Dimensional Analysis for Unit Conversion Using Matlab®

A first order approximation of the definition of a dimension is that it is a measurable extent, such as length, width, depth, and so forth {1}. The mental concept that initiated the first attempt at understanding the creation of a basic set of dimensions initially arose after one of our early wandering ancestors had acquired both an adequate amount of spare energy and sufficient inquisitive desire to attempt trying to comprehend and record the puzzle of dimensions as well as the ongoing demands of surviving each day in their hostile life-threatening environment.

This successful ancestor’s leisure became available after the ancestor’s group developed sufficient foresight and ability to obtain, preserve, and defend surplus quantities of foodstuffs. This required the group to have the ability to defeat, fend-off, or out-think other hungry predator groups, both animal and human. The first true level of new understanding of the functioning of their immediate environment came about slowly when different individuals within the group developed special (unique) skills. Those skilled individuals in each group gradually developed improvements in their intuitive skills. The net result was that each skilled individual aided in raising the overall growth of the long-term survival rate for all the members of their group.

As each skilled individual survived and then, through cultural transmission, passed on their newly acquired observational and survival skills to the next generation, they also passed on the desire to survive to each of the subsequent generations. Those members and groups that did not participate in the skill acquisition and transfer process failed to survive and are not now here. We, the readers of this book, are among the present surviving members of this skill-based process.

One of the first dimensional concepts that impinged on the leisure time consciousness of our earlier ancestors was the concept of time {2}. At least one of some group’s more observant individuals noted that the local environment often became lighter (day {3}) and darker (night {4}). At a later point, some of those same individuals and perhaps their friends additionally noted that during the darker period (night) various patterns comprising lighted {5} objects (moon {6}, stars {7}, planets {8}, meteors {9}, comets {10}, etc.) were visible only during the dark period. Some of said individuals also noted that a few of the patterns were relatively constant, whereas other patterns moved rapidly and some patterns moved very slowly across the dark (sky). The observers additionally noted that the object that they called the moon had a repeating pattern (currently named a lunar month {11}(approximately 30 days)), of light and dark cycles.

Once one of these early ancestors had an initial grasp of the concept of time, the next dimensional concept that this curious ancestor would have also noted, perhaps simultaneously, is that of the change of the environmental temperature {12}. As each observer’s time flowed onward, the light level in the local environment rose and fell with the transition through the day-night cycle. At some point, the early observer would also have noted that the local environment would, over a period of moons (several days = month) get warmer and later get cooler. As the number of lunar months passed, the temperature of the local environment would fluctuate from hotter to colder cyclically (summer {13}, winter {14}) during the yearly {15} seasons {16}. The observation of environmental temperature fluctuations to lower values (cooling) would, in some locations, promote the discovery and use of fire {17}, the development of clothing {18}, and eventually the development of shelters {19}. Perhaps the group would have to migrate to a warmer environment if they knew what dimensions to use to guide travel to that warmer location.

All of the preceding observations require stored memories (either in the head or carved into wood, rock, clay, or metal) of earlier environments and their conditions. They also require that the recorder have sufficient caloric {20} intake and the ability to learn writing {21} and counting {22}. Sufficient caloric intake implies cooking {23} and language {24} skills, and those imply that the earlier ancestor have an adequate brain size {25} with sufficient structural complexity for rational {26} thinking {27}.

Even the most fundamental survival skills require a basic intuitive understanding of the concept of and application of dimensional analysis to our everyday tasks. We need to consider such concepts as size (bigger > same > smaller), flow (faster > same > slower), etc. The universe that we live in requires both a large range and a diverse collection of different dimensional types (see Figure 1-1). In order to understand the workings of this universe, at least to First Order (minimally), we humans must be able to recognize and utilize the relative influence and relative importance of applied dimensional analysis. We must also consider the effects that modifications and/or errors in applied dimensional values will have on the events and opportunities occurring in our real lives on a timely (second-by-second) basis.

Figure 1-1 {28} shows a quick overview of the role of applied dimensional analysis considerations that are required in characterizing the dimensions of various objects that are of interest in this universe. In this case, it is shown by employing the CGS (centimeter (cm), gram (g), second (s)) system to indicate relative size of various well known objects. In this book, only Classical Physics {29} examples are considered. For clarity, however, the location of the non-Classical Black Hole Region and non-Classical Quantum Region are also shown in Figure 1-1.