Understanding dimensions is one of the most important factors. It is the practice of checking relations with aspects to their physical quantities. It uses dimension and its formula of physical quantities. It is used to find inter-relation between them.
It helps in researching the nature of physical quantities mathematically. Two physical quantities are equal if they have similar dimensions. The catch here is that we can only add or subtract the quantities with the same dimension. It helps us derive new relations between physical quantities. So, let’s understand what is Dimensional Analysis in detail.
What is Dimensional Analysis?
Dimension Analysis is a useful technique in all experimentally based areas of engineering. If it is possible to identify the factors involved in a physical situation, dimensional analysis can form a relationship.
The dimension of the physical quantities represents the basic quantities as a product of symbols. These are also known as Powers of Symbols.
Suppose you’re studying oscillations and waves. You know that there is some relationship between the wavelength λ,λ, the frequency f,f, and the speed of propagation v,v, but for the life of you, you can’t remember exactly what that is.
Well, wavelength has dimensions of [length],[length], frequency has dimensions of [time]−1,[time]−1, and speed has dimensions of [length][time]−1.[length][time]−1.
From this, it should be obvious that the correct relation between wave speed, wavelength, and frequency is v=λf. So one reason dimensional analysis is useful is that it cuts down on the need for rote memorization of formulas.
Application and Limitation of Dimension Analysis
It is the method that is used to convert the following:
- To check the dimension correctness (how correct is the dimension) of a given physical equation.
- It also is used in altering (converting) a physical quantity from one to another.
- Establish relations among various physical quantities
The explanation of each is as below –
To check the dimension correctness of a given physical equation.
As explained above, it allows you to add or subtract physical quantities only when they have the same dimensions.
Let us take the equation of motion
v = u + at
Apply dimensional formula on both sides
[LT−1] = [LT−1] [LT−2] [T]
[LT−1] = [LT−1] [LT−1]
Here, we see that the dimensions of both sides are the same. Hence the equation is dimensionally correct.
It also is used in converting a physical quantity from one system to another
The dimensions of the physical quantities are independently used to measure quantity. Suppose M1, L1, & T1 and M2, L2, & T2 are the fundamental quantities.
We will now measure quantity (Q) in both these systems of units, and a,b,c are the dimensions of the quantity, respectively.
Consider a physical quantity with dimensions’ a’ in mass, ‘b’ in length and ‘c’ in time. If the fundamental units in one system are M1, L1and T1and the other system is M2, L2and T2respectively.
Then we can write, n1[M1aL1bT1c] = n2[M2aL2bT2c]. We have thus converted the numerical value of physical quantity from one system of units into the other system.
Establish relations among various physical quantities
The method of dimension analysis will also help in establishing relations between the physical quantities. So, how do we know the relation between physical quantities, and how will they be established?
If we understand that physical quantities depend on each other, we can easily find their relation. This is found by equating equations from both sides. If the physical quantity Q depends upon the quantities Q1, Q2 & Q3. Here, Q is proportional to Q1, Q2 & Q3.
Then,
Q α Q1aQ2bQ3c
Q = k Q1aQ2bQ3c
where k is a dimensionless constant. When the dimensional formula of Q, Q1, Q2and Q3 is replaced, the authorities of M, L, T are made equal on both sides of the equation according to the principle of homogeneity. Out of this, we get the values of a, b, c, and so on.
Dimension Analysis and its limitations
It is self-defined. You can conceptualize a particular into one dimension. It is a tool gesture for mapping an idea into some formal system. A dimension always is defined by what it isn’t – that is, by some other, objective dimension. My hands join and merge into a single hand. I know ‘hand’ and ‘join’, but losing hand and joining in the merge is a grammar move, one defined by what’s realistically probably as an action. So, the limit of a dimension is its contact with the boundary of the realistic dimensions. What we imagine is always within the reality of what we know to imagine.
Let’s have a look into another example for limitation
A dimension is a Universe outside of our own and or on top of our own. Each universe has to have a set of universal laws to exist even if its most basic law exists and functions. Our universe has seven laws of existence.
Another dimension, such as the theorized Superstring 10th dimension, is limitless, and with that, it still has to have a law of existence and a 2nd law stating that no laws govern it. To escape universal laws, you would have to exist outside of a dimensional universe.
This existence is likely possible, and you’d exist in nothingness outside the measurable space of the universe. Nothingness exists as before the universes; there was nothing. Non-existence is the next step, and you can exist and not exist simultaneously in a place that doesn’t exist.
Let space be a 0 dimension. Now, that which tends to fill the space is a positive dimension, and empty is a negative dimension.
Conclusion:
Now, you have understood everything about Dimension analysis. It is checking the relationship between two physical quantities. You also now understand applying dimension analysis by correctness, converting, and relations between physical quantities. We have also learned about the limitations of dimensional analysis. To understand, you will need to go through the details of each aspect.