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DIMENSIONAL ANALYSIS

Is that all there is?

W. J. Wilson

Department of Physics and Engineering

University of Central OklahomaEdmond, Oklahoma 73034-5209

wwilson@ucok.edu

(The Secrets of Physics Revealed)

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Outline

A. Whats the secret of being a Scientist or an Engineer?

B. What are Units and Dimensions anyway?

C. What is Dimensional Analysis and why should I care?

D. Why arent there any mice in the Polar Regions?

E. Why was Gulliver driven out of Lillipute?F. What if Pythagorus had known Dimensional Analysis?

G. But what do I really need to know about Dimensional

Analysis so that I can pass the test?

H. Can I get into trouble with Dimensional Analysis? Theballad of G.I. Taylor.

I. But can it be used in the Lab ?

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How to be a Scientist or Engineer

The steps in understanding and/or control

any physical phenomena is to

1. Identify the relevant physical variables.2. Relate these variables using the known

physical laws.

3. Solve the resulting equations.

Secret #1: Usually not all of these are

possible. Sometimes none are.

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ALL IS NOT LOST BECAUSE OF

Physical laws must be independent of

arbitrarily chosen units of measure. Nature

does not care if we measure lengths in

centimeters or inches or light-years or

Check your units! All natural/physical

relations must be dimensionally correct.

Secret #2: Dimensional Analysis

Rationale

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Dimensional AnalysisDimensional Analysis refers to

the physical nature of the

quantity and the type of unit

(Dimension) used to specify it.

Distance has dimension L.

Area has dimension L2.

Volume has dimension L3.

Time has dimension T.

Speed has dimension L/T

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Why are there no smallanimals in the polar regions?

Heat Loss Surface Area (L2)

Mass Volume (L3)

Heat Loss/Mass Area/Volume

= L2/ L3= L-1

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Heat Loss/Mass Area/Volume

= L2/ L3

= L-1

Mouse (L = 5 cm)

1/L = 1/(0.05 m)

= 20 m-1

Polar Bear (L = 2 m)

1/L = 1/(2 m)= 0.5 m-1

20 : 0.5 or 40 : 1

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Gullivers Travels: Dimensional Analysis

Gulliver was 12x the Lilliputians

How much should they feed him?

12x their food ration?

A persons food needs are

related to their mass

(volume)This dependson the cube of the linear

dimension.

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Gulliver is 12x taller than theLilliputians, LG=12 LL

Now VG (LG)3 and VL (LL)3, so

VG / VL = (LG)3 / (LL)

3= (12 LL)

3 / (LL)3

= 123= 1728

Gulliver needs to be fed 1728times the amount of food eachday as the Lilliputians.

Let LG and VG denote Gullivers linear and volume dimensions.

Let LL and VL denote the Lilliputians linear and volume dimensions.

This problem has direct relevance to drug dosages in humans

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Pythagorean Theorem

Area =F(q) c2

A1 =F(q) b2

A2=F(q) a2

Area = A1 +A2

F(q) c2

=F(q) a2

+F(q) b2

c2= a2+b2

ca

b

qq A1

A2

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Dimensions of Some CommonPhysical Quantities

[x], LengthL

[m], MassM

[t], TimeT

[v], VelocityLT-1

[a], AccelerationLT-2

[F], ForceMLT-2

[r], Mass DensityML-3[P], PressureML-1T-2

[E], EnergyML2

T-2

[I], Electric CurrentQT-1

[q], Electric ChangeQ

[E], Electric Field - MLQT

-2

All are powers of the fundamental dimensions:

[Any Physical Quantity] = MaLbTcQd

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Dimensional Analysis Theorems

Dimensional Homogeneity Theorem: Anyphysical quantity is dimensionally a power lawmonomial - [Any Physical Quantity] = MaLbTcQd

Buckingham Pi Theorem: If a system has kphysical quantities of relevance that depend on

depend on rindependent dimensions, then thereare a total ofk-rindependent dimensionlessproducts p1, p2, , pk-r. The behavior of thesystem is describable by a dimensionless equation

F(p1, p

2, , p

k-r

)=0

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Exponent Method1. List all kvariables involved in the problem

2. Express each variables in terms of [M] [L] [T ]dimensions (r)

3. Determine the required number of dimensionlessparameters (kr)

4. Select a number of repeating variables = r(All dimensions must be included in this set and eachrepeating variable must be independent of the others.)

5. Form a dimensionless parameterpby multiplying one ofthe non-repeating variables by the product of the

repeating variables, each raised to an unknown exponent.6. Solved for the unknown exponents.

7. Repeat this process for each non-repeating variable

8. Express result as a relationship among the dimensionless

parametersF(p

1,p

2,p

3, ) = 0.

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G. I. Taylors 1947 Analysis

Published

U.S.

Atomic

Bomb was

18 kiloton

device

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Nuclear Explosion Shock Wave

r f(E,r,t)

5/15/25/1

5/15/25/1

1

322000

1

5

220:

5

13210:

0:

)()())((

)(

rrp

rp

tE

RtRE

bbaT

acaL

accaM

MLTTMLLTLM

tEr

cba

cba

The propagation of a nuclear

explosion shock wave depends on:E,

r,r, and t.

n = 4 No. of variables

r= 3 No. of dimensions

n r = 1 No. of dimensionless parameters

Select repeating variables:

E, t, andrCombine these with the rest of the

variables: r

E r r t

ML2T-2 ML-3 L T

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11 0)(

rr

pp

tECRCtE

R

CF

5/12

r

EtCR

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r 1 kg/m3

R = (E/r)1/5t2/5

logR = 0.4 log t+0.2 log(E/r)

0.2 log(E/r) 1.56

E= 7.9x1013 J= 19.8 kilotons TNT

Blast Radius vs Time

log R = 0.4058 log t + 1.5593

1

2

3

-2 -1 0 1 2

log (t)

log(R)

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Dimensional Analysis in the Lab Want to study pressure drop as

function of velocity (V1) and

diameter (do)

Carry out numerous experiments

with different values ofV1 and do

and plot the data

5 parameters:Dp,r, V1, d1, do

2 dimensionless parameter groups:

DP/(rV2/2), (d1/do)

p1

p0

V1

A1

V0A0

Much easier to establish functional relations with 2 parameters, than 5

DP r V1 d1 d2

ML-1T-2 ML-3 LT-1 L L

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References1. G. I. Barenblatt, Scaling, Self-Similarity, and Intemediate Asymptotics

(Cambridge Press, 1996).

2. H. L. Langhaar, Dimensional Analysis and the Theory of Models (Wiley,

1951).

3. G. I. Taylor, The Formation of a Blast Wave by a Very Intense Explosion.The Atomic Explosion of 1945,Proc. Roy Soc. LondonA201, 159 (1950).

4. L. D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, 1959).

Section 62.

5. G. W. Blumen and J. D. Cole, Similarity Methods for Differential Equations

(Springer-Verlag, 1974).

6. E. Buckingham, On Physically Similar Systems: Illustrations of the Use of

Dimensional Equations,Phys. Rev. 4, 345 (1921).