Dimensional Analysis Of Physical Quantities And Its Applications
Comparing and converting between different units are very useful and importanat for us.
We do this each and every day without realising it.
Suppose someone is going to follow a recipe.
Then, he needs to do the simple conversion like kilogram to gram or kiloliter to liter, etc.
Similarly, in science and math conversion of units are needed and this conversion is done by dimensional analysis.
Now, we will discuss the dimensional analysis of physical quantities.
Let’s discuss the dimension of physical quantity and dimensional formula.
Dimensions are the nature of a physical quantities and the physical quantities are represented by the derived units.
Now, these all derived units are expressed with the help of seven fundamental quantities. Which are as,
The dimensions of a physical quantity are the exponents to which the base quantities are raised to represent that quantity.
And, square bracket is used to represent the dimension of a physical quantity.
In mechanics, all physical quantities can be represented by three dimensions, [M], [L] and [T]. For Example: Area.
Area is the product of length and breadth. Hence, its dimension will be,
Similarly for the force, its dimension will be,
Now, the dimensional formula is the expression which shows the how and which fundamental quantity represent the dimensions of physical quantity.
For example, the dimensional formula of area, force and acceleration will be as,
Now, the equation obtained from a physical quantity with its dimensional analysis is called dimensional equation of that quantity.
For example: dimensional equation of area (a), force (f) and density (
$ρ$
) is as,
Let’s discuss the principle of homogeinity of dimensions.
The dimensional homogeneity is that the dimensions of each term of a dimensional equation on both sides is the same.
And it is used to check the correctness and incorrectness of the equations.
If the dimensions of all the term in the equation are not same then the equation is incorrect.
For example: dimensions of the expression for the time period of oscillations.
Now, the dimensions of each term is same.
Therefore, this equation is dimensionally homogeneous.
Let’s discuss the uses and limitations of dimensional analysis.
Dimensional analysis has many uses in real life physics. One of them is to check the consistency of a dimensional equation.
Because, we can add or subtract only those quantity which are having same dimension. like the energy cannot be added in the force.
Another use is to change units from one system to another. like meter to centimeter.
There are some limitations of the dimensional analysis.
If the dimensional analysis proved that the equation is not consistent. that means the equation is incorrect.
But, if the dimensional analysis proved that the equation is consistent. Means, this is not necessary that the equation is correct.
For example, we know that Newton's second law of motion is mathematically correct. Also, it is consistent according to dimensional analysis.
Here, this equation is the dimensionally correct. But, we can understand that this equation is mathematically incorrect.
Because, According to Newton's second law of motion, there should be
$2$
in the place of
$4$
.
Thus, we conclude that It is not necessary that a dimensionally proven equation is also mathematically correct.
But, a dimensionally incorrect equation would certainly be mathematically incorrect as well.
Revision
The derived units are expressed with the help of seven fundamental quantities. Which are as,
All physical quantities can be represented by three dimensions, [M], [L] and [T]. For Example: Force.
The dimensional homogeneity is that the dimensions of each term of a dimensional equation on both sides is the same.
Hence, the dimensionally correct equation need not be actually a correct equation. But, the dimensionally incorrect equation must be incorrect.
The End