On 12/19/2022 9:10 AM, Peter J. Holzer wrote:
On 2022-12-19 09:25:17 +1100, Chris Angelico wrote:
On Mon, 19 Dec 2022 at 07:57, Stefan Ram <r...@zedat.fu-berlin.de> wrote:
G = Decimal( 6.6743015E-11 )
r = Decimal( 6.371E6 )
M = Decimal( 5.9722E24 )
What's the point of using Decimal if you start with nothing more than
float accuracy?
Right. He also interpreted the notation "6.67430(15)E-11" wrong. The
digits in parentheses represent the uncertainty in the same number of
last digits. So "6.67430(15)E-11" means "something between 6.67430E-11 -
0.00015E-11 and 6.67430E-11 + 0.00015E-11". The r value has only a
precision of 1 km and I'm not sure how accurate the mass is. Let's just
assume (for the sake of the argument) that these are actually accurate in
all given digits.
So G is between 6.67415E-11 and 6.67445E-11, r is between 6.3705E6 and
6.3715E6 and M is between 5.97215E24 and 5.97225E24. If we compute the
time for those deviations you will find that the differences are many
orders of magnitude greater than the effect you wanted to show. And that
still ignores the fact that a vacuum won't be perfect (and collisions
with a few stray atoms might have a similarly tiny effect), that gravity
isn't constant while the weight falls (it's getting closer to the center
of the earth and it's moving past other masses on its way) that Newton's
law is only an approximation, etc. So while the effect is (almost
certainly) real, the numbers are garbage.
I think there's a basic numeracy problem here. This is unfortunately all
too common, even among scientists. The OP apparently rounded their
numbers to 8 significant digits (thereby introducing an error of about
1E-8) and then insisted that the additional error of 1E-15 introduced by
the decimal to float conversion was unacceptable, showing IMHO a
fundamental misunderstanding of the numbers they are working with.
hp
In a way, this example shows both things - the potential value of using
Decimal numbers, and a degree of innumeracy. It also misses a chance to
illustrate how to approach a problem in the simplest and most
informative way. Here's what I mean -
We can imagine that the input numbers really are exact, with the
remaining digits filled in with zeros. Then it might really be the case
that - if you wanted to do this computation with precision and could
assume all those other effects could be neglected - using Decimals would
be a good thing to do. So OK, let's say that's demonstrated. No need to
nit-pick it further.
As a physics problem, though, you would generally be interested in two
kinds of things:
1. Could there be such an effect, and if so would it be large enough to
be interesting, whether in theory or in practice?
2. Can there be any feasible way to demonstrate the proposed effect by
measurements?
The first thing one should do is to find a way to estimate the magnitude
of the effect so it can be compared with some of those other phenomena
(non-constant gravity, etc) to see if it's worth doing a full
computation at all, or even spending any more time on the matter. There
is likely to be a way to make such an estimate without needing to resort
to extremely high precision - you would only need to get within perhaps
an order of magnitude. Your real task, then, is to find that way.
For example, you would probably be able to estimate the precision needed
without actually doing the calculation. That in itself might turn out
to enough.
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