pressure.

• 1 cm = 0.01 meters
• Understanding different units and converting them properly is crucial in applying formulas correctly.

Finally, the pressure was converted from Pascals to cmH2O, a more meaningful unit for certain applications like respiratory mechanics. Thus, recognizing and converting between these units ensures precise and useful pressure measurement in various fields.