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Chapter 11: Problem 13

Effect of diving on blood. It is reasonable to assume that the bulk modulus of blood is about the same as that of water ( \(2.2 \mathrm{GPa}\) ). As one goes deeper and deeper in the ocean, the pressure increases by \(1.0 \times 10^{4} \mathrm{~Pa}\) for every meter below the surface. (a) If a diver goes down \(33 \mathrm{~m}\) (a bit over \(100 \mathrm{ft}\) ) in the ocean, by how much does each cubic centimeter of her blood change in volume? (b) How deep must a diver go so that each drop of blood compresses to half its volume at the surface? Is the ocean deep enough to have this effect on the diver?

Short Answer

Expert verified
Volume decreases by \(-1.5 \times 10^{-4} \; \text{cm}^3\) at 33 m. The ocean is not deep enough for blood to compress to half its volume.

Step by step solution

01

Understanding the Bulk Modulus Concept

The bulk modulus is a measure of a substance's resistance to uniform compression. In this problem, the bulk modulus of blood is given as that of water, which is approximately \( 2.2 \times 10^9 \; \text{Pa} \). The formula for the bulk modulus is:\[K = -V \frac{\Delta P}{\Delta V}\]where \( K \) is the bulk modulus, \( \Delta P \) is the change in pressure, \( V \) is the original volume, and \( \Delta V \) is the change in volume.
02

Calculating Pressure Increase at 33 Meters Depth

The pressure increase per meter of depth is \(1.0 \times 10^{4} \; \text{Pa/m} \). Therefore, at \(33 \; \text{m}\), the change in pressure is:\[\Delta P = 33 \times 1.0 \times 10^{4} = 3.3 \times 10^5 \; \text{Pa}\]This additional pressure will be applied to each cubic centimeter of blood.
03

Determining Volume Change for Blood at 33 Meters

Using the bulk modulus formula:\[2.2 \times 10^9 = -1 \frac{3.3 \times 10^5}{\Delta V}\]We solve for \( \Delta V \) (given \( V = 1 \; \text{cm}^3 = 1 \; \text{ml} \), the original volume):\[\Delta V = \frac{-3.3 \times 10^5}{2.2 \times 10^9} \approx -1.5 \times 10^{-4} \; \text{cm}^3\]This indicates a small decrease in volume.
04

Finding Depth for Blood Volume to Halve

To find the depth where blood compresses to half its volume, we need \( \Delta V = -0.5 \; \text{cm}^3 \). Using the bulk modulus:\[2.2 \times 10^9 = -1 \frac{P}{-0.5}\]Solving for \( P \):\[P = 2.2 \times 10^9 \times 0.5 = 1.1 \times 10^9 \; \text{Pa}\]The pressure increase per meter is \(1.0 \times 10^4 \; \text{Pa/m} \).\[\text{Depth} = \frac{1.1 \times 10^9}{1.0 \times 10^4} = 1.1 \times 10^5 \; \text{m}\]
05

Final Evaluation of Ocean Depths

Considering the calculation, \(1.1 \times 10^5 \; \text{m} \) is much deeper than any ocean on Earth, as average ocean depth is about 4,000 meters. Thus, it is not possible for the ocean to compress blood to half its volume under normal conditions.

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Key Concepts

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

Pressure Increase in Diving
When diving deep into the ocean, a diver experiences an increase in pressure due to the weight of the water above them. Water is much heavier than air, and as you dive deeper, the amount of water and, consequently, the pressure increases. This increase happens in a very predictable way: every meter deeper you go, the pressure increases by approximately 10,000 pascals (Pa). This means at 33 meters underwater, the pressure is 330,000 Pa greater than at the surface.
This additional pressure can affect different aspects of a diver's physiology, particularly influencing how their body handles internal and external pressure changes during a dive.
Compression of Fluids
Fluids, like blood, have an interesting property known as "bulk modulus," which describes how compressible a substance is under pressure. The bulk modulus of blood is similar to that of water, set at approximately 2.2 GPa.
This measure indicates blood's resistance to being compressed. Utilizing the bulk modulus formula, we can calculate how much a given volume of blood (or any liquid) changes under certain pressures. For example, if a diver goes to a depth of 33 meters, the increase in pressure leads to a tiny decrease in the volume of each cubic centimeter of blood鈥攕pecifically, about 0.00015 cm鲁 per cm鲁.
This change in volume is slight, but it's significant for understanding fluid behaviors under pressure.
Effects of Depth on Volume
As the diver descends deeper beneath the ocean surface, the effect of increasing pressure becomes more pronounced, especially on the volume of compressible fluids like blood. However, it's crucial to note that the changes are small unless extreme depths are reached.
To halve the volume of blood, the pressure needed is enormous. Calculations show that this would require diving to around 110,000 meters deep. Since no part of the Earth's oceans are that deep (they max out around 11,000 meters), such compression to half its volume is practically impossible in natural conditions.
Understanding these effects is vital for divers and people studying fluid dynamics as it addresses safety and physiological impacts on the body in underwater environments.

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Most popular questions from this chapter

Human hair. According to one set of measurements, the tensile strength of hair is 196 MPa, which produces a maximum strain of 0.40 in the hair. The thickness of hair varies considerably, but let's use a diameter of \(50 \mu \mathrm{m}\). (a) What is the magnitude of the force giving this tensile stress? (b) If the length of a strand of the hair is \(12 \mathrm{~cm}\) at its breaking point, what was its unstressed length?

Shear forces are applied to a rectangular solid. The same forces are applied to another rectangular solid of the same material, but with three times each edge length. In each case, the forces are small enough that Hooke's law is obeyed. What is the ratio of the shear strain for the larger object to that of the smaller object?

An astronaut notices that a pendulum that took \(2.50 \mathrm{~s}\) for a complete cycle of swing when the rocket was waiting on the launch pad takes \(1.25 \mathrm{~s}\) for the same cycle of swing during liftoff. What is the acceleration of the rocket? (Hint: Inside the rocket, it appears that \(g\) has increased.)

Weight lifting. The legs of a weight lifter must ultimately support the weights he has lifted. A human tibia (shinbone) has a circular cross section of approximately \(3.6 \mathrm{~cm}\) outer diameter and \(2.5 \mathrm{~cm}\) inner diameter. (The hollow portion contains marrow.) If a \(90 \mathrm{~kg}\) lifter stands on both legs, what is the heaviest weight he can lift without breaking his legs, assuming that the breaking stress of the bone is \(200 \mathrm{MPa}\) ?

A toy is undergoing SHM on the end of a horizontal spring with force constant \(300.0 \mathrm{~N} / \mathrm{m} .\) When the toy is \(0.120 \mathrm{~m}\) from its equilibrium position, it is observed to have a speed of \(3 \mathrm{~m} / \mathrm{s}\) and a total energy of \(4.4 \mathrm{~J}\). Find (a) the mass of the toy, (b) the amplitude of the motion, and (c) the maximum speed attained by the toy during its motion.
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