Name: Limitation of Dimensional Analysis
Uploaded: 2019-07-19
Duration: 5 min 9 s
Description: Dimensional analysis can derive the relation between two physical parameters. But does it work in all the cases? Is it possible? Let us discuss this in this video.

Question

What do you understand by the dimensions of a physical quantity? Explain the principle of homogeneity of dimensions, giving an example.

Accepted Answer

The dimensions of a physical quantity are the powers to which the fundamental units are raised in order to obtain the derived unit of that quantity.

The dimensions of fundamental physical quantities like, the length, mass and time are denoted by $[L], [M] and [T] .$

The dimension of all the term on both side of a physical equation must be the same. This is called the principle of homogeneity of dimensions.

Example : Suppose, we have to check the correctness of the equation $\frac{1}{2} m v^{2} = mgh,$ where $m$ is the mass of a body, $v$ its velocity, $g$ is acceleration due to gravity and $h$ is the height. The dimension formulae of the various quantities in the equation are :

$mass, m = [M] velocity, v = [{LT}^{- 1}] acceleration due to gravity, g = [{LT}^{- 2}] height, h = [L] .$

$\frac{1}{2}$ is a pure ratio, having no dimensions. Substituting these dimensional formulae in the equation $\frac{1}{2} m v^{2} = mgh,$ we have

$[M] {[L T^{- 1}]}^{2} = [M] [L T^{- 2}] [L] [M L^{2} T^{- 2}] = [M L^{2} T^{- 2}] .$

The dimension on both the sides are same.

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