`"Principle of Homogeneity"`
The magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions. This simple principle called the principle of homogeneity of dimensions.
• If the dimensions of all the terms are not same, the equation is wrong.
• Dimensions are customarily used as a preliminary test of the consistency of an equation, when there is some doubt about the correctness of the equation. However, the dimensional consistency does not guarantee correct equations.
• A pure number, ratio of similar physical quantities, such as angle as the ratio (length/length), refractive index as the ratio (speed of light in vacuum/speed of light in medium) etc., has no dimensions.
• Now we can test the dimensional consistency or homogeneity of the equation.
For the distance `x` travelled by a particle or body in time `t`
`x=x_0 + v_0 t + 1/2 at^2`
The dimensions of each term may be written as
`[x] = [L]`
`[x_0 ] = [L]`
`[v_0 t] = [L T^–1] [T] = [L]`
`[(1/2) a t^2] = [L T^–2] [T^2] = [L]`
As each term on the right hand side of this equation has the same dimension, namely that of length, which is same as the dimension of left hand side of the equation, hence this equation is a dimensionally correct equation.
• If an equation fails this consistency test, it is proved wrong, but if it passes, it is not proved right. Thus, a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be
wrong.
`"Principle of Homogeneity"`
The magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions. This simple principle called the principle of homogeneity of dimensions.
• If the dimensions of all the terms are not same, the equation is wrong.
• Dimensions are customarily used as a preliminary test of the consistency of an equation, when there is some doubt about the correctness of the equation. However, the dimensional consistency does not guarantee correct equations.
• A pure number, ratio of similar physical quantities, such as angle as the ratio (length/length), refractive index as the ratio (speed of light in vacuum/speed of light in medium) etc., has no dimensions.
• Now we can test the dimensional consistency or homogeneity of the equation.
For the distance `x` travelled by a particle or body in time `t`
`x=x_0 + v_0 t + 1/2 at^2`
The dimensions of each term may be written as
`[x] = [L]`
`[x_0 ] = [L]`
`[v_0 t] = [L T^–1] [T] = [L]`
`[(1/2) a t^2] = [L T^–2] [T^2] = [L]`
As each term on the right hand side of this equation has the same dimension, namely that of length, which is same as the dimension of left hand side of the equation, hence this equation is a dimensionally correct equation.
• If an equation fails this consistency test, it is proved wrong, but if it passes, it is not proved right. Thus, a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be
wrong.