Problem 58 - 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 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

Water rises ~0.2 m; Gauge pressure needed is 750 kPa.

Step 1: Understanding the Problem

We are tasked with finding the height that water rises in a diving bell when it is submerged to a depth of 73.0 m in seawater, and the gauge pressure needed to expel the seawater. Given parameters include the temperatures at the surface and bottom, and the density of seawater.

Step 2: Calculate the Pressure at the Bottom of the Sea

Pressure at the bottom can be calculated using the formula \( P = P_0 + \rho g h \) where \( P_0 \) is atmospheric pressure (\(1.013 \times 10^5 \) Pa), \( \rho \) is the seawater density (1030 kg/m³), \( g \) is gravitational acceleration (9.81 m/s²), and \( h \) is depth (73.0 m).

Step 3: Use Ideal Gas Law Inside the Bell

Initial condition in the bell is at surface (27.0°C) and atmospheric pressure. When submerged, we assume the temperature inside the bell equals that at the bottom (7.0°C). Use \( P_1V_1/T_1 = P_2V_2/T_2 \) to find \( V_2 \), where \( V_1 = A(2.30) \) and \( V_2 = A(2.30 - h_w) \), \( h_w \) is water height, \( A \) is base area of the bell.

Step 4: Solve for Water Height in the Bell

With the expression \( h_w = 2.30 - \frac{T_2 \cdot P_1}{T_1 \cdot P_2} \times 2.30 \), substitute values \( T_1 = 300 \) K, \( T_2 = 280 \) K, solve for \( h_w \).

Step 5: Calculate Gauge Pressure to Expel Seawater

To expel the water, pressure inside the bell must equal external water pressure. Gauge pressure is \( P_2 - P_0 \), calculated using the formula in Step 2, minus atmospheric pressure.

Step 6: Determine Final Results

Substitute all numerical values into the formulas to compute water rise within the bell (~0.2 m) and gauge pressure needed (750 kPa).

Key Concepts

Pressure Calculation

When dealing with underwater scenarios like the submarine dive, calculating pressure accurately is crucial. The pressure at a certain depth under water is affected by both the weight of the water above it and the atmospheric pressure. To determine the pressure at the bottom of the sea, we can use the formula:
\[ P = P_0 + \rho g h \]where:

• \( P_0 \) is the atmospheric pressure at the surface, about \( 1.013 \times 10^5 \text{ Pa} \) at sea level.
• \( \rho \) is the density of seawater, given as \( 1030 \text{ kg/m}^3 \) in the exercise.
• \( g \) is the standard acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \).
• \( h \) is the depth of the water, in this case, 73.0 m.

This formula calculates the absolute pressure at a given depth. It reflects both the pressure due to the water column and the atmospheric pressure pressing down on the water surface. Understanding this concept is essential for diving operations, engineering, and various calculations in hydrostatics.

Ideal Gas Law

The Ideal Gas Law is a cornerstone in understanding how gas behaves under different conditions of pressure, volume, and temperature. In this scenario, when the diving bell is submerged, it holds air originally at the surface conditions of 27.0°C and atmospheric pressure.To find out how the air volume changes when the bell is submerged 73.0 meters down to a pressure that involves a temperature drop to 7.0°C, we use:
\[ P_1V_1/T_1 = P_2V_2/T_2 \]
This equation allows us to solve for the final air volume (\( V_2 \)) when we know:

• \( V_1 \) is the initial volume of air, related to the height of the bell (2.30 m).
• \( T_1 \) and \( T_2 \) are the absolute temperatures (Kelvin) at the surface and the bottom.
• \( P_1 \) and \( P_2 \) are initial and final pressures.

This fundamental law shows the relationship between physical properties of gases, thus predicting how the gas will behave under varying external conditions, such as those encountered in diving situations.

Gauge Pressure

Gauge pressure is the amount of pressure measured over atmospheric pressure. When calculating the gauge pressure that's needed to expel water from the diving bell, we first determine the absolute pressure at that depth using the pressure calculation formula (Step 2). To find the gauge pressure:
\[ \text{Gauge Pressure} = P_2 - P_0 \]
Where \( P_2 \) is the pressure at the depth we calculated previously, and \( P_0 \) is the atmospheric pressure. The reason for using gauge pressure is straightforward: it helps us understand the additional pressure that needs to be exerted to perform tasks like pushing water out of the diving bell. This extra pressure is important in situations where things are already under atmospheric pressure, as it adds on top of that baseline.

Buoyancy

Buoyancy is the concept that describes why objects submerged in a fluid either float, sink, or stay in place. Archimedes' Principle is key here: it states that the upward buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid the body displaces. In practical terms:

• When the diving bell is submerged, it displaces a volume of seawater equivalent to its own volume.
• The displaced water creates an upward force.
• If this force matches the weight of the diving bell, it will float or stay submerged at a constant depth.

This buoyant force is what counteracts the weight of the diving bell, making it manageable to manipulate the bell safely within the water column. Recognizing and calculating buoyancy effects is critical for the design and deployment of submersible devices and for rescue operations such as the one described.