Problem 16 - Fundamentals of Physics

Question

Snorkeling by humans and elephants. When a person snorkels, the lungs are connected directly to the atmosphere through the snorkel tube and thus are at at- mospheric pressure. In atmospheres, what is the difference \(\Delta p\) between this internal air pressure and the water pressure against the body if the length of the snorkel tube is (a) \(20 \mathrm{~cm}\) (standard situation) and (b) \(4.0 \mathrm{~m}\) (probably lethal situation)? In the latter, the pressure difference causes blood vessels on the walls of the lungs to rupture, releasing blood into the lungs. As depicted in Fig. 14-31, an elephant can safely snorkel through its trunk while swimming with its lungs \(4.0 \mathrm{~m}\) below the water surface because the membrane around its lungs contains connective tissue that holds and protects the blood vessels, preventing rupturing.

Step-by-Step Solution

Answer

(a) 0.00194 atm, (b) 0.387 atm.

Step 1: Understand the Problem

We need to calculate the pressure difference between the internal air pressure and the water pressure against the body while snorkeling. This difference is denoted as \( \Delta p \). The lengths of the snorkel for parts (a) and (b) are given as 20 cm and 4.0 m, respectively.

Step 2: Identify the Relevant Formula

The difference in pressure \( \Delta p \) can be calculated using the formula: \( \Delta p = \rho g h \), where \( \rho \) is the density of water (\(1000 \, \mathrm{kg/m^3}\)), \( g \) is the acceleration due to gravity (\(9.8 \, \mathrm{m/s^2}\)), and \( h \) is the depth of water (the length of the snorkel, converted to meters).

Step 3: Calculate Pressure Difference for 20 cm Snorkel

First, convert 20 cm to meters: \( h = 0.2 \, \mathrm{m} \). Apply the formula: \( \Delta p = 1000 \, \mathrm{kg/m^3} \times 9.8 \, \mathrm{m/s^2} \times 0.2 \, \mathrm{m} = 196 \, \mathrm{Pa} \). Convert this pressure to atmospheres: \( 196 \, \mathrm{Pa} \div 101325 \, \mathrm{Pa/atm} = 0.001935 \, \mathrm{atm} \).

Step 4: Calculate Pressure Difference for 4.0 m Snorkel

For part (b), the length of the snorkel is 4.0 m. Substitute into the formula: \( \Delta p = 1000 \, \mathrm{kg/m^3} \times 9.8 \, \mathrm{m/s^2} \times 4.0 \, \mathrm{m} = 39200 \, \mathrm{Pa} \). Convert this pressure to atmospheres: \( 39200 \, \mathrm{Pa} \div 101325 \, \mathrm{Pa/atm} = 0.387 \mathrm{atm} \).

Key Concepts

Snorkeling Physics

Snorkeling allows humans to explore underwater environments while breathing air from the surface through a snorkel tube. This fascinating activity is grounded in the basic principles of physics. When using a snorkel, the air in your lungs stays at atmospheric pressure since the tube connects directly to the surface air. However, your body is submerged in water, which exerts additional pressure. Understanding how this works is crucial not only for safety but also to grasp the limits of snorkeling physics. The human body is adapted to breathe at atmospheric pressure, which limits the safe length of a snorkel tube. In standard situations, a tube of about 20 cm allows for comfortable and safe breathing. However, as the length of the snorkel increases, so does the pressure difference acting on the body. This is due to the increased depth, where water pressure rises significantly, demonstrating a fundamental concept in snorkeling physics.

Pressure Calculations

Calculating the pressure difference when snorkeling helps to understand how safe it is to use a particular snorkel length. The pressure difference, denoted as \( \Delta p \), can be calculated using the formula: \( \Delta p = \rho g h \). Here:

• \( \rho \) is the density of water (about \( 1000 \, \text{kg/m}^3 \)),
• \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)),
• \( h \) is the depth of water or length of the snorkel converted to meters.

For a 20 cm snorkel, the water pressure adds \( 196 \, \text{Pa} \) or \( 0.001935 \, \text{atm} \) to the atmospheric pressure. For a potentially hazardous 4.0 m length, this pressure jumps to \( 39200 \, \text{Pa} \) or \( 0.387 \, \text{atm} \). This significant increase illustrates why caution is necessary with longer snorkels.

Hydrostatic Pressure

Hydrostatic pressure is a crucial concept when considering the effects of water on submerged objects or bodies. This type of pressure occurs due to the weight of the fluid above the object, increasing linearly with depth. In the context of snorkeling, hydrostatic pressure is what acts on your submerged body while you breathe surface air. This pressure difference is why relief valves or specialized equipment are necessary at greater depths to equalize this force. For snorkeling, understanding hydrostatic pressure helps to appreciate why snorkels longer than about 20 cm are impractical and potentially dangerous. The deeper you go, the greater the pressure difference from atmospheric pressure, as was calculated for various snorkel lengths. This principle not only applies to snorkeling but is a fundamental aspect of fluid mechanics critical for divers and engineers working with submerged systems.

Physics of Survival

The physics of survival touches on the human body's ability to withstand environmental pressures when engaging in activities like snorkeling. When an object, or in this case, a body, is underwater, it experiences increased pressure forces. The human body is designed to handle atmospheric pressure comfortably but struggles against substantial hydrostatic pressure differences. Elephants, on the other hand, have evolved specialized adaptations that allow them to snorkel safely. Their lung structures contain connective tissues that protect blood vessels from rupturing under high-pressure differences. For humans, exceeding safe snorkeling depths, like the 4.0 m scenario, can cause dangerous physiological changes, such as pulmonary barotrauma. Awareness of these survival limits and the physics governing them is essential to safely exploring underwater environments. This knowledge helps to set boundaries to avoid potential injury and promotes responsible enjoyment of snorkeling and similar activities.