Dimensional analysis and its application

In the field of science, dimensional analysis plays a significant role. It determines the relationship between the fundamental physical quantities. It provides a basic understanding of the nature of various objects mathematically. There are various applications of dimensional analysis as the nature of physical quantities can be inferred by their dimensions. Let us read this article to learn all about dimensional analysis and its applications. 

Dimension of physical quantities

All the derived formulas can be written as the dimension of seven fundamental quantities, also known as base quantities. These basic physical quantities are:

• Temperature [K]
• Time [T]
• Mass [M]
• Length [L]
• Electric current [A]
• Luminous intensity [cd]
• The concentration of a substance [mol]

The dimension of any quantity is represented by raising the base to an exponent. For instance, we have the volume of a cuboid is:

V = l bh

Here, length(l), breadth(b), and height(h) are all the measure of the length of the sides of a cuboid. 

Thus, the dimensional formula of volume becomes:

V = [L] [L] [L]

V =[L3]

Another example we take of force. The formula of force is given as:

F=ma

F = m (v/t)

F = m (d/t2)

F = [M L T-2]

Thus, from these examples, we understand that the dimensional formula is used to represent how and which fundamental quantities form the dimension of any physical quantity. Further, the equation that equates physical quantities with their dimensions is called the dimensional equation. 

Principle of homogeneity of dimension

As per the principle of homogeneity of dimensions, only the magnitude of quantities with the same dimension can be added or subtracted. The example mentioned in this study material notes on dimensional analysis; we understand that volume with dimension [L3] cannot be subtracted from the force with dimension [MLT-2]. 

This principle is very useful in determining the correctness of an equation. If all the terms do not have the same dimension, the equation is incorrect. 

For example, we take the equation: 12mv2 = mgh

Here all the symbols have their usual meaning. We can now check whether this equation is correct by finding the dimensions. The dimensions of each term on either side are calculated if it is not found the same, the equation is wrong. 

In this example, the dimension of the left side is:

12mv2 = 12m(d/t) 2 = [M L2 T-2]

The dimension of the right side is:

mgh = m(v/t)h = m(d/t2) h = [M L2T-2] 

Here, we see that the dimensions on the left and right sides are the same; therefore, the equation is correct. 

Deducing relationship among physical quantities 

Using dimensional analysis, we can simply determine the relationship between physical quantities. For finding the relation, the degree of dependence must be known, that is, the degree to which one quantity increases or reduces concerning another quantity. Using the principle of consistency, the dependence can be calculated. 

For example, we have the relation of the time interval of the pendulum given as T=k lx gy mz. Here, T is the time, m is the mass, l is the length of the pendulum, and g is the acceleration due to gravity. 

T=k lx gy mz

Writing the dimensions both sides:

[M0 L0 T] = k [L]x [L T-2]y [M]z

[M0 L0 T] = k [L]x+y [ T]-2y [M]z

Comparing both sides:

x+y=0; -2y=1; z=0

On solving the equations we get: 

y = -1/2 ; x = 1/2; z=0

Substituting the values we get:

T=k l1/2 g-1/2 m0

Therefore the formula of time interval becomes:

T = k lg

Dimensionless quantities

The dimensionless quantities are those that do not have any dimension and only have a constant value. There can be two kinds of dimensionless quantities are:

• Dimensionless quantity without unit: , sin, e, cos , etc.
• Dimensionless quantity with the unit: Joule’s constant, radian, and angular displacement. 

Application of dimensional analysis 

We have already learned how important dimensional analysis is in determining the nature of physical quantities. It is important to know the nature of dimensions of physical quantities as only the quantities with the same dimensions can be added or subtracted. 

The dimensional analysis further helps us determine the relationship among various physical quantities. It also allows us to check whether the derivation of various mathematical formulas is correct or not. 

When the dimensions are multiplied, it is done in the same way as normal algebraic expressions. The same dimension in numerator and denominator can also be cancelled out. 

The dimensional analysis is also used to determine the formula of various physical quantities. Above all, the unit of a physical quantity can be converted from one system to another. 

Though there are various applications of dimensional analysis, some limitations are also present. We can not determine the value of constant using dimensional analysis. 

Above all, dimensional analysis is not used for trigonometric functions, logarithms and exponential functions. 

Conclusion 

Dimensional analysis is the method of expressing the physical quantities as the exponent of fundamental quantities. It has various applications in science and mathematics in deriving the formula and checking its consistency. 

However, even if the equation passes the consistency test, it is not necessarily proved right. Therefore, we learn that dimensionally correct equations may not be the actual formula of the particular physical quantity.