Hydrostatics: should I use P1A1 = P2A2 or P1 = P2?

AI Thread Summary
The discussion centers on the correct application of hydrostatic principles to determine the ratio of densities between two fluids separated by a rigid plate. The confusion arises between using P1A1 = P2A2 and P1 = P2, with the former yielding a density ratio of 1/4 and the latter 1/2. It is emphasized that P1 = P2 is incorrect because pressure varies with depth, meaning equal pressures at the bottom do not imply equal pressures at higher points. The balance of forces on the plate is crucial, and moments do not need to be considered in this specific scenario. Understanding that pressure is depth-dependent clarifies the correct approach to solving the problem.
rbmartel
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Homework Statement
The figure shows the situation in which two fluids (liquids I and II), with densities ρI and ρII, are separated by a rigid plate of height H, supported on a frictionless base. In order for the plate to be on equilibrium on the rigid plate, the ratio between the densities of the liquids must be ρI / ρII. = ?

Hey guys, in order to find ρI / ρII, should I use:

ρI.g.H1 = ρII.g.H2 (P1 = P2)

or

ρI.g.H1.H1.c = ρII.g.H2.H2.c (P1A1 = P2A2)
Relevant Equations
ρI.g.H1 = ρII.g.H2 (P1 = P2)
ρI.g.H1.H1.c = ρII.g.H2.H2.c (P1A1 = P2A2)
I used P1A1 = P2A2 and my professor said that P1=P2 is correct, but some sources on the internet say that P1A1=P2A2 is correct, just like I did, but unfortunately no one explains why one or the other is the correct answer.
For P1A1 = P2A2 you get ρI / ρII = 1/4 and for P1=P2 you get ρI / ρII = 1/2.
 

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rbmartel said:
Homework Statement:: The figure shows the situation in which two fluids (liquids I and II), with densities ρI and ρII, are separated by a rigid plate of height H, supported on a frictionless base. In order for the plate to be on equilibrium on the rigid plate, the ratio between the densities of the liquids must be ρI / ρII. = ?

Hey guys, in order to find ρI / ρII, should I use:

ρI.g.H1 = ρII.g.H2 (P1 = P2)

or

ρI.g.H1.H1.c = ρII.g.H2.H2.c (P1A1 = P2A2)
Relevant Equations:: ρI.g.H1 = ρII.g.H2 (P1 = P2)
ρI.g.H1.H1.c = ρII.g.H2.H2.c (P1A1 = P2A2)

I used P1A1 = P2A2 and my professor said that P1=P2 is correct, but some sources on the internet say that P1A1=P2A2 is correct, just like I did, but unfortunately no one explains why one or the other is the correct answer.
For P1A1 = P2A2 you get ρI / ρII = 1/4 and for P1=P2 you get ρI / ρII = 1/2.
It's just the balance of forces on the plate. If they are not equal the plate will accelerate as per Newton’s laws.
P1=P2 makes no sense since the pressure is a function of depth. If the pressures are equal at the bottom they are not going to be equal half way up, and certainly not equal above that.
 
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In addition to what @haruspex said, do you also need to balance the moments on the barrier, or are the forces not large enough to tilt it?
 
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haruspex said:
It's just the balance of forces on the plate. If they are not equal the plate will accelerate as per Newton’s laws.
P1=P2 makes no sense since the pressure is a function of depth. If the pressures are equal at the bottom they are not going to be equal half way up, and certainly not equal above that.
Thank you very much! "since the pressure is a function of depth" that answers my questions.
 
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Chestermiller said:
In addition to what @haruspex said, do you also need to balance the moments on the barrier, or are the forces not large enough to tilt it?
Nope, it is not required to balance the moments in this specific problem. Thank you!
 
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