# $P = h \rho g$ derivation

Here is a snapshot from textbook, University Physics by Sears and Zemansky : Here only three forces are calculated:

1. The gravitational force
2. The force due to the pressure $$p$$ on the bottom surface
3. The force due to the pressure $$p+\mathrm{d}p$$ on top surface

Why are only these three forces considered? Why is the force due to the pressure $$p+\mathrm{d}p$$ on the top surface downward, and why is the force due to the pressure $$p$$ on the bottom surface upward? Why not vice versa? Why does the downward force only apply on the top surface, it must apply on the bottom surface too, doesn't it?

## 1 Answer

This is a Newton's 3rd law (action/reaction) type situation, similar to what we encounter when we have tension in a rope. The force per unit area (pressure) that the fluid above exerts on your mass is directed downward, and the reaction force (pressure) that the mass exerts on fluid above is directed upward. But only the force exerted by the fluid above on the mass figures in the force balance on the mass. Similar for the bottom surface.

• I can't understand what you mean by "But only the force exerted by the fluid above on the mass figures in the force balance on the mass". – Hardey Pandya Mar 21 '16 at 15:56
• When you do a force balance on a body, you only include the forces exerted on the body by other bodies, not the reaction forces the body exerts on these other bodies, right? – Chet Miller Mar 21 '16 at 18:23