Application of dimensional analysis

Application of dimensional analysis Made by

Gaurav YadavXI-A

Principle of homogeneity The principle of homogeneity is that the dimensions of each the terms of a dimensional equation on both sides are the same .Any equation or formula involving dimensions (like mass, length, time , temperature electricity) have the terms with same dimensions. This helps us, therefore, to convert the units in one system to another system.This also helps us to check a formula or the involvement of the dimensions in a formula.Example : It is conjectured that time of the period of the oscillation of a pendulum is dependent on its mass, length and the acceleration due to gravity.

So time = some constant K*(mass of the pendulum)^a*(length l of the the pendulum)^b* (acceleration due to gravity g)^c.OrT = K*m^a*l^b*g^c. Dimensionally this is like:[T] = [M]^a* [L]^b*[L*T^-2]^c.Comparing the powers of each dimensions on both sides,(K being dimensionless), we get:T: 1 = -2c. Therefore, c =-1/2M: 0 = a. Therefore, a =0L: 0 = b+2c. Therefore, b =-2c = -1. So the formula for the period T of the pendulum is :T = K* m^0*L^(1/2)* g^(-1/2) = K*(l/g)^(1/2).

To check the dimensional correctness of a given physical relation:-

This is based on the principle that the dimensions of the terms on both sides on an equation must be same. This is known as theprinciple of homogeneity. If the dimensions of the terms on both sides are same, the equation is dimensionally correct, otherwise not.Caution:It is not necessary that a dimensionally correct equation is also physically correct but a physically correct equation has to be dimensionally correct.

The factor-label method can also be used on any mathematical equation to check whether or not the dimensional units on the left hand side of the equation are the same as the dimensional units on the right hand side of the equation. Having the same units on both sides of an equation does not guarantee that the equation is correct, but having different units on the two sides of an equation does guarantee that the equation is wrong.

exampleFor example, check the Universal Gas Law equation of PV = nRT, when: the pressure P is in pascals (Pa) the volume V is in cubic meters (m) the amount of substance n is in moles (mol) the universal gas law constant R is 8.3145 Pam/(molK) the temperature T is in kelvins (K)

As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's right hand side are cancelled out, both sides of the equation have the same dimensional units.

Limitations The limitations are as follows:-

(i) If dimensions are given, physical quantity may not be unique as many physical quantities have the same dimension. For example, if the dimensional formula of a physical quantity is [ML2T-2] it may be work or energy or even moment of force.

(ii) Numerical constants, having no dimensions, cannot be deduced by using the concepts of dimensions.

(iii) The method of dimensions cannot be used to derive relations other than product of power functions. Again, expressions containing trigonometric or logarithmic functions also cannot be derived using dimensional analysis, e.g.

s = ut + 1/3 at2 or y = a sin cot or P= P0exp[(Mgh)/RT]

cannot be derived. However, their dimensional correctness can be verified.

(iv) If a physical quantity depends on more than three physical quantities, method of dimensions cannot be used to derive its formula. For such equations, only the dimensional correctness can be checked. For example, the time period of a physical pendulum of moment of inertia I, mass m and length l is given by the following equation.

T = 2(I/mgl) (I is known as the moment of Inertia with dimensions of [ML2] through dimensional analysis), though we can still check the dimensional correctness of the equation (Try to check it as an exercise).

(v) Even if a physical quantity depends on three Physical quantities, out of which two have the same dimensions, the formula cannot be derived by theory of dimensions, and only its correctness can be checked e.g. we cannot derive the equation.

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