What is the principle of homogeneity of dimensional equation ?
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The principle of homogeneity is that the dimensions of each the terms of a dimensiional 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 sytem 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.Or
T = 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 dimensionles), we get:
T: 1 = -2c. Therefore, c =-1/2
M: 0 = a. Therefore, a =0
L: 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).
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