Dimensional Homogeneity and Dimensionless Quantities

We began our discussion of units and dimensions by saying that quantities can be added and subtracted only if their units are the same. If the units are the same, it follows that the dimensions of each term must be the same. For example, if two quantities can be expressed in terms of grams/second, both must have the dimension (mass/time). This suggests the following rule  

Every valid equation must be dimensionally homogeneous that is, all additive terms on both sides of the equation must have the same dimensions. 

This equation is dimensionally homogeneous, since each of the terms u, uq, and gt has the same dimensions (length/time). On the other hand, the equation u = uq + gr is not dimensionally homogeneous (why not ) and therefore cannot possibly be valid. 

Equation 2.6-1 is both dimensionally homogeneous and consistent in its units, in that each additive term has the units m/s. If values of mq, s, and / with the indicated units are substituted into the equation, the addition may be carried out to determine the value of u. If an equation is dimensionally homogeneous but its additive terms have inconsistent units, the terms (and hence the equation) may be made consistent simply by applying the appropriate conversion factors. 

For example, suppose that in the dimensionally homogeneous equation u = uq + gi it is desired to express the time (t) in minutes and the other quantities in the units given above. The equation can be written as 

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