Dimensional Analysis is a way to simplify complex calculations. This method is relatively easy to learn and master, and may seem like a waste of time to the advanced math and/or science brain. I've found it very useful. Especially for Chemistry, Physics or Electronics classes.

In short, Dimensional Analysis is a way of arranging values and their respective units in fractions or ratios. When two units are exactly alike and diagonaly oriented, they are 'crossed out' or eliminated. If, after all the crossing out you end up with the unit(s) desired, then you can simply multiply across and then divide.

Here is a small demonstration (ASCII for your pleasure) of Dimensional Analysis on a Stoichiometric problem:


N2 + 3H2 => 2NH3

             4.3gN2      1mol     3 mol H2   2.02g   0.93g of H2 
  ?g of H2=  ------- x -------- x -------- x ----- = -----------                 1       28.02g    1 molN2    1 mol       1

Thanks to Storm Damage and vivid for pointing out the pre tags! D'oh!

Dimensional Analysis is an extremely important tool for the physicist. The basic principle behind this is that if you have an equation the dimensions of the physical quantity on the right should be the same as the dimensions of the physical quantity on the left.

Consider the equation

x = vt

The dimensions of x(position) are L. (L stands for length). The dimensions of v(velocity) are LT-1 where T stands for Time. The dimensions of T(time) are just T. So the dimensions of the RHS are LT-1 * T = L.
Thus the dimensions of the two sides match and the equation is dimensionally consistent

This is an important tool in situations which are difficult to analyze using other means. Let us say u want to find a relation for one quantity in terms of other quantities and the relation is difficult to derive analytically. So u can just find a combination of the other quantities which has the right dimension. Of course this would only generally give the correct order of magnitude but that is Okay for an estimate.

Also, a very useful tool for mechanical engineers who work anywhere near the field of fluid dynamics. Using dimensional analysis through the buckingham pi-theorem, a problem containing a great number of variables can be reduced to a function dependent on those same variables...this can be tedious if you have more than 3 contiguous equations, but at least it's not navier-stokes!

thnx pope, i forgot

Except for Suvrat, those are all examples of unit analysis. A dimension is something that can be measured, like length, or mass, whereas a unit is a standardized scale which is used to measure it. Unit analysis is a wonderful technique for simplifying equations, as well as double checking your work, to make sure you haven't made a mistake. If the units don't match up, you've either used the wrong equation, or dropped a term in your calculation.

Dimensional analysis, however, is for deciding which basic dimensions a unit actually measures, which is very useful to develop unit conversions, for use in unit analysis. There are many units which don't measure basic dimensions, but rather apply to derived dimensions, like pressure, energy, temperature, etc.

A Newton, for example, is a unit to measure the dimension of force, but what is force?

Let's start with the well known equation: force = mass x acceleration

Mass is a basic dimension, but acceleration is not - it is derived from the rate of change of a velocity, or velocity / time, but velocity itself is the rate of change in distance, or length / time

Using only basic dimensions, we've now got

force = mass x length / time2

Applying this to unit analysis, we know that if the unit for force is the Newton, the units on the right better be

kilogram x meter / second2

Similarly, we know that

E = mc2 . So what is energy?

Using the same technique:

  • m = mass
  • c = the velocity of light, which is length / time
  • and so c2 = length2 / time2
So, energy = mass x length2 / time2

If we are using Joules for units of energy, we'd better have kilogram x meter2 / seconds 2 in our result.


Those are some basics - stuff we already know based on standard use of units. Let's try a weird one:

Most people in the US measure their car's use of gas as miles per gallon, (or kilometers per liter outside of the US) and use the unit Mpg for it. Typical small cars get 30 Mpg. But what dimension does a Mpg measure?

Let's start with the dimensional analysis.
A mile is a unit of length, and a gallon is a unit of volume.
Volume is length3.
Hence, we can see that gas efficiency is 1 / length2 , or 1/area.
This is the dimension used for a cross-section.

What on earth does this cross-sectional area, representing gas efficiency refer to? Let's convert Mph (using unit analysis) into standard CGI units, to see how big it is.

1 gallon ~ 4 liters = 4,000 centimeters3
1 mile = 1.6 kilometers = 1,600 meters = 160,000 centimeters

So, 30 miles / gallon
= 30 x 160,000 centimeters / 4,000 centimeters3
= 1,200 / centimeters 2

But what the heck does 1,200 / cm2 mean?

It's a cross-section, so if we invert the value, we'll have an area:

cm 2 / 1,200
= 100 millimeters2 / 1,200
~ 1/12 millimeter2

...which is an extremely small area - it's the area of a square about 1/3 of a millimeter on a side.

OK, great, we now know that 30 Mpg is a cross section of something, and the area of the cross section is tiny. But what is the something, physically? In terms of operating your vehicle, you know that your car's Mpg means that after a certain amount of travel, you have to refill the gas tank, but what does Mpg actually measure which is 1/3 of a millimeter on a side?

You might be tempted to think of the cross section of the fuel line in the engine, which is a pretty good guess, since it's about the same area, but that's not it.

Imagine your car, traveling over a very long, very thin, very short trough, full of gas. As you travel, the car consumes gas at exactly the rate needed to use up the gas, as it goes: Behind you, the trough is empty, in front of you, it's full.

Mpg is the cross sectional area of this trough. That's why it's thin and short:
30 Mph means a trough about 1/3 of a millimeter on a side.
My old Chevy Suburban, with Mph of about 10, would need a trough about 3 times the cross section: about 1/2 millimeter on a side.

Dimensional analysis can also make rotational velocity much easier to understand.

Dimensional Analysis: A Modest Portrayal of Space-Time

Introduction

Before engaging on a philosophical exposé on the topic of time and its perceived passage, much introductory knowledge is required to be shared with the audience. As a proper preface, the following sections will provide briefings on the qualities of space, the vocabulary concerning time, and a further definition of select concepts which could require clarification. At this point the node will delve into deeper abstraction, relying on basic presupposed scientific principles to suggest a new way of thinking about the fourth dimension (which for the purpose of this node will be at all instances presupposed to be time, or a derivative of space-time). All this will lead up to some attempts to define what time really is in a series of pictorial analogies. The final section will deal with whether events happen when they occur on this empirical time scale or when they are observed.

Three Dimensional Space

The first three dimensions in space are width, height, and depth. These are the basic dimensions that give objects, such as pens, tables, and the computer screen you're reading this off of, their volume and indirectly their mass. As a three dimensional particle is projected through space, it has a position as a function of time (which we will be defining in one minute), as well as a velocity, acceleration, and jerk (the change in acceleration you feel most on theme park rides). As an example, if the position of the particle can be defined as a third order curve (a cubic function), its velocity would be second order (a parabola), its acceleration would be linear, and its jerk constant. Across the globe, humans have very exact, very precise ways of measure space because we can see what we are dealing with, and visual problems are much easier to work with than abstract ones.

As stated, those three dimensions were the ones which grant a substance volume, its mass. It is stipulated that the fourth dimension is the one that grants substance its existence. It is said that unless an object has length, width, and depth for several consecutive instances, it is truly nonexistent. The fourth dimension has a most popular candidate, and it is time. Slightly more formal discussion could also refer to the 4th dimension as space-time. A small example as to the nature of the 4th dimension can be seen in the following excerpt to close this sectioni:

“Imagine a 2-dimensional world, like a piece of paper. If you were to draw a line straight down the page, no 2-dimensional beings would be able to cross from one side of the paper. In a 3-dimensional world, a line is an ineffectual barrier. To impede 3-dimensional creatures a plane is needed. For clarification, a 4-dimensional creature is nearly as impeded by a plane as 3-dimensional ones are by lines.”

Space-Time

Time is a very tricky subject, so the best efforts will be taken to break it down bit by bit. The first step to true understanding is to realize that there are different levels of time. Empirical time is never changing, never ending. It is like the tick of an atomic clock, and not nearly as fickle as manmade clocks, which vary by as little tampering as travel or a change in sea levelii. The time of events is the second layer of time. This layer is linear due to the absurdity of the opposite argument. Imagine a portrayal of time as a path one must walk down to reach (possibly) an end. The current concept of time, limiting myself momentarily and for brevity’s sake to the Western world, is that time would look much like a straight line from birth to death. One must ask, what ever would happen if there was a loop in the middle of this line? Say a young man was walking down this path. There would come a time when he came across an older man, looking quite like himself, who would tell him everything about his life, including his exact thoughts at the moment. The old man would end the conversation by telling the young man they were the same person, and that one day he would understand. Years later the young man, now aged and more wise, sees himself as a young man walking down the street, and the ‘old man's’ prophecy holds trueiii.

Now, if we are to accept time to be linear, it is rational to assume that time has both magnitude and direction. The magnitude we believe to have pretty figured out, on an empirical and event time level at least. The direction is still an open matter of debate which we shall now attempt to resolve.

One concept of direction of time is the popular Arrows of Time ideaiv. In this example, there are three arrows, and their directions point in the same direction as the actual direction of time. The first of these is the thermodynamic arrow, or the arrow of entropy. This arrow points in the direction in which the entropy of the universe is increasing instead of decreasing. Imaging a cup sitting on the edge of a table with a camera recording the time of events. If, through unforeseen or directly applied force, the cup is to leave the table, gravity’s pull would cause the cup to collide with the earth and shatter into a large number of pieces. This display is a fine example of entropy increasing, and we know it as an argument for the direction of time being forward partially because the thermodynamic arrow points forwardv. In order for the arrow of entropy to point in the other direction, such an atmosphere would have to be created in the room such that the cup jumping up and putting itself back together again would increase the entropy of the room, a feat that as of now is not nearly possible.

The second arrow of time is the psychological arrow of time, or the arrow of awareness. This arrow points in the direction in which we remember the past, but we cannot remember the future. One way of examining the effects of this arrow is to imagine a meteorite streaking across the sky. If time points in the direction we currently think it does, we would have no knowledge of the meteorite until we saw it, and then we would remember it for as long as we wished. If the arrow of awareness pointed in the opposite direction, we would know about the shooting star up until the moment that it was visible in the night sky, afterwards having no recollection of it. This is the sense that we can unlearn things, unknow past events, and it is an absurdityvi.

The final arrow of time is the Cosmotological arrow of time. This arrow points in the direction in which the universe is expanding rather than contracting. Due to ancient logic, we assume the universe to be infinite and ever-expanding. Consider the options: If the universe was infinite, and the atoms contained therein finite, all atoms would be in constant motion to equal the distance between them, and there would be no matter. If the void was finite, and the number of atoms infinite, there would be no room and the void would cease to bevii.

Moving on in our portrayal of the 4th dimension, we reach the observational layer of time. This is the passage of time as observed by onlookers. In this sense we define the time of events in the order in which we see them to have happened. In short, If E2 is the effect of E1, then E2, is called later than E1. Furthermore, if E1 is the cause of E2, then E1 is a small part of E2, but a small part of E2 is of no significance to E1viii. It is the observation of this changing in events that allows time to exist. Without such a succession of events, there would be no more time. Since the activity of the mind is a succession of nerve synapses and electric pulses, even if you were to think about the contradiction to this argument and say “succession is an illusion,” your thinking has been itself a succession and your thought is therefore self falsifyingix.

The final layer of time as I will be presenting (as to not exclude future developments in the field) is the layer of perceived time. This is the layer in which one hour’s worth of empirical time can seem like only minutes, while two minutes in some certain cases can seem like an eternity. This is because when events are happening we aren’t observing all of them. Due to selective, or unconscious, ignorance, several events per second can occur unseen to us. Partially, we have selective memories of “past” events, but on the other hand we only have the ability to remember things which we have previously observed directly, which is why time can appear to be slow at some times and quickened in othersx.

The Present and Problems of Time

Of all the different definitions of the present, the one which I found to be most agreeable was presented by Kennith Denbigh. He said the present is a saddle point in time, not a knife’s edge as was popular for the eraxi. A knife’s edge would be metaphorically relative to time speeding up right before an event and then dropping off in a free fall. It would be as if when walking up steps the stairs themselves were instantaneously fabricating themselves as I walked, and becoming nothingness behind me. By defining the present moment as a saddle point in time, we can call the present a more natural progression, one that smoothly flows from event to event. As H. Witrow was quoted as saying in Denbigh’s book, “The past is determined, the present is in the moment of ‘becoming’ when events become determined, and the future is as yet undetermined.” His quote reinforces the concept of a smooth flowing time.

Time still has its problems that need dealing with. The first of which is if we have enough units to measure the change in time. We measure distance in inches and feet or meters, for time we use the second, minute, and hour. Some have said that time shouldn’t measure time, and perhaps the conception of a second unit is needed to deal with the possibility that time is not as simple as we currently think it is.

Pictorial Depiction

Kennith Denbigh once stated that pictorial analysis of time was a bad idea. I would tend to disagree. When trying to prove a point or explain an abstract idea to a large audience, one should always try to tie in visual aides or analysis because it will help ease the high level of abstraction in the theory behind your thoughts, and bring the audience into the midst of your subject. As far as time is concerned, there have been classical imagery and pictorial analysis to help relate the subject to a larger audience. All of these examples have validity and some measure of truth to them, and perhaps when combined with each other will give a true ‘picture’ of time.

The first two which I would like to introduce would be time as a river or desert. When time is perceived as a river it is insinuating that one cannot return to the past. There is an old adage that you can’t step in the same river twice and it’s true, you can’t. Even if you were to step in the same location of a river multiple times, the river, the rushing waters themselves, have already moved on and changed. If you stop moving the river would rush past you, so in order to stay in the present one need to keep moving in the direction of the future. When time is considered as a desert, we are introducing the aspect that time is expansive and cyclic like the dunes of a mighty desert. In this sense, you can be at various different points in time, and have events which are extremely similar to each other, which is reflective of the Eastern portrayal of time as a circle, or more accurately a never-ending spiral.

In his book “Along the 4th Dimension,” Joost Meerloo has an entire section on the different ways time has been pictorially represented during the ages. The first one he deals with is the ouroboros, the Eastern way of thought. The ouroboros is an image of a serpent or dragon swallowing its own tail, reinforcing the circle or never-ending spiral. Secondly Meerloo presents the Western idea of the time arrow, which we dealt with extensively earlier in this paper. With such an emphasis of progress and growth in the western hemisphere, he explains that there is no surprise time would be seen as a straight line here. The final presented view in his book is the Hindu representation of time. In this view, time is a placid pool with occasional ripples. These ripples, they believe, are our lives, and they are as fleeting and as ephemeral as shooting stars. Even those shooting stars which we can remember forever after seeing them.

Conclusion

The final question to answer then is whether time should be accepted as the event time on an empirical scale, or as observed time, on an individualistic scale. Consider a lake with a water bug sitting calmly. If you were to break the surface of the lake with your finger, a ripple would begin to spread out. Graphing the area of the ripple as a function of time would result in the event horizon of the ripple. When the event horizon grows large enough to encompass the water bug, the bug becomes aware of the event, several seconds after it happens. A similar example could be the death of our sun. If the sun were to explode the event horizon of the blast would not reach the earth for several minutes (8 in fact) which would suggest that time should be based on the observational layer of time, because that is when the events become real to rational beings.

As a proper closing remark I would like to present a quandary that I came across indirectly. Stephen Hawking, in his book, dealt greatly with the universe as a whole, not just time. He had some speculation that if the universe had a beginning, it must also have an end, and that it would probably take some catastrophic event to reach that end. Now, referring back to Denbigh’s theory of succession, if there were a catastrophic event that had the potential to bring an end to the universe, it would probably wipe out all atoms in the universe as well. Without atoms or matter there would be nothing to form events. At the precise instant that the universe would be predicted to end through lack of contents, time would halt due to the lack of succession, and the universe would teeter, hypothetically, on the edge of destruction onto infinitum. To obscure an Ashleigh Brilliant quote, an optimist would probably be pleased to hear this in that we would have warning of the universe's destruction, while a pessimist fears it has already happenedxii.


Footnotes
iDrawn out from a late night discussion, later accreditation for this example was given to Stephen Hawking, work unknown.
iiHawking, “A Brief History of Time.” Example shows two clocks synchronized can be defaulted by extensive travel making the clock speed up, and higher altitudes slowing the clovk down.
iiiReichenback, “Philosophy of Space and Time.”
ivHawking, “A Brief History of Time.”
vI originally had conceived this idea for the purpose of tutoring a friend in thermodynamics in high school chemistry; it was not until recently that I discovered Stephen Hawking also made use of a cup example, quite a while before I had the idea. Accreditation once again goes to Stephen Hawking.
viMeteorite example borrowed from Kennith Denbigh, “Three Concepts of Time.”
viiEpicurus, “The First Principle of Materialism.”
viiiReichenback, “Philosophy of Space and Time.”
ixDenbigh, “Three Concepts of Time.”
xGoodfield and Toulmin, “Discovery of Time.”
xi“Three Concepts of Time.”
xii“An optimist thinks our world could become another world’s hell, a pessimist fears it is already true.” –Ashleigh Brilliant.

Log in or registerto write something here or to contact authors.