Question

Write any two applications of dimensional analysis.

Answer 1

Hint
Dimensional analysis is almost important, helping in the study of nature of physical quantities. It is used to find the formula of physical quantities to find interrelation between them. First assume the relation with them using unknown variables to get the actual value. E.g., having the relation with power equal to a and finding the value of a using mass, length and time.

Complete Step By Step Solution
Let’s see, what is dimensional Analysis. It is used to find the formula of physical quantities to find interrelation between them. The facts are based on the physical law which says that they are independent of the units in which quantity is measured.
If ab is the measured value of a physical quantity in one system of units, and cd is the value in another system of units, then ab = cd.
Secondly, the Principle of Homogeneity states that the physical dimension of both sides will be totally equal independent of each and every dimension.
i.e., $[M]^a [L]^b [T]^c = [M]^d [L]^e[T]^f$
then, $a = d$, $b = e$ and $c = f$.
The main Applications are:
Conversion of units where dimensions of a Physical quantity are independent of the System of units used to measure the quantity in.
Let’s suppose,
$q = {x_1}{y_1}{n_1}[{M^a}_1, {L^a}_1, {T^c}_1]$
$q = {x_2}{y_2}{n_2}[{M^a}_2, {L^a}_2, {T^c}_2]$
then $n_1[{M^a}_1, {L^a}_1, {T^c}_1] = n_2 [{M^a}_2, {L^a}_2, {T^c}_2]$
we can substitute and convert the value with the same units to obtain the value of $n_1$ or $n_2$.
Secondly, the other Application can be Checking the consistency of an Equation.
E.g., ${\text{F}} = {{\text{m}}^{\text{2}}} \times {\text{a}}$
${\text{F}} = {\text{ma}}$
These two have different dimensions hence are not the correct form. Which can only be found through dimensional analysis.

Note
The use of mass, length and time dimension is used to find the actual relation between n equations. If you need to add two no, the dimension needs to be the same. You can’t add distance with mass, where someone says, the distance traveled by that person is dependent upon only mass. The dimension helps us to find the real reason why something can’t be related to each other independently.