Applications Of Dimensional Analysis!

Dimensional analysis is a term often used when studying subjects like engineering or any other subjects in the field of Mathematics and Science. Whenever someone thinks of dimensional analysis, the first thing that comes to one’s mind is the physical quantities. Dimensional analysis is crucial to understanding different physical quantities. To understand the three applications of dimensional analysis, it is crucial to understand dimensional analysis and to understand this, one must have a thorough knowledge of the dimensions of a physical quantity. 

The applications and limitations of dimensional analysis are plenty and also important to study the physical quantities. 

Applications of dimensional analysis

The dimensional analysis has proven itself to be a great tool for checking the correctness of equations. Also, this method is a reliable method to form equations if one knows the quantities involved. The method is also utilized to quantify the shape and size of things and to study the mathematical nature of objects. The three applications of dimensional analysis that are usually used are:

1. It is used to find the unit of a physical quantity in a specific unit system

2. The method of dimensional analysis is also used to know the dimensions of physical coefficients and constants

3. It is a reliable method for conversion of a physical quantity from one system to another

Below is a detailed view of each of the above-listed applications with their examples. 

1. Application of dimensional analysis to find the unit of a physical quantity in a specific unit system

Dimensional analysis can be used to write the dimensional formula from the formula to represent a physical quantity. This is one of the three applications of dimensional analysis that are generally used. For example, the physical quantity Work can be represented as the below formula

Work= Force x Displacement

The dimensional formula for Force is [MLT-2] and for Displacement, it is [L]. Therefore, the dimensional for Work will be

[W]= [MLT-2] [L]

[W]= [ML2T-2] 

Now, to find the unit of work, we can replace the base dimensional formulas M, L, and T with their fundamental units (kg, m, s). This will give the unit of Work in the MKS system as kg m2/s2 (named Joules). Also, with this, we can find the unit in the CGS system of units which will be g cm2/s2 (this unit of Work is known as erg).  

2. Using dimensional analysis to find the dimensions of physical coefficients and constants

The dimensional analysis method is also used to find the dimensions of a physical coefficient or constant. To do so, the constant is written with a proportional sign with its formula. Now, the physical quantities in the formula of constant are replaced with their respective dimensional formulas. The below example will help you understand this application in detail. 

• Gravitational constant

G= Fr2/m1m2

Now when we substitute all the physical quantities in the formula with their dimensional formulas, we will get the dimensions for the Gravitational constant. 

[G]= [MLT-2] [L2] / [M] [M]

[G]= [M-1L3T-2]

The above method can be used to find the dimensions of any constant like the Planck constant or the Coefficient of viscosity. 

3. Used for conversion of a physical quantity from one system to another

If one wishes to change a unit from one unit system to another, it can be done easily with the help of dimensional analysis. A physical quantity is represented as 

X= NU

Here, N is the numeral value and U is the unit of the physical quantity. 

Now, to convert a unit from one system to another, the below relation can be used

N1U1= N2U2

Here, N1U1 is the representation of the physical quantity in the known unit and N2U2 is the representation of the physical quantity in the unit which we have to convert. 

For example, Newton can be converted into Dyne using the following method. Here, N1U1 will represent Newton and N2U2 will represent Dyne. 

1N= 1kg m/ s2

N1 [M1a L1b T1c]= N2 [M2a L2b T2c]

N2= N1 [M1/M2]a [L1/L2]b [T1/ T2]c

Now, we will replace a, b, and c with powers of M, L, and T in the dimensional formula of Newton and also replace M, L, and T with units in their respective systems.

N2= 1[kg/g]1 [m/cm]1 [sec/ sec]-2

N2= 1[103g/g]1 [102cm/cm]1 [sec/ sec]-2

N2= 105

Therefore, 1N= 105 Dyne

Conclusion

Dimensional analysis is crucial to understanding the mathematical nature of physical quantities. The applications and limitations of dimensional analysis are significant to studying and understanding different physical quantities and their relationship with their units. Apart from the three applications of dimensional analysis listed above, the method is also used to find new relations between physical quantities and verify the correctness of physical relations.