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Chapter 11: Problem 25

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
Approximately 0.504 meters.

Step by step solution

01

Understanding the Problem

The diver can safely swim until the pressure difference between her lungs and the water equals one-twentieth of an atmosphere. The task is to find this depth in salt water where this pressure difference occurs.
02

Convert Atmospheric Pressure

One atmosphere is equivalent to 101,325 Pascals. Therefore, one-twentieth of an atmosphere is \( \frac{101325}{20} = 5066.25 \) Pascals.
03

Calculate the Pressure Increase with Depth

The pressure increase in water with depth is given by the equation \( \Delta P = \rho gh \), where \( \rho \) is the density of the water (1025 kg/m鲁), \( g \) is the acceleration due to gravity (9.81 m/s虏), and \( h \) is the depth.
04

Set Up the Equation

Set \( \rho gh = 5066.25 \) to find \( h \). Substituting the known values, we have 1025 kg/m鲁 \( \times \) 9.81 m/s虏 \( \times \) \( h = 5066.25 \) Pa.
05

Solve for Depth

Rearrange the equation to solve for \( h \): \( h = \frac{5066.25}{1025 \times 9.81} \). Calculate \( h \).
06

Calculate the Depth

Perform the calculation: \( h = \frac{5066.25}{10060.25} \approx 0.504 \) meters. This means the diver can safely descend to approximately 0.504 meters below the water using a snorkel.

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Key Concepts

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

Atmospheric Pressure
Atmospheric pressure is the weight of the air above us in the atmosphere, pressing down on everything at the surface. It is crucial in many calculations involving gases and liquids. On average, at sea level, this pressure is about 101,325 Pascals or 1 atmosphere. It's what makes breathing possible and holds a lot of influence in how we experience our environment. Small variations in atmospheric pressure can significantly affect physical activities like snorkeling or diving. Moreover, changes in elevation or submersion in water affect the pressure experienced, impacting our bodies differently due to these alterations.
Pressure Difference
Pressure difference refers to the variation in pressure between two points. In the context of diving or snorkeling, it is the difference between the air pressure that the lungs are in contact with and the pressure of the surrounding water on the body. This difference is essential because the human body can only withstand certain limits to ensure efficient and safe functioning. For a diver, it's critical not to exceed the safe threshold of pressure difference. In our example, this limit is a maximum of one-twentieth of an atmosphere, which translates to 5066.25 Pascals. This is the allowable pressure difference that the lungs can manage while snorkeling.
Water Density
Water density is a vital factor in calculations of pressure at different depths. Density is essentially a measure of how much mass is contained in a given volume. For fluids like water, it helps to determine how pressure changes with depth. Saltwater, which has a density of approximately 1025 kg/m鲁, is denser than freshwater due to its dissolved salts. The density of water influences the pressure exerted at lower depths, following the principle that the deeper you go, the more weight of the water is above, increasing the pressure. This is why knowing water density is crucial when calculating how deep a diver can safely go.
Snorkeling Physics
Snorkeling physics revolves around understanding how the body interacts with the water and air while submerged. It involves various principles, with fluid pressure and buoyancy being key components. A diver using a snorkel breathes through a tube extending to the air above the water, aligning with the atmospheric pressure. The diver can safely descend only as far as their lungs can offset the increased water pressure, determined by the physics of the pressure difference and the body's capacity to endure it. By using the formula for pressure increase with depth, snorkelers can calculate safe diving depths and ensure that they are not exceeding the body's natural pressure tolerance limits. Understanding these principles ensures safety and enjoyment while exploring underwater environments.

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Most popular questions from this chapter

A mercury barometer reads \(747.0\) \(\mathrm{mm}\) on the roof of a building and \(760.0\) \(\mathrm{mm}\) on the ground. Assuming a constant value of \(1.29\) \(\mathrm{kg} / \mathrm{m}^{3}\) for the density of air, determine the height of the building.

A patient recovering from surgery is being given fluid intravenously. The fluid has a density of \(1030 \mathrm{kg} / \mathrm{m}^{3},\) and \(9.5 \times 10^{-4} \mathrm{m}^{3}\) of it flows into the patient every six hours. Find the mass flow rate in \(\mathrm{kg} / \mathrm{s}\).

The Mariana trench is located in the floor of the Pacific Ocean at a depth of about \(11000\) \(\mathrm{m}\) below the surface of the water. The density of seawater is \(1025\) \(\mathrm{kg} / \mathrm{m}^{3} .\) (a) If an underwater vehicle were to explore such a depth, what force would the water exert on the vehicle's observation window (radius \(=0.10\) \(\mathrm{m}) ?\) (b) For comparison, determine the weight of a jetliner whose mass is \(1.2 \times 10^{5} \mathrm{kg}\).

(a) The mass and the radius of the sun are, respectively, \(1.99 \times 10^{30} \mathrm{kg}\) and \(6.96 \times 10^{8} \mathrm{m} .\) What is its density? (b) If a solid object is made from a material that has the same density as the sun, would it sink or float in water? Why? (c) Would a solid object sink or float in water if it were made from a material whose density was the same as that of the planet Saturn (mass \(=5.7 \times 10^{26} \mathrm{kg},\) radius \(\left.=6.0 \times 10^{7} \mathrm{m}\right) ?\) Provide a reason for your answer.

Two identical containers are open at the top and are connected at the bottom via a tube of negligible volume and a valve that is closed. Both containers are filled initially to the same height of \(1.00\) \(\mathrm{m}\), one with water, the other with mercury, as the drawing indicates. The valve is then opened. Water and mercury are immiscible. Determine the fluid level in the left container when equilibrium is reestablished.
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