| If we fill water or another liquid into a tank and measure the pressure of the fluid at the bottom, we notice that this value increases as we add more and more of the fluid. Observation demonstrates that the pressure increases linearly with depth (Movie 1). This finding is independent of the size and shape of the container. In fact, we would measure precisely the same linear relation between pressure and depth in a lake, in a narrow tank, or even in the ocean! (See Figure 1.) The slope of the straight pressure–depth relation does depend upon the density of the liquid, however. In water it increases by 1 bar every 10 meters (1 bar = 10^5 Pascals). In a typical vegetable oil this would be reduced to about 0.9 bar for a 10 meter column. It would be 13.6 bar for 10 meters of mercury. In the Earth’s atmosphere, the pressure decreases upward, just as in a lake, but the pressure-height relation is not linear (Figure 2). The pressure at sea level changes by about 0.001 bar every 10 meters, and it changes less quickly at higher altitudes.
Interpretation Fluids press down upon layers below them as a result of their weight. That explains why denser fluids have a steeper pressure gradient. (A gradient measures how quickly a quantity changes in a certain direction. So it is a kind of “spatial rate of change.”) Apparently, the pressure gradient in a fluid due to gravity is proportional to the density of the fluid (Figure 1). In Figure 2 we see that the density of the air in the atmosphere decreases with height above the ground. Experience shows that the pressure-height relations in fluids at rest do not depend upon the form or size of the containers or the horizontal extend of the fluid (see Figure 1, center).The pressure gradient for a fluid is the same in a narrow tank and in a wide one. We conclude that pressure has to be the same in a liquid at a given height since there is no horizontal flow. | |
