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Re: [Phys-L] Halfway point ???

Yes, thank you, you are right. I dropped a factor of 10^3 in the earth' radius (didn't convert from km to m). Geometric mean (corrected) is now about 70 microns (of course, arithmetic mean and harmonic mean are also off by this same error in my original post). Still the geometric mean remains "reasonably" dust particle sized even using the 0.001 to 30 microns range I originally cited for atmospheric dust particles.

Don

-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@www.phys-l.org] On Behalf Of John Denker
Sent: Friday, April 03, 2015 1:23 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Halfway point ???

On 04/03/2015 08:19 AM, Donald Polvani wrote:
range of atmospheric dust particle "sizes" as 0.001 to 30 microns mean
radius of earth = r_e = 6371 km mean "radius" of proton = r_p = 0.86
fm

Then arithmetic mean of r_e and r_p is about 3200 m Geometric mean is
about 2 microns Harmonic mean is about 2 fm

1) I estimated this one in my head: The proton is a little less than one fermi, i.e. 10^-15 meters. The earth is a little less than 10^7 meters (in accordance with the original /definition/ of the meter). So the span there is 22 orders of magnitude. The square root of that is 11 orders of magnitude. So the geometric mean ought to be 10^11 fm, i.e. 10^-4 m, i.e. 100 microns ... or a little less.


2) Whenever I see a mistake, I try to figure out where it came from. Similarly, whenever I see an ill-posed question, I try to figure out where it is coming from, to figure out what question /should have been asked/.

In the present case, it seems plausible that a factor of "kilo" got dropped from the size of the earth, early in the calculations.

My number for the geometric mean is bigger by a factor of √1000; I get 74 microns, which is still a bit too big to qualify as a dust particle.

Also consistent with this hypothesis, my number for the arithmetic mean is bigger by a full factor of 1000; I get 3186 km (not m).


3) Doing order-of-magnitude estimates is a useful technique. In some cases it solves the problem all by itself. In other cases is serves as a valuable check on other methods.


4) BTW, if you take the geometric mean of the /masses/ you get something even bigger, almost 0.1 kg, which is waaay too big to qualify as a dust particle.

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