hey everyone in my
last video I had to
make a small calibration wait for the
microgram balance that I built and so
the challenge is how do you make a 100
microgram test mass if you don't have a
scale that measures micrograms and so I
solved the problem by converting that
difficult measurement challenge you know
100 microgram measurement into a linear
size measurement and the trick is that I
used a piece of paper and made the
assumption that the paper has uniform
density and uniform thickness so the
idea is it's very difficult to measure a
hundred micrograms but it's very easy to
measure a larger piece of paper and cut
it down such that the ratio will give
you something that's very close to a
hundred micrograms this process of
converting a problem into a different
space to make it easier to solve
reminded me of something else that I
heard a while back so in the days before
computers and before calculators even
before calculus itself there is still a
desire to integrate mathematical
functions and we can do this using the
paper trick so to get started I cut out
a bunch of different pieces of paper of
different size and I was mainly
interested to see what the linearity of
this whole system was going to be so I
cut these from different sheets of paper
and the idea is that I just you know
took them one by one and checked their
mass with this balance and this specs a
precision of two milligrams and so I
measured the mass and had the area for
these various different sizes and the
mass divided by area number is actually
pretty consistent it's in fact it's
surprisingly good and the smallest piece
of paper I had was 5x5 which had a mass
of 30 milligrams so even getting all the
way down to 30 milligrams the the mass
over area number was still pretty good
so this gave me confidence that the the
whole method was going to work pretty
well what's interesting is that we don't
have to know anything about milligrams
or millimeters or any units whatsoever
so if we were living in the days before
calibrated balances like this what we
could use is a to pan balance you know
just two plates that are that are set up
on a very delicate pivot
and what we could do is cut out pieces
of paper and use these as the
calibration masses so if we had a whole
set of these we could put these on one
side of the scale and put the piece of
paper that we wanted to mass on the
other side and we know exactly what the
balance is just by having enough of
these calibration masses what's cool is
that we don't even really have to
measure the grid just as long as the
grid is the same on all the pieces of
paper we can make this method work
without knowing anything about Units
okay so let's start with an easy one
this is f of x equals x and so what I
did was I just drew it out with a pencil
and then cut out the actual piece that
we want to integrate and we can check
the mass of that and then divide by that
1.47 constant and we get the area in
terms of squares is equal to 111 and so
what we're really getting here is the
area under the curve and we can check
our math since we have geometry now we
know that the size of this triangle is
one-half base times the height so 1/2
times 15 times 15 is 100 12.5 so pretty
close to the hundred and eleven that we
got through the mass measurement method
and since we also have calculus now we
can do the integral here we end up with
x squared over two plus a constant and
if we evaluate all that of course it's
112 point five and no surprise that the
x squared over two looks an awful lot
like the one half base times the height
and the error turns out to be about one
point three percent so this is pretty
good this is also kind of the best case
scenario since since we're dealing only
with straight lines here really what
we're testing is the the absolute best
possible case for this whole measurement
system here's a more interesting
function we have f of X equals x squared
over 16 and I used the over 16 just to
spread this graph out so that it worked
out better for my graph paper and I
plotted all the points here and just
connected the points with straight lines
so we checked the mass and we divide and
we get eighty seven point one and in
this case there's no easy geometric way
to get the area this we go straight to
calculus to check it and we
have X cubed over three times the 16
which is the original plus constant and
we evaluate all that and we're still
within about 2.1 percent of the
recession of the real value this method
still works for functions that go below
zero two of course what we can do is
just cut out the two pieces separately
measure the mass of this and then
subtract this from it and that will give
us the total area under the curve even
though some of it is negative likewise
if we had two functions and we wanted to
find the area between them we could just
plot both of them out and then cut out
the area that we wanted between them so
this technique is very powerful because
even if we don't have a mathematical
formula for our data we can still
connect all the points with lines and
integrate them for example if you are
taking data on the flow rate into a
reservoir or something like that and it
didn't really follow any particular
mathematical function you could still
connect all these points with lines and
then use the the mass trick to get the
total amount of volume in the reservoir
of course if you're an engineer you
would must build a flow meter that did
the integration mechanically but anyway
okay I hope that inspires you to think
of difficult measurement problems in
terms of converting them into a
different space in which they're easier
to solve see you next time bye
s
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