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

[rbmartel]

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.

 

[haruspex]

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.

 

[Chestermiller]

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?

 

[rbmartel]

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.

 

[rbmartel]

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!