Problem 13 - College Physics

Question

You are designing a diving bell to withstand the pressure of seawater at a depth of 250 \(\mathrm{m}\) (a) What is the gauge pressure at this depth? (You can ignore the small changes in the density of the water with depth.) (b) At the 250 \(\mathrm{m}\) depth, what is the net force due to the water outside and the air inside the bell on a circular glass window 30.0 \(\mathrm{cm}\) in diameter if the pressure inside the diving bell equals the pressure at the surface of the water? (You may ignore the small variation in pressure over the surface of the window.)

Step-by-Step Solution

Answer

Gauge pressure is 2.51125 million Pa; net force is 177,469 N.

Step 1: Identify Relevant Information

To find the gauge pressure at a depth of 250 meters, we use the relationship between pressure, depth, and density of the fluid. The density of seawater is approximately \( \rho = 1025 \, \text{kg/m}^3 \), and acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \). The important relationship is \( P = \rho \cdot g \cdot h \), where \( h = 250 \, \text{m} \).

Step 2: Calculate Gauge Pressure

Using the formula \( P = \rho \cdot g \cdot h \), substitute the known values: \( P = 1025 \, \text{kg/m}^3 \cdot 9.8 \, \text{m/s}^2 \cdot 250 \, \text{m} \). Calculate to find the gauge pressure at this depth.

Step 3: Solve for Gauge Pressure

Calculate \( P = 1025 \times 9.8 \times 250 = 2511250 \, \text{Pa} \). Thus, the gauge pressure at 250 meters is \( 2.51125 \times 10^6 \, \text{Pa} \).

Step 4: Find Area of the Window

The diameter of the circular window is 30.0 cm, which is 0.3 meters. Use the formula for the area of a circle, \( A = \pi r^2 \), to find the area, where \( r = 0.15 \, \text{m} \). Calculate \( A = \pi \times (0.15)^2 \).

Step 5: Calculate Area of the Window

Substitute \( r = 0.15 \, \text{m} \) into the formula: \( A = \pi \times (0.15)^2 = 0.0707 \, \text{m}^2 \).

Step 6: Determine Net Force on the Window

To find the net force, use the equation \( F = P \times A \) where the gauge pressure \( P = 2.51125 \times 10^6 \, \text{Pa} \) and the area \( A = 0.0707 \, \text{m}^2 \).

Step 7: Calculate Net Force

Use the values for pressure and area: \( F = 2.51125 \times 10^6 \times 0.0707 = 177,469.475 \, \text{N} \). The net force acting on the window is approximately \( 177,469.475 \, \text{N} \).

Key Concepts

Gauge Pressure

When we discuss gauge pressure, it's essential to know that it measures how much pressure is exerted by a fluid compared to the atmospheric pressure outside. In simple terms, it's the pressure difference between the fluid you're interested in and the air around us.
Imagine diving deep into the ocean; the deeper you go, the greater the water pressure becomes due to the water's weight above you.

To calculate gauge pressure at a specific depth in a fluid like seawater, you use the relationship:

• \( P = \rho \cdot g \cdot h \)
.

Here, \( \rho \) is the fluid density (1025 \(\text{kg/m}^3\) for seawater), \( g \) is the acceleration due to gravity (9.8 \(\text{m/s}^2\)), and \( h \) is the depth.
In our case, at a depth of 250 meters, the gauge pressure is calculated as: 2.51125 \(\times 10^6\) Pascals.

This gauge pressure tells us how much pressure is exerted over the atmospheric pressure at the surface when you're 250 meters deep.

Net Force

The net force is essentially the overall force acting on an object when all individual forces are combined. In the context of a diving bell, the net force affects the window due to the difference in pressure inside and outside the bell.
When the diving bell is submerged, the water pressure outside is significantly higher compared to the pressure inside which is equal to surface pressure.

To find the net force on the window, we use the formula:

• \( F = P \times A \)
.

Where \( P \) is the gauge pressure (2.51125 \(\times 10^6\) Pascals) and \( A \) is the area of the window (0.0707 \(\text{m}^2\)).
The calculated net force, in this case, is 177,469.475 Newtons.
This substantial force indicates the pressing pressure on the window due to water at such depth, highlighting how strong the diving bell's materials need to be.

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at equilibrium due to gravity. It's what you feel when swimming to the bottom of a pool. The deeper you go, the greater the pressure, because there's more water above pushing down.

In fluid mechanics, hydrostatic pressure at a given depth can be described by the same expression used for gauge pressure:

• \( P = \rho \cdot g \cdot h \)
.

This expression helps calculate how much pressure a fluid relative to its density and depth.
At 250m depth in seawater, the hydrostatic pressure contributes largely to the overall pressure affecting a submerged object like our diving bell. Understanding this concept is vital for designing underwater equipment that can withstand such immense forces without failure.

Fluid Mechanics

Fluid mechanics is the branch of physics dealing with the behavior of fluids (liquids and gases) and the forces on them. It's a crucial subject when studying phenomena like underwater pressure, as it helps explain how fluids move and exert force.

This exercise uses concepts from fluid mechanics to ensure a diving bell can withstand underwater conditions.
Fluid mechanics principles assist in understanding:

• Flow patterns of water
• The impact of pressure variations
• How forces affect submerged objects
.

Understanding fluid mechanics is essential for designing and engineering tools and structures that interact with fluids, ensuring safety and effectiveness in various applications, including underwater exploration.