The deeper an underwater swimmer goes, the greater is the pressure that he experiences due to the added weight of the water above him. Beginning with the equation for Pressure we can easily derive an equation for how pressure changes as the swimmer goes deeper. Imagine the swimmer at a certain depth (h).

P = F / A Force equals the weight (mg) of the water above him.
P = mg / A Mass = Density of fluid x Volume
P = ρVg / A Volume = Area x Height
P = ρAhg / A Area cancels out P = ρgh

The equation above indicates that if a pressure P_{1} is known at a higher level, then a larger pressure P_{2} at a deeper level can be calculated by adding ρgh. In determining the pressure difference ρgh, we assume that the density ρ is the same at any vertical distance h or, in other words, the fluid is incompressible. This assumption CAN be made for liquids, since the bottom layers can support the upper layers with little compression. In a gas, however, the lower layers are compressed much more by the weight of the upper layers, with the result that the density changes with vertical distance. Therefore the density of our atmosphere is much larger near the Earth's surface than it is at higher altitudes and the above equation can only be applied when h is small enough that there is not a big change in density. Another important feature of the above equation is that pressure is affected by the vertical distance (h) but not by the horizontal distance within the fluid.

According to the picture, find the pressure at both points A and B. Do not forget to include the atmospheric pressure into your calculations which is 101.3 kPa.

Since points A-D are at the same distance h beneath the liquid surface in the container on the right, the pressure at each of them is the same.

Mercury Barometer

One of the simplest pressure gauges used for measuring atmospheric pressure is the mercury barometer invented by Evangelista Torricelli. This device is a tube sealed at one end, filled completely with mercury, and then inverted, so that the open end is under the surface of a pool of mercury. The space in the tube above the mercury that results is a vacuum and the pressure is zero. In the diagram, points A and B are at the same level so they are at the same pressure. Clearly the pressure at point B is atmospheric pressure. If the density of mercury is 13,600 kg/m^{3}, what will be the height of the mercury column in millimeters? inches?