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Chapter 11: Problem 24

The human lungs can function satisfactorily up to a limit where the pressure difference between the outside and inside of the lungs is one-twentieth of an atmosphere. If a diver uses a snorkel for breathing, how far below the water can she swim? Assume the diver is in salt water whose density is \(1025 \mathrm{~kg} / \mathrm{m}^{3}\)

Short Answer

Expert verified
The diver can swim approximately 0.507 meters below the water using a snorkel.

Step by step solution

01

Understand the Problem

The problem requires finding out how far below the water a diver can swim while using a snorkel, given the maximum allowable pressure difference between the outside and inside of the lungs is one-twentieth of an atmosphere. In saltwater, with a known density, we need to find the depth at which this pressure difference is reached.
02

Pressure Difference Condition

The difference in pressure the diver can handle is one-twentieth of an atmosphere. Since standard atmospheric pressure is approximately 101325 Pa, divide by 20: \( \frac{101325}{20} = 5066.25 \) Pa.
03

Using the Pressure Equation

The pressure due to a liquid column is given by the formula \( P = \rho gh \), where \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity, and \( h \) is the depth. Substitute \( \rho = 1025 \) kg/m鲁, \( g = 9.8 \) m/s虏, and \( P = 5066.25 \) Pa into the equation to solve for \( h \): \( 5066.25 = 1025 \times 9.8 \times h \).
04

Solve for Depth

Rearrange the equation \( 5066.25 = 1025 \times 9.8 \times h \) to find \( h \): \[ h = \frac{5066.25}{1025 \times 9.8} \]Calculate \( h \).
05

Calculation Result

After performing the division, \( h \approx 0.507 \) meters. This is the maximum depth below the water the diver can swim using a snorkel while maintaining a safe pressure difference.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pressure Difference
The concept of pressure difference is vital in understanding how different pressures impact activities like diving. Pressure difference refers to the change in pressure between two points. In this problem, it is particularly important because the human lungs can only withstand a limited pressure difference when breathing through a snorkel underwater.

Pressure is usually measured in pascals (Pa), and standard atmospheric pressure is about 101,325 Pa. For the diver, the pressure difference she can tolerate is one-twentieth of the atmospheric pressure. Therefore, the diver can handle a pressure difference of approximately 5,066.25 Pa. This calculation is essential in determining how deep the diver can safely go while using a snorkel.

Knowing this allowable pressure difference helps us use the pressure equation to find out how deep underwater one can swim without exceeding this limit.
Atmospheric Pressure
Atmospheric pressure is the force exerted by the weight of the air above us in the Earth's atmosphere. It plays a central role in various phenomena, including weather changes and our ability to breathe comfortably.

  • At sea level, atmospheric pressure is approximately 101,325 Pa.
  • This standard atmospheric pressure serves as a reference point for measuring other pressures, such as the pressure a diver experiences underwater.
When a diver uses a snorkel, atmospheric pressure helps push air down the snorkel into the lungs. However, as the diver goes deeper into the water, the outside pressure increases due to the weight of the water above, making it harder to breathe.

Understanding atmospheric pressure's influence is foundational for realizing why there is a limitation on how deep a diver can safely go while using a snorkel. The difference in pressure that the body can withstand is significantly determined by how the atmospheric pressure changes against the pressure underwater.
Density of Salt Water
The density of a fluid is another integral factor to consider when studying fluid mechanics, as it directly affects buoyancy and pressure.

  • In this exercise, the density of salt water is given as 1,025 kg/m鲁.
  • Density is defined as mass per unit volume and affects how much pressure a fluid exerts.
When submerged in water, the diver experiences an increased pressure due to the density of the water. The denser the water, the more pressure it exerts at a given depth.

The provided density number allows us to use the formula for calculating pressure under a liquid, which is \( P = \rho gh \). This formula incorporates the density of the fluid, \( \rho \), gravity, \( g \), and depth, \( h \). Understanding how density interacts with these variables lets us solve for the maximum depth the diver can safely achieve using a snorkel under specific conditions. This demonstrates the real-world application of understanding fluid density.

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Most popular questions from this chapter

A suitcase (mass \(m=16 \mathrm{~kg}\) ) is resting on the floor of an elevator. The part of the suitcase in contact with the floor measures \(0.50 \mathrm{~m}\) by \(0.15 \mathrm{~m} .\) The elevator is moving upward, the magnitude of its acceleration being \(1.5 \mathrm{~m} / \mathrm{s}^{2}\). What pressure (in excess of atmospheric pressure) is applied to the floor beneath the suitcase?

An airplane wing is designed so that the speed of the air across the top of the wing is \(251 \mathrm{~m} / \mathrm{s}\) when the speed of the air below the wing is \(225 \mathrm{~m} / \mathrm{s}\). The density of the air is \(1.29 \mathrm{~kg} / \mathrm{m}^{3} .\) What is the lifting force on a wing of area \(24.0 \mathrm{~m}^{2} ?\)

A paperweight, when weighed in air, has a weight of \(\mathrm{W}=6.9 \mathrm{~N}\). When completely immersed in water, however, it has a weight of \(W_{\text {in water }}=4.3 \mathrm{~N}\). Find the volume of the paperweight.

A hollow cubical box is \(0.30 \mathrm{~m}\) on an edge. This box is floating in a lake with one-third of its height beneath the surface. The walls of the box have a negligible thickness. Water is poured into the box. What is the depth of the water in the box at the instant the box begins to sink?

A ship is floating on a lake. Its hold is the interior space beneath its deck and is open to the atmosphere. The hull has a hole in it, which is below the water line, so water leaks into the hold. (a) How is the amount of water per second (in \(\mathrm{m}^{3} / \mathrm{s}\) ) entering the hold related to the speed of the entering water and the area of the hole? (b) Approximately how fast is the water at the surface of the lake moving? Justify your answer. (c) What causes the water to accelerate as it moves from the surface of the lake into the hole that is beneath the water line? Explain.
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