11.4 Variation of pressure with depth in a fluid

If your ears have ever popped on a plane flight or ached during a deep dive in a swimming pool, you have experienced the effect of depth on pressure in a fluid. At the Earth’s surface, the air pressure exerted on you is a result of the weight of air above you. This pressure is reduced as you climb up in altitude and the weight of air above you decreases. Under water, the pressure exerted on you increases with increasing depth. In this case, the pressure being exerted upon you is a result of both the weight of water above you and that of the atmosphere above you. You may notice an air pressure change on an elevator ride that transports you many stories, but you need only dive a meter or so below the surface of a pool to feel a pressure increase. The difference is that water is much denser than air, about 775 times as dense.

Consider the container in [link] . Its bottom supports the weight of the fluid in it. Let us calculate the pressure exerted on the bottom by the weight of the fluid. That pressure    is the weight of the fluid $\text{mg}$ divided by the area $A$ supporting it (the area of the bottom of the container):

We can find the mass of the fluid from its volume and density:

The volume of the fluid $V$ is related to the dimensions of the container. It is

where $A$ is the cross-sectional area and $h$ is the depth. Combining the last two equations gives

If we enter this into the expression for pressure, we obtain

The area cancels, and rearranging the variables yields

This value is the pressure due to the weight of a fluid . The equation has general validity beyond the special conditions under which it is derived here. Even if the container were not there, the surrounding fluid would still exert this pressure, keeping the fluid static. Thus the equation $P=\mathrm{h\rho g}$ represents the pressure due to the weight of any fluid of average density $\rho$ at any depth $h$ below its surface. For liquids, which are nearly incompressible, this equation holds to great depths. For gases, which are quite compressible, one can apply this equation as long as the density changes are small over the depth considered. [link] illustrates this situation.