Hydraulic press equilibrium equation inconsistency

Suppose we have a hydraulic press with a smaller area $$A_1$$ and a bigger area $$A_2$$, with the smaller area being higher with a height difference of $$\Delta h$$.

We first calculate the pressure at point A. By Pascal's law, the increase in pressure at point A comes from the pressure created by forces $$F$$ and the weight $$G_m$$ of the mass $$m$$. These pressures get added at each point in the liquid, so for point A we get $$p_A = F_1/A_1+F_2/A_2$$.

With similar reasoning for point B we get $$p_B = F_1/A_1+F_2/A_2+\rho g \Delta h$$ where we have now taken into account the hydrostatic pressure from the height difference.

However, if the system is now in equilibrium, at the area $$A_2$$ we must have equality with pressure from above and pressure from below, so that the forces acting on both sides are equal. This would give $$F_1/A_1$$ for the pressure from above and $$p_A = F_1/A_1+F_2/A_2$$ for the pressure from below. This would imply that $$F_2=0$$ which is a contradiction.

Where am I going wrong? I'm stuck with this and there must be a fundamental misunderstanding somewhere in there. I would really appreciate any pointers.

1 Answer

These pressures get added at each point in the >liquid, so for point A we get 𝑝𝐴=𝐹1/𝐴1+𝐹2/𝐴

This is where you go wrong. In static equilibrium, the pressure is the same in all directions, and neglecting the weight of the fluid, at all points in the fluid. This means that the two pressures must be equal, but never that they add together.

If you wanted to find the pressure by considering the total influence of both sides, you could do so, but the formula would be P=(F1+F2)/(A1+A2) , not F1/A1+F2/A2. It is as if you are trying to find the density of a composite object, and you have approached it by adding up the densities of each component.