Problem 14 - College Physics

Question

\(\bullet\) 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 \(1.0 \times 10^{4}\) 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?

Step-by-Step Solution

Answer

(a) Volume changes by -0.00015 cm³. (b) 110 km depth needed; ocean not that deep.

Step 1: Understand the Bulk Modulus Formula

The bulk modulus, denoted as \( K \), is a measure of a material's resistance to uniform compression. It is defined by the formula \( K = - V \frac{\Delta P}{\Delta V} \), where \( \Delta P \) is the change in pressure, \( \Delta V \) is the change in volume, and \( V \) is the initial volume.

Step 2: Calculate the Change in Pressure for 33m Depth

The change in pressure \( \Delta P \) for every meter is given as \( 1.0 \times 10^{4} \, \text{Pa/m} \). For a depth of 33 meters, \( \Delta P = 33 \times 1.0 \times 10^{4} = 3.3 \times 10^{5} \) Pa.

Step 3: Calculate Volume Change at 33m Depth

Using the bulk modulus formula \( K = 2.2 \times 10^{9} \) Pa for blood, we rearrange to find \( \Delta V = - V \frac{\Delta P}{K} \). Assuming the initial volume \( V \) is 1 cm³:\[ \Delta V = -1 \times \frac{3.3 \times 10^5}{2.2 \times 10^9} \, \text{cm}^3 = -1.5 \times 10^{-4} \, \text{cm}^3. \] Hence, each cubic centimeter decreases by approximately \( 0.00015 \) cm³.

Step 4: Calculate the Depth for Blood to Compress to Half Volume

To compress the blood to half its volume \( \frac{V}{2} \), from the bulk modulus, we need \( \Delta V = \frac{V}{2} \). Thus, \(-\frac{\Delta P}{K} = \frac{1}{2} \). Solving for \( \Delta P \):\[ \Delta P = \frac{K}{2} = \frac{2.2 \times 10^9}{2} = 1.1 \times 10^9 \, \text{Pa}. \]Checking the depth for this pressure increment, \( \Delta P = 1.0 \times 10^4 \times \text{Depth in meters} \). Thus depth = \( \frac{1.1 \times 10^9}{1.0 \times 10^4} = 110,000 \, \text{m} = 110 \, \text{km}. \)

Key Concepts

Pressure in Fluids

In the world of physics, pressure refers to the force exerted by a fluid per unit area on its container or any object submerged within it. The ocean is a vast body where such pressure is markedly felt as one goes deeper.
Each meter of depth in the ocean adds

• 10,000 Pascals (Pa) more pressure

to the diver's environment. This increase is due to the weight of the water above pressing down.
Understanding this pressure is crucial for divers, engineers, and anyone working with fluid systems. It not only affects the equipment and physical endurance of humans but also plays a significant role in the study of fluid dynamics and marine biology.

Volume Change

Volume change in a fluid can occur when it is subjected to pressure, temperature, or other external influences. In this context, the focus is on the compressive change in volume due to pressure.
Compressing a fluid like blood doesn't yield huge changes in volume due to its bulk modulus, yet small adjustments impact calculations and operations significantly.

• For blood descending 33 meters, find that the volume of each cubic centimeter decreases by about 0.00015 cm³.

This value demonstrates the resilience and slight compressibility of fluids like blood, where massive pressure changes correlate to minute volume adjustments.

Compression

Compression refers to the reduction in volume of a substance when pressure is applied. Understanding the compression behavior using concepts like the bulk modulus is vital in many fields, including underwater exploration and medical applications.
The bulk modulus measures how resistant a material is to compression. For fluids,

• such as blood, the bulk modulus is similar to water, around 2.2 gigapascals (GPa).

This means that even though blood can withstand pressure, it does experience compression that alters its original volume slightly.
Knowledge of how materials compress is pivotal for designing medical devices and allocating safety parameters for divers.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases proportionally with depth, meaning the deeper you descend into a body of fluid, the greater the pressure exerted on you.
In diving scenarios, hydrostatic pressure determines how deep one can safely dive without experiencing adverse effects due to pressure changes.

• A critical understanding is that to compress an object to half its initial volume, immense depths (e.g., 110 km as calculated for blood) are required.

While practically infeasible in bodies like the ocean, it underscores the importance of hydrostatic pressure management in design and safety measures for sub-aquatic activities.