Container A is sealed (not open to the atmosphere) filled with water with an unlimited width and depth. Container A consists of Section 1 and 2. The upper part of Container A, Section 1 has 5 feet of air at 21.83 psi
Tube B is 80 feet long and full of air at 14.7 psi; B is closed on the top and open on the bottom. Tube B is now inserted into Container A (as shown in the illustration) and water rises into tube B .
Problem:
How much will water rise into Tube B?
What will the PSI of the air column in Tube B now be?
To determine how much water will rise into Tube B and the PSI of the air column in Tube B, we can apply the principles of fluid pressure and hydrostatics.
Step 1: Understand the setup
From the given information, we know that:
- Container A has two sections, Section 1 (filled with air) and Section 2 (to be filled with water).
- Section 1 of Container A has an air column that is 5 feet tall and has a pressure of 21.83 psi.
- Tube B is inserted into Container A, with a length of 80 feet and air at 14.7 psi.
- Tube B is closed at the top and open at the bottom.
Step 2: Apply Pascal's Law
Pascal's Law states that when a pressure is applied to a fluid in a confined space, it is transmitted equally in all directions. This means that the pressure at any point in a fluid is the same in all directions.
Since Section 1 of Container A is sealed, the pressure at its bottom (underneath the air column) is also 21.83 psi.
Step 3: Determine the water level in Container A
Water will rise in Container A until the pressure at the bottom of Section 1 matches the pressure at the bottom of Section 2.
Given that the atmospheric pressure is 14.7 psi, the pressure of the water column in Section 2 (at its base) is 14.7 psi.
To find the height of the water column in Section 2 (and thus the height it will rise into Tube B), we can use the equation:
Pressure (Section 1) + Pressure (Section 2) = Pressure (Water column)
Substituting the known values:
21.83 psi + 14.7 psi = Pressure (Water column)
Thus, the pressure in the water column is 36.53 psi.
Step 4: Calculate the water height in Tube B
Since the pressures in Section 1, Section 2, and the water column are equal, we can use the equation:
Pressure = Density × Gravity × Height
In this equation, Density represents the density of the fluid, Gravity is the acceleration due to gravity, and Height is the height of the fluid column.
Since the fluid in the tube is air, its density is constant. Assuming that the density of air (ρ) remains the same, we can write:
Pressure (Tube B air column) = ρ × Gravity × Height (water level in Tube B)
Solving for the water level in Tube B:
Height (water level in Tube B) = Pressure (Tube B air column) / (ρ × Gravity)
Step 5: Calculate the PSI of the air column in Tube B
The PSI of the air column can be calculated by subtracting the pressure of the water column from the atmospheric pressure.
Pressure (Tube B air column) = Atmospheric pressure - Pressure (water column)
Substituting the known values:
Pressure (Tube B air column) = 14.7 psi - 36.53 psi
Step 6: Calculate the final results
By performing the calculations in Step 4 and Step 5, we can obtain the following results:
1. The water level in Tube B (height) measured from the bottom will be equal to the height (water level in Tube B) calculated in Step 4.
2. The PSI of the air column in Tube B will be equal to the pressure (Tube B air column) calculated in Step 5.
Please note that to obtain accurate results, you need to know the density of air (ρ) and the acceleration due to gravity. Make sure to use the appropriate units when substituting these values into the equations.