Saturday, January 5, 2013

Day 10 - Fundamentals of Physics - 1.4 - Dimensional Analysis

You are welcome to check out my previous blogs within the series “Fundamentals of Physics

Dimensional Analysis in physics is one of the most common sense tools to gain access to a problem, and is an essential tool to find mistakes or prevent them within a solution.

Firstly lets see how physics defines dimensions, as this word can have several meanings depending on the back round you are coming from. Dimensions in physics denotes the physical nature of a quantity, thus for instance, the dimension of distance, the physical nature of distance, is length, and as we've seen in chapter 1.1 it is symbolized as [L]. The dimension of volume would then be [V] = [L]^3 because the nature of volume is the nature of distance in the power of 3 - as height, width and depth.

Knowing the physical nature of a quantity would then mean knowing to express it in dimensional, thus using the most basic quantities, as [M], [L] and [T] in mechanics, and other quantities as temperature and electrical current in other fields of physics, but the same principle of dimensional analysis holds in all fields of physics.

Dimensional analysis requires that both sides of an equation can be expressed as the same dimensions, which makes sense because we have stated an equality within placing an equation. Thus, the relationship within the equation can be correct only if the dimensions are the same on both sides of the equation. Quantities can be added or subtracted only if they have the same dimensions, for instance if you have 2 apples and 3 oranges you cannot add them together in an equation, because they are not represented by the same dimensions, you could however in this example declare them as fruit and the you will have 2 fruit and 3 fruit which you can now add because they all have the same dimension of fruit. Dimension thus can be treated as algebraic quantities and all it entails.

Dimensional analysis doesn't solve the problem entirely, but it gives you the main properties of it and is great to use when you have forgotten the relationship and can use this procedure as a reminder. I shall give an example to illustrate how this works, take a deep breath, I'll brake this down in words first so you know what you are about to see - I'm going to find the relationship between distance time and acceleration, using dimensional analysis, I will place them in an equation as proportion to each other because I will not be able to find any factor that is dimensionless, I am going to loose that part of information from the equation, that's way I cannot place the symbol for equality (=), you will also see that I place the quantities in power ofclip_image001[52] I don't know what they are yet, and I will find them out through this procedure where the only restriction I have is that I am required to have the same dimensions on both side of the equation, the rest is basic algebra which I will not explain because that would be an entire blog in itself, here goes:

I want to know the relationship between the distance of an object, in a specific time frame moving in a specific acceleration. I would say then the distance (x) must be proportional to the acceleration (a) in the power clip_image001[54] and to the time (t) in the power ofclip_image001[56].
The equation will look something like this:


where clip_image001[50]  is a symbol for proportionality

Now we represent the quantities in their nature, thus in the form of their dimension and we get this:

Now we need to make sure there are the same dimensions on both sides of the equation and that will determine the value of clip_image001[54] andclip_image001[56]

On the one side we have the dimension of distance [L] in power of 1, thus that is what we must have on the other side for this equation to be valid. We see thus, that the dimension of time must be canceled, thus, for the equation will be equalized within the dimensions of it, we must determine the values

Placing the values we found through basic algebra, we get that the equation

some of you may know that the accurate equation is:

but since 1/2 is dimensionless, there is no way of determining it using dimensional analysis.
So we see here the limitation of this technique is that it will not give us all the information regarding dimensionless constants in the equation, but it will give us a good stepping stone to work with, and as I said it is a great way to check for mistakes within an equation, because if the dimensions don't hold up there is defiantly something wrong that must be fixed and resolved for the equation to be valid.

What is essential for this technique is to understand the nature, thus the dimensions, of the quantities.

I'll give you another example to see how this analysis can go into different solutions, lets say I throw an object from a certain height, and I want to see how the time is dependent / proportional to the height.

I can assume that the time it takes will be proportional to the mass at a certain power, to the height at a certain power and to the pull of gravity at a certain power, the equation will look as follows:

where: h = height, m = mass, g = gravity

and clip_image001[34]  are to be determined through the equation as befor.

On the right side of the equation I have the dimension of time and thus I must have the same dimension on the left side as well. The dimensions of height is [L], the dimensions of mass is [M], and of gravity is acceleration and thus will have the dimensions of:

If you continue with the algebra as before you will find that the dimension of mass must be canceled out and thus =0, which means that the time it takes for an object to fall from a certain height is independent on it's mass, this is considered to be true and we have found it through the simple procedure of dimensional analysis.

Though, we could have created the original equation in such a way that would leave us at a dead end, and that there might be a dependency on mass after all, if we would have said that the time must be proportional to the mass of the object and the mass of the earth, which is still within common sense because the larger the earth the grater the pull is and thus it might effect the time of the fall. If we had done it in that way we would end up with an equation that has the dimension of time on the one side and the dimension of length and mass on the other - which would indicate we had made a mistake or there might be something more to the problem that we haven't considered.

This example was again, to show the limitation of this procedure, to recap, it cannot indicate the factor of the proportion, that is, it cannot indicate an equality because there might be a dimensionless component that the method is not capable of finding, and we can place relationships of quantities in such a way that will bring us to a dead end, which would be a cool opportunity to ask some more questions and get a clearer perspective, maybe discover new relationships that scientist have not thought of yet.

How can we now take this knowledge we have about dimensional analysis and the common sense within it and apply it on our life as a whole, to extract it from the confined world of physics into our entire existence?

What I am going to discuss is a point that we can derive from this regarding equations in general, as it is now clear that both sides of any equation must have the same dimensions to it, what this means is that the input will be equal to the output, in other words, the starting point of ones actions for example, will have the same dimensions as the outflow and consequences of ones actions - As within so without, as above so below. Whatever I put into one end of the equation will have to manifest something of the same dimension as the result on the other end of the equation. This then implies that we must come to know ourselves to such an extent that we can take responsibility for all the input we add into the equation of our lives - so if I am trying to figure out something about myself and my reality I must come to know the nature of myself, as the dimensions I exist as and participate with, dimensions in this context can be thoughts, fears, imaginations, emotions, feelings, reactions - and thus, when I am trying to figure myself out as to understand the relationships in my world that I have allowed to direct me and determine who I am, I must realize that what I bring as myself to the equation is what I will receive on the other side - this demonstrates the principle of "give as you would like to receive", as an actual law of physics, because what we give must be what we receive, within the common sense of dimensional analysis.

And looking at it from a broader perspective, not just in relation to my individual life, but rather within realizing that we live in a reality that is one, one reality shared by all of us, we then realize that there is in fact only one equation that is the accumulation of all our individual equations, as we exist within a reality that all and everything is interconnected and intertwined, and thus what I, the individual, put into the equation effect all in existence and will have an outflow as manifested consequence in the same dimension as the input I had added to the equation. Within this we can come to realize that all dimensions that exist, actually exist in everything, because we are all in fact living in the same reality, and thus the same equation, and thus within and as the same dimensions. Within realizing this, we must take responsibility for who we are within it all, and thus change our relationships within it all to create ourselves within the principle of "give as you would like to receive" and as "love thy neighbor as yourself" because only that way we will ensure that our input results in the output that is most beneficial to ourselves, and to all as equals. It's a big responsibility but it is the simple truth, all of who I am is part of the equation, and thus the dimensions I add to the equation - so I have the opportunity to decide who I am as to determine who I will influence and effect my life and reality as a whole.

An interesting point to consider, as we've seen above, this dimensional analysis can expose mistakes we have made in our equations, where we have equated two quantities that are of different dimensions - lets take a look at walking the desteni process: many beings within process have made the equation of process = time, they have been in process for 2, 3, 4 years and expect themselves to change, and to see results, but they neglected to realize that process and time are not of the same dimension, what one must realize is that it's not about how much time one walk process, but rather it's about who one is within walking process, and so, the accumulation of who one is, as how one apply oneself will equate to who one has become through the walking of the process in time, and then the equation would look like "who I am within walking process" = "who I am becoming" - where both sides of the equation have the dimension of self as who self is. In this way the dimension analysis can expose such a mistake in our understanding of process and help us change our relationship to it, so that we can correct ourselves within the realization that who I am in every moment within my life, and thus within walking process, will determine who I become, as the accumulation of 1+1+1+1… where components of the accumulation equation must be within the same dimension - this equation is not actually dependent on time, yes, we live in a space time reality, but what determines who I am, is who I am - as what I have accepted and allowed myself to exist as, as who I am.

Another example for a mistake in dimensional analysis we are all living is expressed in the current economic system, where we have created, and are living, the equation of life = money. we have equated the value of life to the value of money, thus, we have given life the value of money, but money and life do not have the same dimension, money is not alive, and doesn't in fact, as a physical necessity, support life. Thus we must reevaluate our equation and see what is the source of our mistake, and to take self responsibility in allowing ourselves to see the outflow as manifested consequences of our mistake.

The result of the current equation as money = life has created a world that is driven through profit, as we have equated money to life, and thus the more money one has the more value their life has, and the less money one has the more struggle they will experience through trying to survive in a world that has taken the value of life and has diminished it to the value of money. As if to say that life has value only if you have money, if you don't have money your life has no value. In the current system we can see the problem very clearly as so many people struggle to survive, as so many people are in fact slaves to the system, slaves to corporations that exist in the sole interest of making as much profit as they can, because that gives their life more value in the current equation we have all been living within.

Within realizing this, we must now take responsibility for our mistake and create an equation that equates life to life, as suggested in the equal money system, where the value of life is not diminished to value money, but rather that money is a tool to support the value of life, where all the relationships within the system are driven within the starting point of supporting the value of life, and then we will have an equation that all relationships exist within and as the dimension of life, and thus will equate life - at the moment all the relationships in the current system exist within the dimension of greed and profit, which allows as a manifested consequence the existence of manipulation, abuse and suffering as an integral part within the basic equation we have accepted and live according to, while not realizing that our equation is simply false, clearly evident through a simple dimensional analysis.

We must correct the equation so that the value of life will be equated in each and every one of the relationships that exist, within realizing that the accumulation of all the relationships that exist is represented as what we have in front of us as our reality - the result of all the relationships is what we see as the value of life - we must then change the value of our relationships, so that each and every relationship will support life, will be of life, and then we can have a corrected equation with life on the one side, and the accumulation of all relationships that support life and thus equate to life, on the other side.

The answer to the question above may have many more dimensions to it, lol, so, if any of you can find a cool and insightful perspective you are welcome to share in a comment or a response blog - I would love to see more perspectives, insights and understandings on this point, and so, together, we can connect the dots.

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