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[Phys-l] real-world fuel-tank calibration ... also yardstick balance



How'bout we discuss the real solutions, rather than fixating
on what's wrong with the wrong solutions.


As I interpret it, the task as set forth at
http://www.cartalk.com/content/puzzler/transcripts/201103/index.html
is to calibrate the tank without using calculus, and without
using a computer.

1) The REAL real-world solution is to run the tank down
to nearly empty ... then refuel it gallon by gallon,
sticking it repeatedly, so as to establish the entire
calibration curve empirically.

I've done this more than once, usually on airplanes, where
there is a very real benefit to having accurate and reliable
fuel gauges.

This works for tanks of any shape, not just ideal cylinders.

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

2) The low-tech method for predicting the 1/4 point in a
cylinder, in accordance with the hints given by the cartalk
guys, involves converting the yardstick into an equal-arm
balance.

This particular application of the yardstick-balance is
completely hokey, but there are plenty of situations where
a yardstick-balance makes perfect sense. I've made many
of the things over the years. It's "appropriate technology"
for little kids. Most families don't provide their kids
with laboratory-grade balances. Mine certainly didn't.

I assume an ordinary /wooden/ yardstick.

Drive a sewing needle through the "18" line on the yardstick.
Put it 1/8th of an inch above the midline of the yardstick so
that the balance exhibits slight positive stability. Beware:
if you do it wrong the needle may break, so hold the needle
with pliers (not your hand) and push on it with something
solid (not your hand). Wear eye protection.

Drive pins into the ends of the yardstick to hold threads
that hold the stuff you want to weigh.

Balance the thing between a pair of salt shakers borrowed
from the truck stop. Or beverage cans. Or whatever.

You can calibrate it using a US "nickel" coin, which is
a metric coin. It weighs 5 grams and is 2 mm thick.
The mnemonic is 5 cents, 5 grams, and 5 per centimeter.
OTOH for the present application, we don't even need to
calibrate it; all we need is a comparator.

Even in non-expert hands, such a balance can resolve 25
milligrams ... i.e. less than the mass of a postage stamp.
It is dramatically more sensitive than an ordinary kitchen
scale or postage scale. You may need to shield it from
drafts.

So, if you are going for 1% accuracy on the fuel tank, you
can do it using a circles laid out on plain old inkjet
printer paper, which weighs about 5 grams per sheet. Fancy
expensive stationery weighs less per unit area, but you're
not likely to find that at the truck stop. You can get
additional accuracy by using a larger disk cut from heavier
stock, such as poster board or last night's pizza box.

Plan A is to start with a disk. Cut the disk in half. Weigh
each half against the other to verify that the cut is accurate.

Plan B is better if you are starting with 8.5x11 paper:
Use two pieces of paper, and lay out an 11-inch diameter
semicircle on each one.

From here on, plan A and plan B are the same:

Cut one of the semicircles in half the easy way, to make
quarter-circle sectors. Weigh them against each other to
verify accuracy. This gives you an accurate reference for
the "quarter tank" point.

Take the other semicircle and cut it in half (as accurately
as you can) the hard way, i.e. with a straight cut parallel
to the existing straight edge. I suggest that placing the
cut 0.3 of the diameter in from the curved edge would be a
very good guess. That corresponds to a tank filled to depth
of 0.3 of diameter, i.e. 0.3 of the distance from empty to
full.

Using calculus and/or a computer, you can figure out the
answer to one part in 10^16 or so, but that is not the
point of the exercise, so let's not go there.

At this point you have a lune (piece "A") and another piece
("B"). Keep both pieces.

Weigh the lune ("A") against the 1/4 tank reference sector.

If piece "A" is too light, cut a sliver off of piece "B"
and add it to the scale. If piece "A" is too heavy, cut
a sliver off of piece "A" and set it aside.

Do a binary search. This will converge in less time than
it takes to tell about it. Add up the "depth" of all the
pieces needed to balance the reference.

I've left out a lot of details. You can figure it out.