To determine the depth \( h \) of the lower edge of the pipe when it is submerged in the freshwater lake, we can apply the principles of fluid mechanics and the behavior of gases in an open-closed pipe system. The key here is to understand how the pressure changes with depth in the water and how that affects the air column inside the pipe.

Understanding the Pressure Dynamics

In an open-closed pipe, one end is open to the atmosphere, while the other end is closed. When the pipe is submerged, the air inside the pipe is compressed by the water pressure. The pressure at a certain depth in a fluid can be expressed using the hydrostatic pressure formula:

• P = P₀ + ρgh

Where:

• P = total pressure at depth
• P₀ = atmospheric pressure (approximately 101,325 Pa)
• ρ = density of the fluid (for freshwater, about 1000 kg/m³)
• g = acceleration due to gravity (approximately 9.81 m/s²)
• h = depth of the fluid above the point of measurement

Setting Up the Equation

When the water rises halfway up the pipe, the length of the air column above the water is \( L/2 = 25.0 \, \text{m} / 2 = 12.5 \, \text{m} \). The pressure at the bottom of the air column can be expressed as:

• P_{air} = P₀ - \frac{ρgh}{2}

Here, \( P_{air} \) is the pressure exerted by the air column, and it must equal the atmospheric pressure plus the pressure due to the water column above it. Therefore, we can set up the equation:

• P_{air} = P₀ + ρgh

Equating the Pressures

Since the air pressure inside the pipe must balance the pressure from the water column, we can equate the two expressions:

• P₀ - \frac{ρgh}{2} = P₀ + ρgh

Rearranging this gives us:

• - \frac{ρgh}{2} = ρgh

Combining terms leads to:

• 0 = \frac{3ρgh}{2}

Solving for Depth

From the equation, we can see that we need to isolate \( h \). To find \( h \), we can use the fact that the total pressure at the bottom of the air column must equal the atmospheric pressure plus the pressure from the water column:

• P₀ + ρgh = P₀ + \frac{ρgL}{2}

Solving for \( h \) gives us:

• h = \frac{L}{2} = 12.5 \, \text{m}

Final Result

Thus, the depth \( h \) of the lower edge of the pipe is 12.5 meters. This means that when the pipe is submerged, the water rises halfway up the pipe, compressing the air column above it and maintaining equilibrium with the external water pressure. This example illustrates the balance between fluid pressure and gas pressure in a practical scenario.