Density and pressure as scalar point functions

SolitaryRoad.com

Website owner:  James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

Density of a substance. The density of a substance of uniform density is its weight per unit volume and is given by the formula D = w/V where D is the density, w is the weight and V is the volume of the substance. However, the density of a substance may vary from point to point within the substance. How then is the density of such a substance at a particular point P within the substance computed since a single point contains no mass? Imagine a cube centered at point P with edges of length Δx. Using the formula D = w/V compute the density for a succession of smaller and smaller cubes centered at point P for a succession of smaller and smaller values of Δx. Example. Compute density for Δx = 1mm, 0.1 mm, 0.001mm ,etc. Doing this gives a sequence of values for the density, D, which have as a limit the density of the substance at point P.

Pressure. Pressure may vary with location and thus be a point function. For example the air pressure in the atmosphere may vary with location.

The concept of pressure is associated with the concept of a load distributed over an area. A pile of sand would be an example of a load distributed, or spread out, over an area. Pressure is defined as the force per unit area (pounds per square inch, for example) in a situation where you have a load distributed over an area. Consider a pile containing 2000 pounds of sand. The 2000 pounds of weight is not concentrated at one point on the ground but is distributed over the area the sand is sitting on. At each point under the pile of sand there is some force acting on the ground measured in, say, lbs/in², due to the weight of the column of sand above it (envision a vertical column of sand measuring 1 inch x 1 inch in cross section extending from the ground to the top of the pile). Now if we view the entire pile of sand as consisting of 1 inch x 1 inch columns then the sum of all the downward acting forces from all the columns total to 2000 pounds and the force at any point under the pile comes from the weight of the column of sand above it. The force at a particular point under the pile is the pressure at that point. Consider the pile of sand to be lying in the x-y plane of an x-y-z Cartesian coordinate system. Then the pressure p at any point x, y is given by p = f(x, y). What is the pressure at a particular point (x, y)? Since a point has no dimensions, we must define what we mean by the pressure at a point. We imagine the point (x, y) to be centered in a small square with sides of length Δu. We then compute the pressure for a succession of smaller and smaller squares centered at point (x, y) for a succession of smaller and smaller values of Δu. Example. Compute pressure (in lb / sq. in) for Δu = 1 in, 0.1 in, 0.01 in , 0.001 in etc. Doing this gives a sequence of values for the pressure which have as a limit the pressure in lb / sq. in. at point (x, y).

Another example of a load distributed over an area: the load on the bottom of a container containing a liquid such as water. If there is 20 lbs of water in the container and the container bottom has an area of 10 in² there is a pressure of 2 lbs/in² at each point on the bottom due to the weight of the column of water above it.

Envision a vertical column of liquid 1 inch x 1 inch in cross section. The pressure at any point in the column is due to the weight of the liquid above it. In general, the pressure, p, exerted by a liquid of density, D, at depth h is given by

p = hD.

Liquids exert pressure in all directions. The pressure exerted by a liquid is directly proportional to the depth of the liquid and to the density. It is independent of the area or shape of the container; it also is independent of direction.