University Physics Volume 1 — Mechanics · Problem 89

Units and Measurement — OpenStax

Physics — University · dimensional-analysis

Problem

The purpose of this problem is to show the entire concept of dimensional consistency can be summarized by the old saying ``You can't add apples and oranges.'' If you have studied power series expansions in a calculus course, you know the standard mathematical functions such as trigonometric functions, logarithms, and exponential functions can be expressed as infinite sums of the form

\sum_{n=0}^{\infty} a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots,

where the

a_n

are dimensionless constants for all

n = 0, 1, 2, \cdots

and

x

is the argument of the function. (If you have not studied power series in calculus yet, just trust us.) Use this fact to explain why the requirement that all terms in an equation have the same dimensions is sufficient as a definition of dimensional consistency. That is, it actually implies the arguments of standard mathematical functions must be dimensionless, so it is not really necessary to make this latter condition a separate requirement of the definition of dimensional consistency as we have done in this section.

Related theory

dimensional-analysis

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