Dimensional Analysis and Atomic Bombs - mosaic-web.orgBuckingham Pi Theorem Some \philosophical"...

$: Dimensional Analysis and Atomic Bombs - mosaic-web.orgBuckingham Pi Theorem Some \philosophical" ideas... You can only compare (and add and subtract) quantities with the same dimension$
Dimensional Analysis and Atomic Bombs

Chad M. Topaz

Department of Mathematics, Statistics and Computer ScienceMacalester College

St. Paul, Minnesota 55105

[email protected]

April 15, 2011

Overview of this talk

1 Welcome to Dimensional Analysis

2 The Buckingham Pi Theorem

3 Example: Vertical Projectile

4 Atomic Bomb Assignment

5 Final Thoughts

C. M. Topaz (Macalester) Dimensions and Bombs April 2011 2 / 19

Welcome to Dimensional Analysis

What are dimensions?See Kaplan & Flath MCASTs from Feb. 11 and Mar. 4 2011

Relate physical quantities to each other

Reveal truths about the universe (powerful and surprising!)


http://www.mosaic-web.org/MCAST/videos/MCAST-2011-02-11/lib/playback.html

http://www.mosaic-web.org/MCAST/videos/MCAST-2011-03-04/lib/playback.html


There are seven primary dimensions (controversial).

Dimension Symbol Example unitLength L metersMass M kilogramsTime T secondsAmount of stuff N moleTemperature Q KelvinElectric Current I AmpereLuminosity C Candela



All other dimensions are called derived dimensions

Formed as integer-power products of derived dimensions

Examples include:

Quantity Dimensions IntuitionVelocity LT−1 distance/timeAcceleration LT−2 change in velocityForce MLT−2 F = maFrequency T−1 how oftenEnergy ML2T−2 force × displacementArea L2 length × widthVolume L3 length × width × heightDensity ML−3 mass per volume


Buckingham Pi Theorem

Some “philosophical” ideas...

You can only compare (and add and subtract) quantities withthe same dimension (e.g. 7 miles + 3 oranges is meaningless).

Sensible laws of nature don’t care what unit system they aremeasured in.

Π Theorem: Sensible laws of nature can be put in the formf (Π1,Π2, . . .) = 0 where the Πi are dimensionless quantities.

Proof comes from linear algebra.

e.g., F = ma→ F/ma = 1→ F/ma − 1 = 0. DefineΠ ≡ F/ma, f (Π) ≡ Π− 1.

How on earth is this useful?


Example: Vertical Projectile

How high does the ball go? Don’t use physics

m v

h

g

m = mass of projectilev = launch velocityh = maximum heightg = acceleration due to gravity

1 Choose what quantities enter the problem.

We choose m, v , h, and g .




m v

h

g


2 List dimensions of all quantities.

[m] = M

[v ] = LT−1

[h] = L

[g ] = LT−2




m v

h

g


3 Construct dimensionless quantities from original quantities.

Let Π = mavbhcgd .

Then [Π] = Ma(LT−1)bLc(LT−2)d = MaLb+c+dT−b−2d .

For Π to be dimensionless, a = 0, b + c + d = 0, −b − 2d = 0.

Solution is a = 0, b = −2c , d = c (one free parameter).

For convenience, choose c = 1. Then a = 0, b = −2, d = 1.

Thus Π = hg/v 2.




m v

h

g


4 Apply Buckingham Pi theorem and solve for quantity of interest.

f (Π) = 0 (function f is unknown)

f (hg/v 2) = 0

hg/v 2 = k (constant k is unknown)

h = kv 2/g




m v

h

g


4 Interpret the results.

h = kv 2/g

Height does not depend on mass

Height only depends on v 2/gI e.g, Doubling initial velocity will increase height 4-foldI e.g, Throwing ball on the moon will increase height 6-fold



Other choices of c lead to the same answer (for this example).

More complicated situations may well yield more than one Π.

In my calculus class, I stick to examples with one Π.

Interpreting the answer is critical.

Students benefit from practice.

Some useful sourcesBarenblatt, G., Scaling, Self-Similarity, and IntermediateAsymptotics. Cambridge University Press, 1996.

Giordano, F. et al., A First Course in Mathematical Modeling.Brooks Cole, 2008.

Adam, J., Mathematics in Nature: Modeling Patterns in theNatural World. Princeton University Press, 2006.


Atomic Bomb Assignment

The Royal Society is collaborating with JSTOR to digitize, preserve, and extend access toProceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences.

www.jstor.org®

on August 8, 2010rspa.royalsocietypublishing.orgDownloaded from



Historical re-enactment course project in Calc I.

G.I. Taylor, The Formation of a Blast Wave by a Very IntenseExplosion, Proc. R. Soc. Lond. A (1950).

Students apply mathematics to study real-world issues; practiceconfronting open-ended problems; and communicate scientificresults clearly and professionally.

Project SummaryThe year is 1945. The United States has just detonated its firstatomic bomb in the Trinity test at White Sands, New Mexico.Details of the atomic bomb are highly classified for reasons ofnational security. Your scientific team wishes to deduce the energy ofthe explosion, which is one of the most closely guarded of all secrets.Your team of three needs to perform the requested analysis and writea full report.



A nuclear blast creates a clearly visible wavefront.



Your team collected the photographs obtained by the press andother entities (foreign governments, for example). Then, youestimated the wavefront radii (in meters) off of each photographand arranged the data in a table with the corresponding times(in seconds). Thus, you now have a data set where the first fewentries look like this:

Time t (sec.) Blast Radius R (m)0.00038 25.40.00052 28.80.00066 31.90.00080 34.20.00094 36.30.00108 38.9



A reasonable starting assumption is that the radius R of thewavefront depends on time t (as seen in the series of pictures),on the ambient density ρ of the air (whose value can be found inmany sources) and on the energy E of the explosion. Othereffects will be ignored.

Dimensional analysis might help in determining an expression forthe wavefront radius R .

You remember that an expression arising from dimensionalanalysis will have an unknown constant in it. However, thephysicists on your team assure you that using physicalarguments, they’ve determined that this constant isapproximately one.


Atomic Bomb Assignment (Solution)

Dimensional analysis yields that R = k (Et2/ρ)1/5

.

Set k = 1 (given), use ρ ≈ 1 and take log to obtainlogR = 1/5 log E + 2/5 log t.

Plot (log t, logR) and find best-fit line.

3 2 11.2

1.6

2

2.4

log10t

log 10

R

log10R = 0.39 log10t + 2.7

(I removed a few early-timedata points to facilitatecurve-fitting for students.)

E follows from intercept; find E ≈ 3.2× 1013 J .

Actual value was E ≈ 9× 1013 J .


Final thoughts

Dimensional analysis is...A powerful application of simple, fundamental ideas.

Often surprising to students.

Accessible to calculus students having 1 - 2 hrs. of preparation.

More complicated examples appropriate for upper-level courses.

Useful in modeling, differential equations, . . . ...

Buckingham Pi proof is interesting exercise for a linear algebraclass.

The End.


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