Calculating Pressure Difference in an Equilibrium Hydrostatics System

[Lavabug]

Homework Statement

See attached image. Two cylinders of the same diameter D contain the same incompressible (green) gas of density rho. Their bases are connected via a tube of diameter d containing an incompressible fluid of density rho m. The system is in equilibrium if the pressure on both cylinders is P atm.

If we supply different pressures P1 and P2 to each of the cylinders respectively, the fluid on the right hand side will rise by a height of Hm.

Express the difference P1 - P2 in terms of the given variables.

The Attempt at a Solution

I'd say P1 = P2 + rhom*gHm, so I'd just solve for P1-P2 and I'd have the answer, but something tells me I'm neglecting something. I neglected the contributions of the gas of density rho on both sides since I figure they would cancel out, am I doing this correctly?

 

[ideasrule]

That's the right answer, because an incompressible gas doesn't behave differently from the incompressible fluid. However, are you sure it's supposed to be an incompressible gas? I don't think those exist--by definition, the molecules of a gas interact very weakly, so the gas is very compressible.

 

[Lavabug]

That's the right answer, because an incompressible gas doesn't behave differently from the incompressible fluid. However, are you sure it's supposed to be an incompressible gas? I don't think those exist--by definition, the molecules of a gas interact very weakly, so the gas is very compressible.

Whoops, you're right, the green substance in question is also a liquid, so no gases. Thanks.