Units and Dimensions USES OF DIMENSIONAL ANALYSIS 3
Uses of Dimensional Analysis The Fourth primary use of Dimensional Analysis is to determine the dimensions and therefore Unit of an unknown undescribed quantity by analyzing the equation. For this purpose, we utilize the Principle of Homogeneity According to the Principle of Homogeneity, the quantities separated by the following symbols :- +,-, = , 2,#, <, > necessarily have same dimensions. For example, if in the expression. S = A + Bt + Ct2, S represents displacement and t represents time, then the dimension of A is same as the dimension of S as both are separated by an = ,. Therefore the Dimensions of A will be M L1TO and its SI Unit will be meter.
Similarly the dimension of Bt is equal to the dimension of A (as both are separated by a '+' sign) which in turn is equal to the dimensions of S as described earlier. Therefore, the dimensions of Bt is also equal to M 1ITO. And henceforth, the dimensions of B = Moll T-1 Similarly. We can derive the dimensions of C. As Ct2 and Bt are again separated by a +' sign, therefore the dimensions of Ct2 and Bt are same. But the dimensions of Bt are n L1To (as discussed before), therefore dimensionally Ct2=M"L"To Therefore dimensionally, ML1T
Similarly, we can utilize the Principle of Homogeneity, to further analyse different situations. Find out the unit and dimensions of the constants a and b in the van der Waal's equation ILlustration OL P+ V-b)RT. where p is pressure, v is volume, R is gas constant, and Tis temperature Sol. We can add and subtract only like quantities Dimensions ofP Dimensions of 2 and dimensions of v- Dimensions of b From (i), Dimensions of a Dimensions of P Dimensions of V" 4 Unit of a-Unit of P Unit of v2- 2 ron 11 So unit of bUnit of V m
Nikhil Mishra is an engineer and has been a Faculty of Physics in some of the most premier institutes of the country and has been involved