Principal of Homogeneity of Dimensions

Consider the equation: s = vₒt + ½ at² … (1)

Where the letters have their usual meaning. Since it is meaningless to equate two quantities of different nature, e.g., mass to a length or a temperature to a force, it is evident that for the above equation to have any physical meaning, the three terms combining it must have the same dimensions. We know that the dimensional formula of s is [L], that of t is [T], that of vₒ is [LT⁻¹] and that of a is [LT⁻²¹]

Hence, the dimensional equation corresponding to eqn. (1) which neglects all numerical coefficients may be written as [L] = [LT⁻¹] [T] + [LT⁻²¹] [T²] = [L] + [L]

i.e., the dimensions of all the terms on the two sides of the equation are the same. This property is true of all the equations representing different physical phenomena and may be stated as:

\(\left\{ P \right\}=\left\{ \Pr essure \right\}=\left\{ \frac{Force}{Area} \right\}=\left\{ Mass\frac{Length}{Tim{{e}^{2}}Lengt{{h}^{2}}} \right\}=\left\{ \frac{m}{{{t}^{2}}L} \right\}\).

\(\left\{ \frac{1}{2}\rho {{V}^{2}} \right\}=\left\{ \frac{Mass}{Volume}{{\left( \frac{Length}{Time} \right)}^{2}} \right\}=\left\{ \frac{Mass\times Lengt{{h}^{2}}}{Lengt{{h}^{2}}\times Tim{{e}^{2}}} \right\}=\left\{ \frac{m}{{{t}^{2}}L} \right\}\).

\(\left\{ PgZ \right\}=\left\{ \frac{Mass}{Volume}\frac{Length}{Tim{{e}^{2}}}Length \right\}=\left\{ \frac{Mass\times Lengt{{h}^{2}}}{Lengt{{h}^{2}}\times Tim{{e}^{2}}} \right\}=\left\{ \frac{m}{{{t}^{2}}L} \right\}\).

“Physical equations must be dimensionally homogeneous.” This is known as the principle of homogeneity of dimensions and according to this principle.

When a physical equation consists of a number of terms, each of these terms must be of the same dimensions in each of the fundamental units.

One can state the principle in a much simpler way as follows: The powers of mass, length and time in each term on one side of a dimensional equation must always be equal to their respective powers in each term on the other side of the dimensional equation.

According to the principle of homogeneity, the dimensions of each term on the LHS must be equal to the dimensions of the term on the RHS.

Clearly [ML²T⁻²] a = [L²]

Or a = [L²]/[ML²T⁻²] = [M⁻¹T²]

And b = [L²]

Dimensional Analysis and its uses: Sometimes we might be required either to derive or to check a specific relation. Even if we have forgotten the details of the derivation, there is a useful and powerful procedure that can help us in this matter. This procedure, which treats dimensions as algebraic quantities is called dimensional analysis. Thus, dimensional analysis is the process of analysing a physical problem with the help of dimensional equations.

Following are the three main uses of dimensional analysis (or of dimensional equations).

(a) Checking the dimensional correctness of a physical equation: To check the correctness of a physical equation, we use the principle of homogeneity of dimensions.

We calculate the dimensions of the quantities on both sides of the equation in terms of M, L and T. If the dimensions obey the principle of homogeneity, the equation is dimensionally correct, otherwise not.

(b) Deriving relationship between different physical quantities: Using the principle of homogeneity of dimensions, in many cases, we can find the expression for a physical quantity, if we know the factors on which it depends.

(c) Conversion of one system of units into another: The magnitude of a quantity is the same, whatever may be the system of its measurement. This fact helps us to change one system of units into another.

Let [MᵃLᵇTᶜ] be the dimensional formula of that quantity whose units are to be converted from one system into another.

Further, let M₁, L₁, T₁ represent the fundamental units of mass, length and time in one system of units and M₂, L₂, T₂ be the corresponding units in another system.

If u₁ and u₂ represent the units of the quantity in the two systems, then

u₁ = [M₁ᵃL₁ᵇT₁ᶜ] and u₂ = [M₂ᵃL₂ᵇT₂ᶜ]

If n₁ (known) is the numerical value of the quantity in one system and n₂ (unknown) is the numerical value of the quantity in the other system, then

n₁u₁ = n₂u₂

Or n₁ [M₁ᵃL₁ᵇT₁ᶜ] = n₂ [M₂ᵃL₂ᵇT₂ᶜ]

Or n₂ = n₁ [M₁ᵃL₁ᵇT₁ᶜ]/[M₂ᵃL₂ᵇT₂ᶜ]

Or n₂ = n₁ [M₁/ M₂]ᵃ[L₁/ L₂]ᵇ[T₁/ T₂]ᶜ … (1)

Thus, if the fundamental units in both the systems, dimension of the quantity and its numerical value in one system are known, we can easily calculate n₂, i.e., the numerical value of the quantity in the other system.

It should be clearly understood that eqn. (1) is to be applied only after expressing the quantity in the absolute units.