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

[KEØM]

Homework Statement

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.

Homework Equations

P = P(initial) + $$\rho$$gd
$$\Sigma$$F = ma
P=F/A

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.

 

[Doc Al]

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

 

[KEØM]

Isn't it equal to the atmospheric pressure?

 

[Doc Al]

Isn't it equal to the atmospheric pressure?

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

 

[KEØM]

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

 

[Doc Al]

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

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.

 

[KEØM]

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

 

[Doc Al]

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

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

 

[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.

 

[Doc Al]

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?