4.1.1 Hydrostatic Pressure (hρg)

Consider 5 weights, each of weight 100 N and cross-sectional area 0.10 m², stacked one on top of another. If an unfortunate spider somehow got itself wedged between the 2 weights at the bottom, it is going to be subjected to a pressure of $\displaystyle p=\frac{F}{A}=\frac{{400}}{{0.10}}=4000\text{ Pa}$ .

Now consider a cylindrical volume of water (held in a beaker, perhaps). At a depth of h below the water surface, the pressure must be equal to the weight of the water mg above that height, divided by the circular cross-sectional area. Writing the expression in terms of the density of water ρ, we get

$\displaystyle p=\frac{F}{A}=\frac{{mg}}{A}=\frac{{\rho Vg}}{A}=\frac{{\rho hAg}}{A}=h\rho g$

This is called the hydrostatic pressure or static fluid pressure.

Obviously, water is different from a stack of weights. Unlike a solid where molecules are strongly bounded and fixed in position in a rigid molecular structure, a liquid has molecules which are loosely bounded and are free to “wander around” in a fluid molecular structure. As such, water always flows downward and sideward, until it fills the container and takes the shape of the container. The container is required to provide the horizontal forces to “hold up” the water.

Pressure acts in all direction

This means that besides vertical compression caused by gravity (fun fact: seawater is 4% denser at 10, 000 m depth), a fluid is also subjected to horizontal compression. As such, every molecule in a fluid exerts repulsive forces against all the surrounding molecules in all directions. This leads to the unique property of fluids: pressure “acts in all directions”. (Do realize that pressure, as a scalar quantity, has no direction. It is the pressure forces that act in all directions).

Fluid pressure depends only on vertical depth

Remember that the compression of the fluid is caused by the weight of the fluid. Since gravity acts vertically, the compression should only increase vertically with depth. Horizontally, at any given height, the compression should be uniform. (Keep in mind that the molecules in a fluid are free to “wander around”. If the compression at any given horizontal level is not uniform, there will be unbalanced horizontal forces to move the fluid around until the horizontal pressure is equalized.) This results in another property unique to fluids: fluid pressure (at any point in the fluid) is only dependent on the vertical depth (from the top surface), regardless of the shape of the fluid. In other words, the formula $\displaystyle p=h\rho g$ holds even if the container does not have a uniform cross sectional area.

Atmospheric Pressure

We are literally submerged at the bottom of a deep ocean of air, aka the Earth’s atmosphere. The atmosphere’s own weight is causing itself to be pressurized. What the atmosphere lacks in density (of the order of 1 kg per m³ at sea level, dropping with altitude) is more than compensated for by its sheer size (roughly 20 km of air above us). This results in a formidable atmospheric pressure (at sea level) of about 100,000 N per m².