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Old 03-02-2004, 09:46:16   #1
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A moderately interesting maths thing

Was having dinner at my parents house on Sunday (mmm... roast beef) and the following subject came up:

If you had a string that went all the way round the earth how much longer would you have to make it to have a string that was a meter above the surface all around the earth. It's almost a trivial little proof and I think we'd heard the idea before but all forgotten it. Anyway I found the answer quite interesting.
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Old 03-02-2004, 09:49:14   #2
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Old 03-02-2004, 09:51:02   #3
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Nah, I'd go and work that out to find out what was interesting, which is what I did.
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Old 03-02-2004, 09:52:11   #4
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2pi m
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Old 03-02-2004, 09:54:30   #5
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string1=2pi r m where r is the radius of the earth in meters.
string2=2pi (r+1) m
string2-string1=2pi (r+1) - 2pi r = 2pi (r+1-r) = 2pi(1) = 2pi m.
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Old 03-02-2004, 09:56:20   #6
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Yeah! The interesting thing is that the radius cancels so it doesn't matter how big the initial circle is, to increase the radius 2pi times the distance you want to increase it by, which is the same as the circumference of a circle with that radius.

Well I think it's interesting.

And you only need 6.3m of string to make the string round the entire earth ( 40 075 KM long) a meter higher. Amazing!


This is probably something that I used to know and forgot.
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Old 03-02-2004, 09:59:19   #7
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Not really. 6.3m would be a trivial change on a string that long, and 1m would be equally trivial next to the earth's radius. It would actually be quite difficult to make a string that was exactly the earth's circumference. It would always be a few meters too long or too short, and that would equate to a few meters in altitude.
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Old 03-02-2004, 10:02:40   #8
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What are you? Some kind of engineer?
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Old 03-02-2004, 10:05:05   #9
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Old 03-02-2004, 10:15:32   #10
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surely you make it deliberately too long, wrap it round the earth thencut off the excess?
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Old 03-02-2004, 10:23:52   #11
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Equatorial or polar?
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Old 03-02-2004, 10:24:46   #12
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That's fine if you want it tight around the earth. What if you want it a meter above it though? How do you figure on getting the whole string the same height above the earth at the same time?
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Old 03-02-2004, 10:33:20   #13
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Old 03-02-2004, 10:35:15   #14
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Multiply it by 3.141592653589793238462643383279502884197169399375 10582097494459230781640628620899862803482534211706 79821480865132823066470938446095505822317253594081 28481117450284102701938521105559644622948954930381 96442881097566593344612847564823378678316527120190 91456485669234603486104543266482133936072602491412 73724587006606315588174881520920962829254091715364 36789259036001133053054882046652138414695194151160 94330572703657595919530921861173819326117931051185 48074462379962749567351885752724891227938183011949 12983367336244065664308602139494639522473719070217 98609437027705392171762931767523846748184676694051 32000568127145263560827785771342757789609173637178 72146844090122495343014654958537105079227968925892 35420199561121290219608640344181598136297747713099 60518707211349999998372978049951059731732816096318 59502445945534690830264252230825334468503526193118 81710100031378387528865875332083814206171776691473 03598253490428755468731159562863882353787593751957 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34551050080190869960330276347870810817545011930714 12233908663938339529425786905076431006383519834389 34159613185434754649556978103829309716465143840700 70736041123735998434522516105070270562352660127648 48308407611830130527932054274628654036036745328651 05706587488225698157936789766974220575059683440869 73502014102067235850200724522563265134105592401902 74216248439140359989535394590944070469120914093870 01264560016237428802109276457931065792295524988727 58461012648369998922569596881592056
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Old 03-02-2004, 10:35:24   #15
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Wrong height.
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Old 03-02-2004, 10:36:02   #16
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Old 03-02-2004, 10:37:49   #17
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Old 03-02-2004, 10:37:52   #18
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Why multiply it by pi?

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Old 03-02-2004, 10:38:08   #19
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You tie it tight around the earth (make it too long then cut off the excess). Then you add 2Pi meters onto it (fairly easy to do very acurately, and hey presto you have a rope that would stand 1m from the earth all the way round. The difficult part would then be to raise it that one meter.
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Old 03-02-2004, 10:38:47   #20
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We've already solved that problem. Pay attention.

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Old 03-02-2004, 10:39:41   #21
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That's assuming you could keep it around the equator and it wouldn't creep up toward the poles as you tightened it.
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Old 03-02-2004, 10:41:32   #22
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...or cut it in half, if it was monofilament.
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Old 03-02-2004, 10:41:33   #23
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Quote:
Originally posted by TV4Fun
That's fine if you want it tight around the earth. What if you want it a meter above it though? How do you figure on getting the whole string the same height above the earth at the same time?
repulsive subspatial fields
or you give it a negative graviton charge

BTW, do we have an earth max circumference that doesn't cross any land?
Otherwise, you'll have to dig a tunnel thru any >1m elevation, to let the string pass thru it at the proper height.
THIS would be a much more interesting challenge (although practical rather than theoretical):
how long is the shortest string that you can wrap around an earth max circumference, GIVEN the actual elevations?
(you're allowed to make it float over the sea - but don't forget tidal motion....)
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Old 03-02-2004, 10:42:24   #24
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Quit bringing reality into the equation.

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Old 03-02-2004, 10:42:37   #25
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Nobody said anything about sea level - just "the surface".
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Old 03-02-2004, 10:43:34   #26
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Old 03-02-2004, 10:44:13   #27
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You're assuming that the radius of the earth we use would be the radius at sea level. It would make much more sense to use the average altitude of the whole earth, which would make the tunneling issue a moot point.
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Old 03-02-2004, 10:44:38   #28
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well, sea surface IS "planet EARTH" surf-ace, ain't it?

Or are you one of those who equate "earth" with land, ground, soil?????
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Old 03-02-2004, 10:45:05   #29
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Old 03-02-2004, 10:46:17   #30
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the sea level is only to do with ho mucvh water us on the earth in liquid form at this particular time, it hasvg littles to to with tha eactiul eatht
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Old 03-02-2004, 10:46:38   #31
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Quote:
Originally posted by TV4Fun
It would make much more sense to use the average altitude of the whole earth, which would make the tunneling issue a moot point.
By definition, if you take an AVERAGE altitude, there will be some points actually lower, AND some points actually HIGHER.
Thus the tunneling issue is NOT moot.
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Old 03-02-2004, 10:46:56   #32
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Add another 5 km to the altitude, and you only need to accelerate it to 28,450.3 km/h!

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Old 03-02-2004, 10:47:58   #33
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Old 03-02-2004, 10:48:15   #34
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Quote:
Originally posted by TV4Fun
it hasvg littles to to with tha eactiul eatht

eactiul eatht to you!!!!
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Old 03-02-2004, 10:48:31   #35
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there will be some points lower yesr, but there will be a lot more points higher, so the average will still bef well above sea lev..
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Old 03-02-2004, 10:49:01   #36
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the fact that it's called SURFace must mean something, tho....
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Old 03-02-2004, 10:49:12   #37
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Old 03-02-2004, 10:50:27   #38
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SURF referd to ted SRUFACE of the water. tehadf SURFasdce of the ratrhg includes the SIRFAC fo the land ant the SURFACe of the watun.
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Old 03-02-2004, 10:50:33   #39
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Quote:
Originally posted by TV4Fun
there will be some points lower yesr, but there will be a lot more points higher, so the average will still bef well above sea lev..
indeed, thank you for helping me proving my point against yourself!

You'd still have to tunnel thru the points where the elevation is ABOVE average...
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Old 03-02-2004, 10:50:35   #40
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Quote:
Originally posted by TV4Fun
the sea level is only to do with ho mucvh water us on the earth in liquid form at this particular time, it hasvg littles to to with tha eactiul eatht
Are you channeling jpm Mukkef now?
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Old 03-02-2004, 10:52:45   #41
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Quote:
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indeed, thank you for helping me proving my point against yourself!

You'd still have to tunnel thru the points where the elevation is ABOVE average...
But it's a lot easier to find a path where you don't have to tunnel than it is to do that at sea level. The only altitudes where you won't have that worry are those more than about 10 km above sea level.

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Old 03-02-2004, 10:54:40   #42
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No if you go down at the points below the afvreerage thenf thteys'll be wwsxtara slack tfoet rthe potines abouive avereage.
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Old 03-02-2004, 10:55:33   #43
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SP: prove your statement!
I double dare you!!!

Remember that you can't divert your path from an exact earth max circumference.
Which brings us back when I first put reality into the equation, as you said.

TV: if you also GO DOWN, you'll make it even longer...
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Old 03-02-2004, 10:56:51   #44
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Why should I prove my statement, when you're the one who insists on using reality as a model?

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Old 03-02-2004, 10:57:33   #45
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and, do you want me to start typing withOUT looking at the keys?

Likje tjids???
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Old 03-02-2004, 10:58:13   #46
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Old 03-02-2004, 10:58:41   #47
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SP, your statement makes no sense under whichever theoretical model
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Old 03-02-2004, 10:59:16   #48
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Quote:
Originally posted by Funkodrom
The earth is a perfect sphere.
just as my balls
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Old 03-02-2004, 11:00:10   #49
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no, you're assuming that all pionts below avegaeea ewoll aftake the form aof lietalsa e dips that will take up nmoere slackef than tsahey efigve. if for ecample you were to go over a large body of ocean than the initial dip woul d be trivial compared to the saceingfas fromg ahfcefing a large length of string wiell below the avetage height.
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Old 03-02-2004, 11:01:42   #50
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I can't object to something I don't understand....
you win!
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