B What is the pressure acting on the air?

AI Thread Summary
The discussion centers on understanding the pressure acting on air in a glass submerged in water, particularly why hydrostatic pressure is calculated using the height of the air column (Δx) rather than the total length of the glass (L). The pressure exerted by the water on the gas is expressed as p = patm + Δx ρg, leading to confusion about the use of Δx instead of L. Participants clarify that the pressure at the interface must balance, which is determined by the depth of the air column relative to the water surface. The conversation emphasizes the need to apply the ideal gas law and consider the incompressibility of fluids to resolve the pressure relationship intuitively. Ultimately, the key takeaway is that the pressure calculations depend on the specific height of the air column, not the total length of the glass.
Lotto
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TL;DR Summary
If I take a glass and dip it (perpendicularly to a water surface) with its opened part into water, there should be a thin layer of compressed air. How to determine its pressure?
Let us say that the glass isn't whole under the water. If the height of the air layer under the water is ##\Delta x##, then the pressure the water acts on the gas is ##p=p_{\mathrm {atm}}+\Delta x \rho g##. But my confusion is why the "hydrostaic pressure" is ##\Delta x \rho g##, not ##L\rho g##, where ##L## is the total lenght of a dipped glass. It would be more intuitive for me. I imagine it the way that the hydrostatic pressure ##L\rho g## pushes water up into the glass. It seems strange to me that it is different.

How to explain it?
 
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Lotto said:
TL;DR Summary: If I take a glass and dip it (perpendicularly to a water surface) with its opened part into water, there should be a thin layer of compressed air. How to determine its pressure?

Let us say that the glass isn't whole under the water. If the height of the air layer under the water is ##\Delta x##, then the pressure the water acts on the gas is ##p=p_{\mathrm {atm}}+\Delta x \rho g##. But my confusion is why the "hydrostaic pressure" is ##\Delta x \rho g##, not ##L\rho g##, where ##L## is the total lenght of a dipped glass. It would be more intuitive for me. I imagine it the way that the hydrostatic pressure ##L\rho g## pushes water up into the glass. It seems strange to me that it is different.

How to explain it?
1702399659850.png


Something like this?
 
erobz said:
View attachment 337132

Something like this?
Yes, exactly. My question is why the part of the total pressure is ##(h-\delta)\rho g## instead of ##h\rho g##.
 
Lotto said:
Yes, exactly. My question is why the part of the total pressure is ##(h-\delta)\rho g## instead of ##h\rho g##.
The absolute pressures of the gas above section 1-1 and the absolute hydrostatic pressure of the fluid below 1-1 must balance at their interface (if this is happening as a quasistatic equilibrium process).
 
erobz said:
The absolute pressures of the gas above section 1-1 and the absolute hydrostatic pressure of the fluid below 1-1 must balance at their interface (if this is happening as a quasistatic equilibrium process)
Yes, I know, but why is the part of the pressure ##(h-\delta)\rho g##? It is just unintuitive to me.
 
Lotto said:
Yes, I know, but why is the part of the pressure ##(h-\delta)\rho g##? It is just unintuitive to me.
I'm not getting the issue. It is that, because that is where the interface is - depth of section 1-1 relative to surface. The fluid is incompressible (virtually).
 
erobz said:
I'm not getting the issue. It is that, because that is where the interface is - depth of section 1-1 relative to surface. The fluid is incompressible (virtually).
Ah, OK, I can see my weird thoughts now.
 
Lotto said:
Ah, OK, I can see my weird thoughts now.
Maybe if the container of water is relatively small surrounding the cup (such that the height of water inside the container would significantly change as you dipped the cup) you can argue there is some less obvious relationship there. It still would be measured from the surface, but the surface height would have some dependency on ##\delta##.

Edit: Disregard, I got myself twisted up.
 
Last edited:
If the total pressure is ##p##, I would derive it from

##p_{atm}+h\rho g=p+\delta\rho g##.
 
  • #10
You don't need to mess with all that. Just use the ideal gas law assuming that the temperature is constant. If ##A## is the area of the vessel and no air escapes, the gauge pressure inside the vessel will be given by $$p_1V_1=p_2V_2\implies p_{atm}~\cancel{A}~l=p~\cancel{A}~(l-\delta)\implies p=\left(\frac{l}{l-\delta}\right)p_{atm}.$$I assume that by "total" pressure you mean absolute pressure.
 
  • #11
kuruman said:
You don't need to mess with all that. Just use the ideal gas law assuming that the temperature is constant. If ##A## is the area of the vessel and no air escapes, the gauge pressure inside the vessel will be given by $$p_1V_1=p_2V_2\implies p_{atm}~\cancel{A}~l=p~\cancel{A}~(l-\delta)\implies p=\left(\frac{l}{l-\delta}\right)p_{atm}.$$I assume that by "total" pressure you mean absolute pressure.

Lotto said:
Let us say that the glass isn't whole under the water.

In order to find ##P(h)## we need to first find ##\delta(h)##

$$ P_{atm} \cancel{A} \ell = P \cancel{A} ( \ell - \delta ) \tag{1} $$

$$ P_{atm} + \rho g h = P + \rho g \delta \tag{2} $$

##(1)## and ##(2)## combine to give ##\delta (h)##

Then sub back into ##(1)## for ##P(h)##
 
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