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Chapter 8: Fluids
4.

How Pressure Varies with Depth

As an object descends into a body of fluid, the pressure that the fluid exerts on the object increases. To see why, consider a cubic region of fluid with height Δy, whose top and bottom surfaces have area A. Fluid exerts pressure on everything it touches, including other bits of fluid, and so the cube feels force on it from the fluid that surrounds it. If the fluid pressure at the top of the cube is Pt, then the cube feels a force Ft=PtA on the top surface. The bottom of the cube feels a pressure Pb and an upward force of Fb=PbA. The cube experiences a third force, due to the weight of the fluid: Fw=mg. The volume of the cube is V=AΔy and its mass is m=ρfV=ρfAΔy (where ρf is the density of the fluid); therefore the weight of the cube is Fw=ρgAΔy. These forces must balance each other, so ρgAΔy+PtA=PbAPbPt=ρgΔy In other words, the change in pressure between any two points in a fluid depends on how deep one point is as compared to the other:

ΔP=ρfgΔy

That lets us calculate changes in pressure, but how can we calculate the actual pressure? To do that, we need to know what the pressure P0 is at some reference point. Then the pressure at some point a distance d below the reference point is

P=P0+ρfgd

For example, if the fluid is in contact with a gas such as the atmosphere, then the pressure at the surface of the fluid is the same as that of the gas: if the pressure in the fluid were larger, than the fluid would rise; if smaller, it would fall. Such a surface is often a useful reference point: the pressure of the open atmosphere, for instance, is typically Patm=1.01×105Pa, as mentioned earlier.