Problem 16 - College Physics

Question

\(\bullet\) \(\bullet\) Compression of human bone. The bulk modulus for bone is 15 GPa. (a) If a diver-in-training is put into a pressurized suit, by how much would the pressure have to be raised (in atmospheres) above atmospheric pressure to compress her bones by 0.10\(\%\) of their original volume? (b) Given that the pressure in the ocean increases by \(1.0 \times 10^{4}\) Pa for every meter of depth below the surface, how deep would this diver have to go for her bones to compress by 0.10\(\%\) ? Does it seem that bone compression is a problem she needs to be concerned with when diving?

Step-by-Step Solution

Answer

The pressure must increase by 148 atm, equivalent to diving to 1500 m depth; bone compression is not an issue for typical diving.

Step 1: Understanding Bulk Modulus

The bulk modulus (\(K\)) is defined as \(K = - \frac{\Delta P}{\frac{\Delta V}{V_0}}\), where \(\Delta P\) is the change in pressure, \(\Delta V\) is the change in volume, and \(V_0\) is the original volume. For a compression of 0.10% of the original volume, \(\frac{\Delta V}{V_0} = -0.001\).

Step 2: Calculate Change in Pressure

Use the bulk modulus equation \(15 \times 10^9 \text{ Pa} = - \frac{\Delta P}{-0.001}\) to find \(\Delta P\). Simplify to get \(\Delta P = 15 \times 10^9 \times 0.001 = 15 \times 10^6 \text{ Pa}\).

Step 3: Convert Pressure to Atmospheres

Since 1 atmosphere is approximately \(1.013 \times 10^5\) Pa, convert the change in pressure to atmospheres: \(\frac{15 \times 10^6 \text{ Pa}}{1.013 \times 10^5 \text{ Pa/atm}} \approx 148 \text{ atm}\).

Step 4: Calculate Depth in Ocean

Pressure increases by \(1.0 \times 10^4\) Pa per meter of depth, so the depth required to reach \(15 \times 10^6\) Pa is \(\frac{15 \times 10^6 \text{ Pa}}{1.0 \times 10^4 \text{ Pa/m}} = 1500\) meters.

Step 5: Comparing Bone Compression Concerns

The calculated depth is extremely deep for typical scuba diving, which usually does not exceed 40 meters. Hence, bone compression is not a concern for typical diving depths.

Key Concepts

Pressure Conversion

When dealing with various problems related to pressure, understanding how to convert between different units is crucial. In the study of physics, we often use Pascals (Pa) to measure pressure. However, in everyday settings, pressure is often measured in atmospheres (atm) because it provides a more intuitive sense of the pressure change relative to the normal atmospheric pressure we experience.

• Pressure in Pascals: The Pascal is a very small unit compared to atmospheric pressure. Therefore, to express larger pressures, we often use megapascals (MPa) or gigapascals (GPa). For example, 1 GPa equals 1 billion Pascals.
• Pressure in Atmospheres: 1 atmosphere (atm) is equivalent to the air pressure at sea level and is approximately equal to 101,300 Pascals. This conversion factor is crucial for transforming results from scientific equations into more relatable figures.

To convert pressure from Pascals to atmospheres, you divide the pressure value in Pascals by this conversion factor. For instance, if we have a change in pressure of \(15 \times 10^6\) Pa, we convert it to atmospheres by dividing by \(1.013 \times 10^5\). This results in approximately 148 atm, indicating an extremely high-pressure change from an everyday perspective.

Bone Compression

Bone compression, especially in high-pressure environments, is a fascinating application of physics in biology. When external pressure increases significantly, it can cause a slight reduction in the volume of bones.
The bulk modulus of a material such as bone is a key property in these calculations. It is denoted as \(K\) and it measures how much the material can be compressed by a given amount of pressure. A high bulk modulus means that a material is less compressible.

• Expressing Compression: To express how much a bone compresses, we use the volume change ratio \(\frac{\Delta V}{V_0}\), which represents the change in volume \(\Delta V\) as a fraction of the original volume \(V_0\).
• Calculating Pressure Needed for Compression: By rearranging the bulk modulus formula \(K = - \frac{\Delta P}{\frac{\Delta V}{V_0}}\), one can calculate the required pressure change to achieve a certain percentage of volume compression, like the 0.10% given in this problem.

For instance, if you want to compress bone by 0.10% which equates to \(\frac{\Delta V}{V_0} = -0.001\), and knowing the bulk modulus of bone is 15 GPa, you can solve for the necessary pressure change. This illustrates how bones are incredibly resilient to pressure changes, scarcely compressing under usual conditions.

Ocean Depth Pressure

Understanding ocean depth pressure is essential for anyone studying marine life or preparing for activities like deep-sea diving. As you descend into the ocean, the pressure increases by approximately \(1.0 \times 10^4\) Pa for every meter.
This rate of pressure increase means that even a relatively shallow dive exposes a diver to significantly increased pressure compared to the surface.

• Effects of Depth: As the diver descends, the atmospheric pressure combined with the pressure of the water above increases dramatically. This is why specialized equipment is necessary for deep sea exploration.
• Calculating Needed Depth for Specific Pressure: In scenarios such as this exercise, we calculated the necessary depth to achieve a pressure increase of \(15 \times 10^6\) Pa, resulting in 1500 meters of ocean depth. This is significantly deeper than standard diving limits.

For typical scuba diving, which does not usually exceed 40 meters, this depth is not feasible, allowing us to infer that bone compression due to pressure changes during regular dives is not a risk. This highlights the importance of understanding pressure changes both for safety and practical assessments of human limits underwater.