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#26 2016-09-26 09:31:11

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: The Earth and The String

Civil Engineering Onsite Handbook.

Btw, my notation in post #21 is a bit different from theirs, as I changed theirs to fit notes I'd been making before finding their site.

My   
   is their   

And my 
    is their   

One I didn't mention in post #21 is the formula for the arc length between the tangents, which I used to help find c (central angle) in M's FindRoot.

My arc length formula 
   is their   


My r & their R = earth's radius
My c & their I = central angle;
My h & their E = height of string apex above top of earth
My t & their T = length of tangent from earth to string apex

My current M code, and yours:

In[1]:= 
(*Mine*)
r=3959*5280;
c=c/.FindRoot[2r Tan[c \[Degree]/2]-100-r*c \[Degree]==0,{c,2.2},WorkingPrecision->100]
h=r*Sec[c \[Degree]/2]-r

(*Bobby's*)
R=3959*5280;t=(150/R)^(1/3);N[(2 R*Sin[t/2]^2)/Cos[t],100]

Out[2]= 2.210149733171091920100374337118113893066411007878966450185575166794692294275531251879912977959844808
Out[3]= 3888.614460593728239463379167602010692449187441461292338724354316092744378397068689244933991315452952
Out[4]= 3889.000315217547355956190858271992382571341435999597543245020468161721458410729392937937972940735646

In Geogebra, when I plug your result into point E's y value, I get height = 3889.000315219164, and central angle = 2.210259366456653 degrees. That angle figure differs from that of Out[2] in the M code above, and must be the reason for the height discrepancy.

Is FindRoot giving me the wrong central angle? If so, is that because of the formulas I gave it? dizzy

Last edited by phrontister (2016-09-26 13:18:01)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#27 2016-09-26 13:59:18

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: The Earth and The String

Hi;

I do not think FindRoot is making a mistake.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#28 2016-09-26 14:05:39

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: The Earth and The String

I wouldn't have thought so, either. It's probably me...but I haven't found out yet where my error is.

It seems odd, though, that FindRoot's value for c (the central angle, theta) results in nearly exactly the same answer (3888.614460593728239463379167602`...see Out[3] in my previous post) as thickhead's 3888.6144 in post #14, which he said in post #23 was incorrect due to truncation. But that may be coincidental.

I've been working in degrees, in which theta values are:
Your 2t = 2.210259366455766942706949`;
G's γ = 2.21025936645665`;
FindRoot's c = 2.2101497331710919201003743`

Your t value (either in radians or converted to degrees) plugged into my h (string height) formula, gives the correct height (3889.0003152175473559561908582`); so my h formula is correct. The error source here, as thickhead said earlier about his error, is in the theta calculation. 

I'll look again, but there's probably something going on somewhere that I don't have any understanding of and won't be evident to me.

Last edited by phrontister (2016-09-26 15:16:37)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#29 2016-09-26 16:04:09

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: The Earth and The String

Hi;

I think I see what your problem is... I do not know if your answer is wrong. You are misinterpreting what I was saying in that earlier post.

I was saying that

and that

I never meant to imply that this formula is the solution to that problem. It may be or it may not. Having not worked on this problem at all I have no idea what the right answer is.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#30 2016-09-26 21:20:36

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: The Earth and The String

Hi Bobby;

Sorry...my mistake. I'd read into what you'd been writing that you were saying that the second answer was the right answer to the problem.


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#31 2016-09-26 21:42:09

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: The Earth and The String

Nope, you know how much I hate geometry questions. To me, geometry is done with the Geebra or M.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#32 2016-09-26 22:21:41

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: The Earth and The String

When I saw you leap into this thread I thought you were out to surprise me!

Without Geebra I'd've been nowhere on this problem...and using it was a pleasure. smile

M too.

Last edited by phrontister (2016-09-26 22:22:57)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#33 2016-09-26 22:25:19

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: The Earth and The String

You know I am one of those crazy ole guys who does not want to do math the same way Euclid did it. For some nutty reason I want to use electricity to assist me in mathematics. Yea, I am a screwball. Why not write up how you did it in Geogebra in computer math?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#34 2016-09-26 22:51:07

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: The Earth and The String

Well, it's not actually a solution as such...but it's not far off and does enable a reasonable graphical appreciation of this huge-scale problem.

I don't know G well enough to do anything clever, like somehow using (coding?) it to actually give the solution. Would be nice to include that, if it can be done.

Until I saw that I could plug figures into point E (the string vertex), I just grabbed E and manipulated it vertically (while also adjusting display size) towards or away from the earth until the sum of the length of the two tangents was as close to 100 feet greater than the length of the earth's arc between the tangents as I could get it.   

I'd have to think about it...


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#35 2016-09-28 04:41:56

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: The Earth and The String

Hi phrontister,


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#36 2016-09-28 23:12:41

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: The Earth and The String

Hi thickhead;

Well, that's very interesting. That's pretty much my post #21's result, where I'd used M's FindRoot to get 0.01928719490295426 (ie, radians...my formula was in degrees), resulting in 3888.614460591227.

By increasing working precision in my M to 100 digits, I now get 0.019287194902954844824414790282500 and 3888.61446059372823946337916760 (just showing the first few decimal places for both).

W|A gives the same result as above for theta with 41807040Tan[th]-100-41807040th=0 (41807040 being 2*3959*5280), and, after plugging in 100-digit theta, also gives the same result as above for the string height with 20903520*Sec[th] - 20903520 (20903520 being 3959*5280).

Sorry, I don't understand your series approach...my maths never got that far.

In Geogebra I've not been able to work out how to code exactly 100 feet for the extra bit of string in order to get Geogebra's most accurate value for the string height above the earth. By grabbing the string vertex and manually dragging it to &/or from the earth, I've got as close to 100 as 99.9999999984866 (by zooming right in on the vertex and adjusting it), which gives 3888.6144606396556. And that's near enough to close enough for me! smile

Plugging 3888.61446059372823946337916760 into Geogebra gives 99.99999999988358. Your 3888.614461, which looks like being either truncated or rounded, gives a slightly less accurate result.

I think our results show that my Geogebra method is good, so that gives me more confidence in doing as Bobby has suggested, which is to write it up in the Computer Math forum.

So, thanks for reporting your latest finding. smile

Last edited by phrontister (2016-09-29 00:41:33)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#37 2016-09-28 23:46:11

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: The Earth and The String

I am just zero in Geogebra and mathematica. I just want to enter the field but inertia persists.Wolfram alpha widgets are easy to use taht too I have adopted recently but transferring data from one module to another module is manual and messy as far as i am concerned. Copy and paste does not work.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#38 2016-09-29 00:36:03

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: The Earth and The String

Sorry, can't help you with widgets (knowledge = zero). I've always used W|A online, here, and copy & paste works well there. I have no need for any other W|A vehicle.

I'd highly recommend downloading and installing Geogebra...even I can use it! It can do so much, is easy and intuitive to use, and caters for a very wide range of abilities and interests (etc, etc, etc). It's free, frequently updated, continually being improved, has wide community interest and support, an active forum, and a good online instruction manual and tutorials. 

You may have seen some of what it can do already, in post #9...but that's only poking at its abilities.


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#39 2016-09-29 03:08:20

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: The Earth and The String

Hi;

You may have seen some of what it can do already, in post #9...but that's only poking at its abilities.

The new capabilities in M for geometry are rivalling Geogebra in firepower.

As this fellow says:

https://www.youtube.com/watch?v=-1TTN-Ev2KI

Unlimited Power!

Sorry, I don't understand your series approach...my maths never got that far.

M got much further than that and is your friend. Supposing you want the series expansion around a point for some function like my integrand...

To get terms up to x^5  just enter

Series[1/(x^2 + Cos[x]), {x, 0, 5}]

M delivers,

The O(x^6) is its estimation of the truncation error.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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