Pressure is the same in every direction in a fluid at a given depth; if it weren’t, the fluid would be in motion. (This is a well known fact to scuba divers!
If a fluid is NOT flowing, it is at rest, then the P on all sides must be equal to keep it static.
The force due to the fluid’s pressure is ⊥ to any surface it is in contact with.
If there were a component of the force that is צ to the surface, then according to Newton’s 3rd Law, the surface would exert a force back on the fluid that would
also have a component צ to the surface. If that were the case, it would cause the fluid to flow and it couldn’t be at rest!
∴ the F due to the P is ⊥to the surface
How does the pressure of a liquid of uniform density vary with depth?
Consider a point at a depth , h, below the surface of a liquid [surface is at
height, h, above this point]
P due to fluid at h is due to the weight of the column of fluid above it
∴ P= F/A = ρAhg/A
P = ρgh
∴ P ∝ ρ ∝depth
∴ P at EQUAL depths are the SAME!
Incompressible – density is constant and does NOT change with depth.
Obviously the oceans are exceptions, this applies to MUCH smaller fluid samples! [gases are VERY COMPRESSIBLE and ρ varies with depth]
IF ρ varies only slightly, then ΔP = ρgΔh