## Dimensional Analysis

It is not uncommon to hear the following:

"I need twelve yards of carpet." "Put thirty pounds of air in your tires."

"The temperature is seventy-eight."

In everyday conversation, many people are, . . . shall we say "sloppy" , with their use (and misuse) of units. It is far clearer to say

"I need twelve squareyards of carpet.""Put air in your tires at a pressure of thirty pounds

per square inch."It may be unnecessary to be so clear as to say

"The temperature is seventy-eight degrees, Farenheight."In Physics, however,

units are not an option!They're necessary. Nothing has a length of 1.2. There is no mass of 4.5. The time for something to happen is never 0.00234.While we will almost alwayse use SI units, there may always be the question as to whether a length has been measured in millimeters, centimeneters, meters, or kilometers. While the basic, fundamental unit of time is the second, it is often more convenient to state time in minutes or hours or even years. The basic, fundamental unit of mass is the kilogram but we may also measure mass in grams, milligrams, micrograms, or mass units. The unit following a number is

necessary!We can use this to our advantage.

You can not

add"apples" and "oranges". Three "apples" can never equal four "oranges". Every term in anequationmust have the same unit. The equation for the position of an object undergoing uniform acceleration is

s = s _{o}+ v_{o}t + (^{1}/_{2}) a t^{2}s is a length, typically measured in meters, so every term on the other side of the equation must also be a length. We can indicate length by

L. Initial velocity v_{o}will be measured in length per time, typically in meters per second (m/s); we can indicate this byL/T. Acceleration a, as we will learn soon, is measured in (length per time) per time, typically in (meters per second) per second or (m/s)/s or m/s^{2}; we can indicate this byL/T. We will indicate the units of a quantity with square brackets, [ ]. Then we can "check" the units or dimensions of this equation by writing^{2}

s = s _{o}+ v_{o}t + (^{1}/_{2}) a t^{2}

[s] = [s _{o}] + [v_{o}] [t] + (^{1}/_{2}) [a] [t^{2}]

L = L + [L/T] [T] + [L/T ^{2}] [T^{2}]

L = L + L + L All the terms on the right side of the equation are, indeed, lengths. It is also common--though not quite correct--to write this same sort of analysis with

unitsinstead ofdimensions, as

s = s _{o}+ v_{o}t + (^{1}/_{2}) a t^{2}

[s] = [s _{o}] + [v_{o}] [t] + (^{1}/_{2}) [a] [t^{2}]

m = m + [m/s] [s] + [m/s ^{2}] [s^{2}]

m = m + m + m In either case, note that numbers like

^{1}/_{2}have dimension (or unit) of unity. The final equation does not mean L = 3L (or m = 3 m); it is a statement about thedimensionsorunitsof every term.To add two terms, they must have the same dimensions or units. The two sides of an equation must have the same dimensions or units.

Standards Units Conversion Return to Table of Contents, Ch 1 Introduction(c) 2005, Doug Davis; all rights reserved