Dimenional Analyi A Tool for Guiding Mathematical Calculation Dougla A. Kerr Iue 1 February 6, 2010 ABSTRACT AND INTRODUCTION In converting quantitie from one unit to another, we may know the applicable converion factor but be uncertain a whether to multiply or divide. The ame uncertainty oen arie in other baic mathematical calculation, uch a thoe involving ditance, time, and velocity. A technique called dimenional analyi can give foolproof guidance in thee cae, and can even help u develop the formula needed to deterne a certain quantity. Thi article decribe the principle involved and illutrate the practical application of the technique. BACKGROUND Dimenionality The centerpiece of the technique decribed here i attention to the dimenionality of quantitie. Dimenionality i the property that tell u the nature of the quantity being dicued. It ditinguihe, for example length, time, area, and volume. The dimenionality of a quantity doe not dictate the pecific unit to be ued to denonate it quantitative expreion; oen, there are a number of choice. But the pecific unit ued mut be one whoe dimenionality i conitent with the dimenionality of the quantity itelf. We can denonate an area with the unit quare inch, quare meter, or acre; we cannot denonate an area with the unit inch, cubic inch, or volt. It i thi fact that give the technique decribed here it name: dimenional analyi. Editorial practice It i common in narrative writing, when the full form of a unit i involved (uch a meter ), to preent the unit in it plural form: When the numeric value of the quantity i greater than one ( the length i 1.05 llimeter ). (Sometime) when mentioning the unit by itelf ( velocity can be denonated in the unit feet per econd ) Copyright 2010 Dougla A. Kerr. May be reproduced and/or ditributed but only intact, including thi notice. Brief excerpt may be reproduced with credit.
Dimenional AnalyiError! Reference ource not found. Page 2 For compound unit (whoe name i made of two or more other unit name, which I call unit factor ), the plural ending i applied to the lat unit factor mentioned in the numerator. Example are: volt ; foot-pound ; watt-econd per acre-foot. There i no correponding notion when unit are preented with their ymbol ( abbreviation ). Here we will not apply the plural to unit name in any context. Compound unit In baic textual writing, baic ( all-numerator ) compound unit name are preented uing a hyphen: watt-econd (W-). In equation, where the unit are actually embedded (a we will be doing here), oen no joining mark i ued, the adjacency of the factor implying multiplication, a it hould: L= 2.65 W. If a mark i felt to be deirable, the centered dot i recommended, rather than the hyphen: L=2.65 W. Thi make implicit that the operation of multiplication i involved, not (although thi would be aburd) ubtraction. The ame convention i ometime followed in highly-formal mathematical textual writing. Here, we will ue the hyphen in text, and generally ue no ymbol in equation (or the centered dot if a ymbol i helpful for clarity). For ratio unit, we can ue either of two form: / We will alway ue the latter form in equation, ince it i uually leat ambiguou there. DIMENSIONAL ANALYSIS Baic The technique of unit analyi involve: For unit converion tak, putting the converion factor into unit ratio form. Showing the unit of all quantitie in equation or ilar form of calculation.
Dimenional AnalyiError! Reference ource not found. Page 3 Noting that unit, jut a numerical quantitie, can be manipulated with baic algebraic rule. Being alert to the dimenional appropriatene of the unit of a reult given by a candidate procedure. Unit converion We will firt look at the application of the technique to the converion of quantitie from one unit to another. By way of example, we will work with a ituation in which length need to be converted between the unit llimeter (mm) and inch (in). We can expre the relationhip between thee two unit with thi equation: 1 (1) Note that the complete mathematical ignificance of thi i that the length 1 in i the ame length a. By the application of baic algebraic manipulation principle, we can rewrite thi a: 1 (2) Or, if we wihed, we could rewrite it intead a: 1 (3) 1 in which we could in turn write a: 1 (4) The le ide of equation 2 and 4 (and the right ide of equation 3) can be poken of a unit ratio. The term i a clever pun: unit becaue they involve the unit of quantitie, and unit becaue their value i one ( unity ). Thi mean that, in any mathematical equation or expreion, we can multiply any factor (or a whole ide of the equation) by a unit ratio 1 Note that thi preentation i unambiguou a to the direction of the ratio, unlike the form oen ued in table of converion factor.
Dimenional AnalyiError! Reference ource not found. Page 4 without actually changing anything mathematically. (It ght not be profitable, or even enible, but it i alway correct.) Let ee thi at work in a imple cae (in which we of coure could readily ee how to do the unit converion without thi tool). We tart with a imple aertion of the width of omething: w 63.5mm (5) Suppoe we want to retate thi uing the unit inch. We know the converion factor ( 25.4 ) but don t remember whether we multiply or divide. But hopefully we have available the equivalence in the form of equation 1, or we can put it in that form. Then we can deterne the unit ratio that ght be needed (equation 2 and 4). Let try the converion, uing our technique, with the unit ratio from equation 4. We multiply the right-hand ide of the equality by that unit ratio (treating the unit of the right ide of the original aertion a an integral part of the equation ): w 63.5mm (6) Thi doe not in any way dirupt or invalidate the equality; we jut multiplied the right-hand ide by a pecial form of one. We ve hown a dot to emphaize that multiplication i involved here. Now, we rearrange, recalling that unit name can be treated jut like numerical value or variable name from an algebraic manipulation tandpoint: mm mm w 63.5 25.4 (7) in We ve hown the dot between the two part to clarify that multiplication i involved (not jut a tatement of the unit, although there no mathematical difference jut a difference in viewpoint). We can conolidate thi a: 2 mm w 1612.9 (8) in Here, we ve le out the dot, uggeting the interpretation a a tatement of the unit (although again there no mathematical difference jut a difference in viewpoint).
Dimenional AnalyiError! Reference ource not found. Page 5 The bottom line i that the unit mm 2 /in i not dimenionally appropriate for a length. Thu we mut have proceeded the wrong way up. So we ll do it the other way. We tart again with: w 63.5mm [5] Thi time, we multiply the right-hand ide of the equality by the unit ratio per equation 2: w 63.5mm (9) Rearranging, we get: 63.5 mm in w (10) The two factor mm cancel, and diappear. Conolidating, we get: w 2.5in (11) The unit of the reult, in, i the deired unit (and i dimenionally conitent with the quantity). Thu we are done. Now thi eem very elaborate, for a cae where we would have certainly (?) known whether to multiply or divide 63.5 by 25.4. But of coure thi will work for cae that ght confue u, and it i not nearly o difficult to actually do a ght be inferred from my rigorou explanation. Now let ee thi work with compound unit. We have a vehicle velocity expreed in the unit / (popularly called MPH, but that doe u no good in rigorou mathematical work), but would like to know it in the unit /. So we tart with: v 37.5 (12) We alo know thi equivalence: 60 88 (13)
Dimenional AnalyiError! Reference ource not found. Page 6 Thu, our unit ratio can be either: 60 88 or 88 60 (14) Thee of coure could be rewritten variou other way; I have le them in thi obviou form for clarity a to what i going on. Note that either of thee ha the value one, o we can multiply by either without dirupting the equation. We tar, a before, with a wrong gue a to how to proceed. Recall that we begin with: v 37.5 [12] Now, we multiply the right-hand ide by the firt unit ratio per equation 14: 60 v 37.5 (15) 88 Now, if we go tough all the tep (I won t detail them here), we get: 2 v 25.56 (16) 2 Clearly the unit of the reult i not dimenionally appropriate (and certainly not the unit we were expecting). So let try it the other way up: 88 v 37.5 (17) 60 Now, aer all the manipulation, we get: v 55 (18) which i credible, in the expected unit, and preumably correct.
Dimenional AnalyiError! Reference ource not found. Page 7 With an undertanding of thi in nd, we hould be able to exane the problem and, the firt time, figure which of the two way up of the unit ratio to ue. It alway the one where the exiting unit i in the denonator. But note that the dicipline of the technique make the proce eentially foolproof, even in le-obviou cae. Another kind of calculation Next we will apply the technique to a calculation not involving converion of unit. The example i a imple one: calculating the velocity of on object when we are given a ditance and the time to travel it. A before, thi would be imple to do correctly, but tough it we can ee the application of our procedure, which would guide u in a le-obviou cae. We aume that we know the ditance, 84, and the time to cover it, 12. We can try the calculation (incorrectly) thi way: V 84 12 (19) Collecting factor, we get: or V 84 12 (20) V 1008 (21) Clearly thi unit i not dimenionally conitent with a velocity (we would be expecting the reult in the unit / ). So let try it the other way up: 84 V (22) 12 Rearranging, we get: or 84 V (23) 12
Dimenional AnalyiError! Reference ource not found. Page 8 V 7 (24) which ha a dimenionally-appropriate unit and i preumably correct. A BROADER USE The technique of dimenional analyi can oen help guide u to the formula by which ome quantity can be deterned. We may not even be certain what factor go into it, or how they are to be multiplied or divided. We may try a certain formula we think i correct, only to find that (in a dummy run, with dummy number) it give a reult in the wrong unit that i, in a unit of inconitent dimenionality. We may notice that, if there had been a quantity with the dimenionality of area in the denonator, the dimenionality of the dummy reult would have been proper. Thi may be the cue we need to realize, Oh, of coure! The reult i affected by the area of the region from which the lunou flux i etted. And I gue I will need to put that in the denonator. Naturally, thi will not alert u to the need to have uch contant a in the formula. But it will rend u (to give a trivial example) that the area of a circle involve the quare of the radiu, not the radiu itelf. SUMMARY The technique of dimenional analyi can lead u to the proper way to do unit converion, to perform many other practical calculation, and even to develop the formula for a calculation. Once the concept i graped, we can oen ee how to proceed without actually doing all the manipulation needed by the technique in it full-blown form. # For more article by thi author, viit The Pumpkin at: http://doug.kerr.home.att.net/pumpkin