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Series P: Piecemeal

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Celebaelin
1347735.  Sat May 09, 2020 3:41 am Reply with quote

In the tedium of lockdown I decided to check the maths on the 'for a piece of string around the equator increase the height above ground by 16cm and find the increase in length of the string' question.

So here it is...

The metre was originally defined in 1793 as one ten-millionth of the distance from the equator to the North Pole along a great circle, so the Earth's circumference is approximately 40000km; at the equator the distance is closer to 40075km.

C= 40075km = 40075000m

C = πD = 2πr

=> D = C/π

π ≈ 3.14159265359

D = 40075000m/3.14159265359 = 12756268.6888m

C = 2πr = πD

=> r = D/2

r = 12756268.6888m/2 = 6378134.3444m

r + 0.16m = r*

6378134.3444m + 0.16m = 6378134.5044m = r*

C* = 2πr*

C* = 2 x 3.14159265359 x 6378134.5044m = 40075001.0053m

C*-C = 40075001.0053m - 40075000m = 1.0053m

Rather more interesting is what happens when you compare that result to the result for a sphere of circumference 1m...

C= 1m

C = πD = 2πr

=> D = C/π

π ≈ 3.14159265359

D = 1m/3.14159265359 = 0.31830988618m

C = 2πr = πD

=> r = D/2

r = 0.31830988618m/2 = 0.15915494309m

r + 0.16m = r*

0.15915494309m + 0.16m = 0.31915494309m = r*

C* = 2πr*

C* = 2 x 3.14159265359 x 0.31915494309m = 2.0053m

C*-C = 2.0053m - 1m = 1.0053m

I know this, and that it holds for all the circumferences I've tested to date, but I still don't understand it really except to say

For

Increase in radius x

and

Increase in circumference = i

C* = 2π(C/2π + x)

=> C*-C = [2π(C/2π + x)]-C

=> i = 2πC/2π + 2πx C

=> i = C + 2πx C

=> i = 2πx

but it still seems intuitively wrong.

 
PDR
1347747.  Sat May 09, 2020 5:31 am Reply with quote

Yes, they made us do that at school and it still feels wrong! In my 2nd year uni maths module they made us do a similar, but more realistic. analysis in which we used aluminium alloy wire of a specified diameter and raised it a distance x off the ground with posts spaced sm apart and the wire tensioned to t% of the elastic limit of the material. The assignment required an analytical rather than numerical solution (ie an equation rather than just a number) which had wire diameter, material properties (density, elastic limit and young's modulus), wire tension, distance between posts as variables. This really fixed my understanding of catenery (hypoerbolic sine) curves which described the sag between the posts.

We were then required to re-express the resulting equation to give numbers in terms of each variable (eg to give tension per increase in wire diameter, or %age elastic limit required per increase in post distances) and extract the first derivative of each to look at the shape of the curves as they were varied. After that my grasp of partial differentiation and calculus of hyperbolic trig functions was significantly improved.

It was at this point the penny dropped - this was the analysis needed to understand the mechanical properties and limits of long-distance electricity cables. So we were then given the relatively simple task of implementing all of that in software to make a calculator - something that a power company could use to understand the options (and available trade-offs) for a given run of national grid cable. For this they then required us to include both resistivity, thermal expansion and cost parameters with the cable properties so that a user could evaluate the benefits of different cable materials. Some of us gained extra credit for making provision for the deterioration of strength due to cable damage (scratches) using the "stress raiser" equations.

The whole thing was a 2 week assignment, intended to mimic the way an engineer might have to work in a real job. It started out with "calculate the extra wire length due to curvature of the earth". An hour later we got an update - "we've been thinking about this - you need to add in the catenery calculations for sag". The next day they added another update about needing to know the tension and material characteristics and so on. So we learned that in the real world there is a thing called "Requirements Creep" where the first question is never the final one, so general solutions are always safer than specific ones*.

This was then left for a couple of months, but we were then tasked with revisiting the project and adding in calculations for wind-loadings on the cable (this is a non-trivial addition). A month after that they asked us to include a model of resonant frequencies in the cable to see if there was a risk of harmonics being excited and magnifying the force loadings.

So what started out as a simple "one of eight questions in the overnight maths quiz" grew into the whole year cross-module project involving about 200 hours of work.

The following year they asked us to go back to this work and add in stuff about the properties of the "posts" (ie the electricity pylons). Only half of us twigged that the thermal expansion coefficients needed to be linked to the pylon heights and thus interacted with every other aspect of the model. Fully incorporating, verifying and testing this very late-breaking requirement was about 100 hours of work - another lesson about real world projects and the consequences of late changes...

PDR

* If you want to know why costs always spiral upwards on large projects then this is a large part of it - the initial cost is usually based on a simplistic understanding of the problem, but as the work proceeds the greater complexity/opportunity becomes visible so the size of the job snowballs.

 
Celebaelin
1347807.  Sat May 09, 2020 5:12 pm Reply with quote

PDR wrote:
Yes, they made us do that at school and it still feels wrong!

It's just simple algebra for sure but the last time I posted anything like this I got some feedback to the effect that some found it unintelligible. With that in mind I posted the 'big print' version in the hope that some readers might be able to follow the path through it.

Call it home schooling if you like!

 

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