Peter Wittich

Back of the Envelope Physics

Reminder: Lab reports due TODAY• Turn in your lab report by 5 pm today at the box next

to the entrance to the lab on the 3rd floor of Clark Hall– Late reports only with prior arrangement– Otherwise, loose 1 point (10%) per late day

• You should turn in a printout of your report as well as your lab notebook

• Sign out of your 1st round lab tomorrow if you have not already done so

• Once you have signed out turn in your driver’s license to me and I can sign you into your next lab– Web page (Google doc) shows what labs are available

for round 2 2

Back of the envelope goal: know what to expect• I want to measure a new quantity• I need to know roughly what to expect

• For 510 lab:–should my DVM be set to mV, V, 10V scale?

• Two related tools to answer questions like this–back-of-the-envelope physics–dimensional analysis

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Famous physics example: Enrico Fermi• Italian-American• “father of the bomb”

– first sustained nuclear chain reaction under U of Chicago squash court

• pre-eminent experimentalist of his time (and 1st rate theorist too)

• Fermi method: simple estimates to get order-of-magnitude idea for value of a parameter

• Classic Example: How many piano tuners in Chicago?

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wartime Los Alamos ID

Example estimate broken down.• about 3 million people

in Chicago –(no longer Second

City!)• most in four-family

households– 750,000 households

• one in five has a piano–150,000 pianos

• Gets tuned once/year–150,000 piano tunings

a year

• Each piano tuner can tune 5 pianos/day–30,000 tunings/year

• Work year is ~250 work-days–120 tuners needed to

fill this need.• Ergo, there are

roughly 120 piano tuners in Chicago.

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My estimate • about 3 million people

in Chicago –(no longer Second

City!)• most in four-family

households– 750,000 households

• one in five has a piano–150,000 pianos

• Gets tuned once/year–150,000 piano tunings

a year

• Each piano tuner can tune 5 pianos/day–30,000 tunings/year

• Work year is ~250 work-days–120 tuners needed to

fill this need.• Ergo, there are

roughly 120 piano tuners in Chicago.

• Key: fact, estimate, result

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I’m happy if this is within a factor of ten of the true number.

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!!!

Common estimation tricks• Remember, we want to simplify our calculations..• Volume: everything is a square or a sphere• π = 3, 3^2 = 10, 3 π = 10, e = 3 • Use the right units

– example: for energy: joules for macroscopic scales; eV for nuclear physics scales

• round, round, round– if you think your number is only good to an order of

magnitude it doesn’t make sense to drag along many significant digits

• know some appropriate numbers– if calculating g on the moon, compare it to g on earth

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http://www.stsci.edu/institute/Copyright

constants in units appropriate for modern physics

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� = 6.582× 10−22 MeV · sc = 2.998× 108 m/s

�c = 197.3 MeV · fm1 eV = 1.602× 10−19 J

me = 0.511 MeV

mp = 938 MeV

µB = 5.788× 10−11 MeV T−1

µN = 3.152× 10−14 MeV T−1

kB = 8.617× 10−5 eV K−1

Class problems: from everyday life (Swartz)• What is the force or acceleration you receive in a car crash

(and when is it fatal)?

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• F = ma; a = F/m = Δv/Δt• Δv = 40 mph = 18 m/s → 20 m/s• Δt = ?

• assume car stops in 1 m (crumple zone) and that the average speed is 20 mph (9 m/s)• → time is t = s/v = 1 m / 9 m/s = 111 ms

• So a = 18 m/s / 0.111 m/s = 162 m/s2 = 16.5 g• Assumptions?

• Gradual stopping (air bag.) • What happens if no air bag?

• Δt → smaller, force goes up, life expectancy goes down.

What’s different compared to the piano tuners?• You still need to know physics

–You can’t use ‘back of the envelope’ to forgo knowledge of physics

–You need to know what the base equations are.

• This is where practice comes in.–Learn what the relevant approximations are–Learn what you can ignore and what you cannot

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Class problems: Hydrogen atom• QM 101 problem: estimate the ground state

energy of hydrogen and the size of the hydrogen atom.

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E =p2

2m− e2

r

p = �k,λ ≈ 2πr ⇒ E ≈ −e2

r+

�22mr2

dE

dr

����a0

= 0 ⇒ a0 =�2me2

= 0.53 A

E0 = −me4

2�2 = −13.6 eVGreat estimate - consistentwith measured value.

Rotational energy spectrum of H2.

• Estimate the energy required to increase H2 angular momentum by one unit. Can I use microwave radiation?

• Δ(angular momentum) = Δ(I ω) = ℏ

• Moment of inertia of hydrogen molecule I = 2mpr2.• Rotational kinetic energy

–Er= ½Iω2= ½ (Iω)2/I = ½ ℏ2/I ≈ 1/100 eV.

• this is not microwave energies - but microwave works for heavier molecules.

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Preliminary conclusions

• Doing this kind of estimates is a good tool to have in your toolbox

• It requires a willingness to estimate quantities and a knowledge of enough physics to know what’s important

• it’s a skill that can be practiced–We’ll do more of that next week.

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References for further reading

• V. Weisskopf, “Modern Physics From an Elementary Point of View”. CERN 70-8 (1970)

• V. Weisskopf, “Search For Simplicity.” Series of articles in AJP from 1985/1986

• Clifford Swartz, “Back of the Envelope Physics” (Johns Hopkins U, 2003)

• A. Santos, “How Many Licks?: Or, How to Estimate Damn Near Anything” (Running, 2009)

• S. Mahajan, “Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving” (MIT, 2010) 15