Problem 56 - University Physics with Modern Physics

Question

During a test dive in \(1939,\) prior to being accepted by the U.S. Navy, the submarine Squalus sank at a point where the depth of water was 73.0 \(\mathrm{m}\) . The temperature at the surface was \(27.0^{\circ} \mathrm{C},\) and at the bottom it was \(7.0^{\circ} \mathrm{C}\) . The density of seawater is 1030 \(\mathrm{kg} / \mathrm{m}^{3} .\) (a) A diving bell was used to rescue 33 trapped crewmen from the Squalus. The diving bell was in the form of a circular cylinder 2.30 \(\mathrm{m}\) high, open at the bottom and closed at the top. When the diving bell was lowered to the bottom of the sea, to what height did water rise within the diving bell? (Hint: You may ignore the relatively small variation in water pressure between the bottom of the bell and the surface of the water within the bell. \((b)\) At what gauge pressure must compressed air have been supplied to the bell while on the bottom to expel all the water from it?

Step-by-Step Solution

Answer

(a) Water rises 0.9 m inside the bell. (b) Gauge pressure needed is 739940 Pa.

Step 1: Understand the Problem

We need to calculate two things: (a) the height to which water rises in a submerged diving bell, and (b) the gauge pressure required to expel water from the bell. We will apply concepts related to pressure, temperature, and volume to solve these problems.

Step 2: Analyze the Diving Bell's Water Level

When the diving bell is submerged, water rises inside because of pressure differences. At the bottom, the pressure is higher due to water depth, and the air inside is compressed. The problem assumes that the small variations in water pressure are negligible due to the height of the bell being relatively small.

Step 3: Apply the Ideal Gas Law

We use the ideal gas law to handle the compressed air change. At the surface and at the bottom, calculate the initial and final volumes based on temperatures and pressures: \( P_1 \times V_1 = nRT_1 \) and \( P_2 \times V_2 = nRT_2 \). Here \( V_1 \) is the original air volume inside the bell (2.3 m height), and \( V_2 \) is the reduced volume due to incoming water.

Step 4: Calculate Initial and Final Conditions

Initially, the volume is full inside the diving bell: \( V_1 = A \times 2.3 \). At depth, the temperature drops from 27°C to 7°C, and pressure increases by water pressure (\(P_2 = P_0 + \rho g h\)). Determine \( V_2 = A \times x \) where \( x \) is the height air occupies now, and the remaining volume is filled with water.

Step 5: Solve for Water Height Rise in Bell (Part a)

Using the relation for the ideal gas law and substituting pressure and temperature changes: \( (P_0 \times 2.3) = (P_0 + \rho g h) \times x \times \frac{T_{bottom}}{T_{surface}} \). Solve for \( x \), then calculate \( 2.3 - x \) to find the water rise.

Step 6: Calculate Gauge Pressure to Expel Water (Part b)

The gauge pressure needed at the bottom to expel water completely is the difference between external seawater pressure and internal air pressure. With complete expulsion, internal and external pressures are equal. Thus, it equals the hydrostatic pressure due to the water column. \( P_{gauge} = \rho g h \).

Step 7: Use Known Values and Solve Equations

Use given values: \( ho = 1030 \, \text{kg/m}^3 \), \( g = 9.8 \, \text{m/s}^2 \), \( h = 73 \, \text{m} \). Substitute into equations to find the numerical solution for \( x \) and the gauge pressure. Assume standard atmospheric pressure \( P_0 = 101325 \, \text{Pa}. \) Calculate \( T \) in Kelvin from Celsius for accuracy.

Key Concepts

Pressure

Pressure is a crucial concept in hydrostatics and plays a significant role when analyzing diving bells. Water exerts pressure due to its weight, and this pressure increases linearly with depth. At any depth in a fluid, the pressure exerted can be calculated using the formula: \[ P = P_0 + \rho g h \] where:

• \( P \) is the total pressure at depth,
• \( P_0 \) is the atmospheric pressure at the surface,
• \( \rho \) is the density of the fluid,
• \( g \) is the acceleration due to gravity,
• \( h \) is the depth below the surface.

This equation shows that deeper you go under water, the higher the pressure. For the diving bell's problem, the pressure difference between the water's surface and depth is what causes water to rise inside the bell.

Ideal Gas Law

The Ideal Gas Law is essential for understanding how the volume of air in a diving bell changes with varying pressure and temperature. The law is expressed by the equation: \[ PV = nRT \]where:

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature in Kelvin.

When a diving bell is submerged, the pressure inside changes as water enters, affecting the volume of air trapped inside the bell. In this scenario, temperature also plays a role, as it impacts the pressure and volume relationship. To solve our problem, we assess the initial and final conditions influencing these variables to determine the volume of air remaining as water enters the bell.

Temperature

Temperature plays an important role when using the Ideal Gas Law to understand changes in a diving bell. It's important to remember that temperature should be converted to Kelvin in these calculations to maintain consistency with absolute temperature scales. The conversion from Celsius to Kelvin is straightforward: \[ T(K) = T(°C) + 273.15 \]In the diving scenario, the temperature at the surface is given as \(27.0 \degree \text{C}\), and at the bottom, it's \(7.0 \degree \text{C}\). These temperatures affect both the air volume and pressure inside the diving bell. As the temperature drops, so does the energy of the gas molecules, resulting in reduced pressure and thus influencing the volume calculations within the bell during submersion.

Diving Bell

The diving bell, a fascinating tool used in underwater exploration, operates on principles of pressure and buoyancy. Its cylindrical form, open at the bottom and sealed at the top, allows it to be lowered into the water while air remains trapped inside. When submerged, the rising water inside the bell is a result of the hydrostatic pressure, which increases with depth. In our problem, the height to which water inside the bell rises must be determined. This is done by analyzing the pressure balance and employing the Ideal Gas Law to understand how air behaves under different conditions.
The bell's internal air pressure must be equalized with the underwater pressure to expel the water completely. By understanding these dynamics, one appreciates the engineering precision in rescue operations using tools like diving bells, ensuring the safety and effectiveness of such endeavors.