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Re: [Phys-l] radio-entertainment-world problem



On 02/02/2011 12:12 PM, John Mallinckrodt wrote:

I see that Tom and Ray did fall for the COM-based solution,

It seems that two MIT bachelor degrees are not sufficient
to ensure infallibility.

http://www.cartalk.com/content/puzzler/transcripts/201103/answer.html

but I still doubt that any car company engineer would have gone that
route. Interestingly, Tom and Ray were likely attracted to that
solution precisely because they thought it elegantly illustrated some
fundamental physics which is precisely NOT what it does.

Indeed it does NOT.

I wrote to them, but I imagine others have already done the same.

Here is the complete calibration curve.
http://www.av8n.com/physics/img48/cylindrical-tank.png

It is somewhat remarkable that at no point does the curve differ
from the diagonal by more than 6% of full scale. The diagonal is
the obvious "zero parameter" linear approximation. I reckon this
is related to the celebrated "sphere hardening" phenomenon.

In case anybody is wondering where the curve came from, the solution
is
normalized volume = (θ - 0.5 sin(2 θ)) / π
normalized depth = 0.5 - 0.5 cos(θ)

Yeah, I know you can make that into a single equation for volume
as a function of depth, but for every purpose I can think if, you
are better off leaving it in parametric form, i.e. Vnorm as a
function of θ and Dnorm as a function of θ. I'm thinking of
purposes including (a) plotting and (b) solving.

For Vnorm = 0.25 the solution for Dnorm is
Dnorm = 0.29801362335024135(10)

=========================

I changed the Subject: line to emphasize that this is not even
remotely a real-world problem. The objective is contrived and
the method of solution is not practical.

OTOH the idea of the yardstick-balance is a keeper. See previous
note.