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If a scuba diver descends too quickly into the sea, the internal pressure on each eardrum remains at atmospheric pressure, while the external pressure increases due to the increased water depth. At sufficient depths, the difference between the external and internal pressures can rupture an eardrum. Eardrums can rupture when the pressure difference is as little as \(35 \mathrm{kPa} .\) What is the depth at which this pressure difference could occur? The density of seawater is \(1025 \mathrm{kg} / \mathrm{m}^{3}\).

Short Answer

Expert verified
The depth is approximately 3.49 meters.

Step by step solution

01

Understanding the Problem

We need to find the depth at which the pressure difference between the external and internal pressure on the eardrums equals 35 kPa. We know that pressure increases with depth in a fluid like seawater.
02

Using the Pressure Formula

The pressure due to a fluid column is given by the formula \( P = \rho g h \), where \( P \) is the pressure, \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity \((9.81 \mathrm{m/s}^{2})\), and \( h \) is the depth. We will rearrange this to solve for \( h \).
03

Rearranging the Formula

Rearrange \( P = \rho g h \) to solve for \( h \):\[h = \frac{P}{\rho g}\]
04

Substituting Known Values

Substitute the known values into the equation: \( P = 35,000 \mathrm{Pa} \), \( \rho = 1025 \mathrm{kg/m}^{3} \), and \( g = 9.81 \mathrm{m/s}^{2} \).\[h = \frac{35,000}{1025 \times 9.81}\]
05

Calculating the Depth

Perform the calculation:\[h = \frac{35,000}{1025 \times 9.81} \approx 3.49 \mathrm{m}\]Thus, the depth at which the pressure difference could rupture an eardrum is approximately 3.49 meters.

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Key Concepts

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

Eardrum Rupture
Eardrums are critical for our hearing as they vibrate in response to sound waves, allowing us to perceive sounds. When scuba diving, or even when flying, our eardrums experience differing pressure conditions. While underwater, the external pressure increases with depth, whereas the internal pressure stays the same unless equalized. If the pressure difference becomes too high, an eardrum can rupture.

This rupture occurs because eardrums are delicate, thin membranes. If the external pressure is greater than the internal pressure by about 35 kPa, the stress can make the eardrum burst. This is why divers are taught to ascend and descend slowly, allowing the pressure in their inner ear to equalize with the external pressure. Equalizing is crucial to prevent discomfort and injury to the eardrums.
Pressure Difference
Pressure difference plays a significant role in various physical interactions, especially in fluids like seawater. When you dive into the ocean, the deeper you go, the more water is above you, and the greater the pressure. This increased pressure results from the weight of the water column pressing down.

Mathematically, the pressure increase at a certain depth can be calculated using the formula:
  • \( P = \rho g h \)
Here:
  • \( P \) is the pressure,
  • \( \rho \) is the density of the fluid (seawater in this example),
  • \( g \) is the acceleration due to gravity,
  • \( h \) is the depth.
This formula shows that pressure difference directly correlates with how deep you dive and the density of the fluid.
Seawater Density
Density is a key attribute of any fluid and plays an essential role in calculating pressure changes in seawater. Seawater density, typically around 1025 kg/m³, indicates that each cubic meter of seawater has this mass. This higher density than fresh water is due to the salts and minerals dissolved in it.

This density affects pressure measurements because water exerts a greater force per unit area than less dense fluids like air. Therefore, as a diver descends, they experience a rapid increase in pressure. It's crucial to remember that different oceans may have slightly varying seawater densities due to temperature and salinity changes, but the average is fairly consistent for general calculations.
Scuba Diving
Scuba diving is a popular recreational activity that allows people to explore underwater worlds. However, diving safely requires understanding the basics of how pressure affects the body. As a diver goes deeper, the water pressure around them increases significantly.

To maintain safe diving practices, divers must be aware of the risks of pressure changes, such as eardrum rupture. Proper diving techniques include descending and ascending at modest rates, allowing the body time to adjust the pressure inside the ears and other air spaces.

Additionally, divers are taught to equalize pressure - often by pinching the nose and gently blowing or by swallowing. This process balances the external pressure with the internal body pressure, preventing injuries like eardrum rupture. Scuba diving is exhilarating but requires respect for the physical laws governing pressure differences.

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