Find the depth of a tube submerged in water that is half filled with air

1. Dec 5, 2008

KEØM

1. The problem statement, all variables and given/known data

A tube of length L = 25 m that is open at one end contains air at atmospheric pressure. This is done in Denver so atmospheric pressure P = .667 X (1.10 X 105 Pa). The tube is thrust vertically into a freshwater lake until water rises halfway up in the pipe. Find the depth of the tube in the water. Note: air can be treated as an ideal gas.

In the attachment there is a picture on problem #5. The picture will really help clarify the problem.

2. Relevant equations
P = P(initial) + $$\rho$$gd
$$\Sigma$$F = ma
P=F/A

3. The attempt at a solution
I know the pressure of the air inside the tube and I also know that the forces exerted by the air and the water are the same but I am just not sure how to tie all of these things together to find the depth.

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2. Dec 5, 2008

Staff: Mentor

Hint: What's the pressure of the compressed air in the tube?

3. Dec 5, 2008

KEØM

Isn't it equal to the atmospheric pressure?

4. Dec 5, 2008

Staff: Mentor

Not after being compressed. (How did its volume change?)

5. Dec 5, 2008

KEØM

By putting it in the water the volume was halved so the pressure is the atmospheric pressure divided by 2?

6. Dec 5, 2008

Staff: Mentor

No. Use the hint that the air can be treated as an ideal gas. (What's the ideal gas law?) You can assume the temperature is constant.

7. Dec 5, 2008

KEØM

But I don't know the volume or the number of moles either.

8. Dec 5, 2008

Staff: Mentor

You don't care about the actual volume, only that it went from V to V/2. The number of moles is constant.

9. Dec 5, 2008

KEØM

Ok so I solved for pressure now can I use this in the formula P = P(initial) + $$\rho$$gd but I am not sure if that will work.

10. Dec 5, 2008

Staff: Mentor

Make use of the fact that the pressure must be the same at the same height in a fluid. What must the water pressure be right at the air/water interface in the tube?