Annoying truths

[bluedog225]

I read the math explanation. Makes sense. Don’t like it.

The Rope Around The Earth Problem
Take a rope tied tautly around a basketball. Now the rope must be lengthened so that there is a one foot gap between the ball and the rope at all points, as if the rope is hovering a foot away around the entirety of the ball. How much must the rope be lengthened to accomplish this? 6.28 Feet.

Now take a rope around tied tautly around the equator of the earth. We have the same goal for the one foot hovering gap around the entirety of the earth. How far must the rope be lengthened? 6.28 Feet.

[Flying Orca]

2ϖr don't lie.

[ShorelineJohn]

I get that, what I can't ever figure out is how to get the pi symbol out of my keyboard.

[L.W. Baxter]

that can't be right

[Too Little Time]

I think it is really hard to put a rope across an ocean. And dangerous to put one across a road.

But that is outside the scope of the problem or solution.

[Rich Jones]

We will need Nick's or CW's input for the absolute proof, but here is my attempt:


Circumference = 2 * pi * R

In this instance R is in units of feet

Adding 1 foot to the radius gives: 2 * pi * (R+1)

Multiplying out gives: 2 * pi * R + (2 * pi)

Subtracting 2 * pi * R gives the differences in the circumferences of the two spheres:

[2 * pi * R + (2 * pi)] - [2 * pi * R] = 2 pi

Thus, for any given R, adding 1 foot to the R adds 2 * pi to the circumference, or, 6.283 feet.

My head hurts just looking at this post.

[ShorelineJohn]

Basketball
(2 * pi * (radius of the basketball + 1)) - (2 * pi * (radius of the basketball)

The world
(2 * pi * (radius of the world + 1)) - (2 * pi * (radius of the World)

the difference between the 2 equations are subtracted out.

But I give up on ALT + 227

So now I am not sure this is right, I look at the equations and to me they look correct but I keep trying it in SketchUp, letting it do the math and the numbers are close but not quite right. Might be rounding errors but....
I am going to have to keep working on this.

[CWSmith]

We will need Nick's or CW's input for the absolute proof, but here is my attempt:


Circumference = 2 * pi * R

In this instance R is in units of feet

Adding 1 foot to the radius gives: 2 * pi * (R+1)

Multiplying out gives: 2 * pi * R + (2 * pi)

Subtracting 2 * pi * R gives the differences in the circumferences of the two spheres:

[2 * pi * R + (2 * pi)] - [2 * pi * R] = 2 pi

Thus, for any given R, adding 1 foot to the R adds 2 * pi to the circumference, or, 6.283 feet.

that can't be right

It doesn't seem right, but I can't fault the math.

[Keith Wilson]

Cute. Simple math, and an amusing result.

[sandtown]

Annoying truths ??

Nuclear weapons do not keep us safe

Covid is real

Expensive water treatment plants prevent cholera

Oligarchic rule is worse than the "mob" of democracy

I could go on . .

[JayInOz]

I think it is really hard to put a rope across an ocean. And dangerous to put one across a road.

But that is outside the scope of the problem or solution.

I once heard someone on TV say that if you got every person on earth and laid them end to end- two thirds of them would drown

[sandtown]

...to...annoying half-truths.

Not so wild about those ??

Here is another one for ya . . .

countries with strong labor unions have a higher quality of life.

[Nicholas Carey]

We will need Nick's or CW's input for the absolute proof, but here is my attempt:


Circumference = 2 * pi * R

In this instance R is in units of feet

Adding 1 foot to the radius gives: 2 * pi * (R+1)

Multiplying out gives: 2 * pi * R + (2 * pi)

Subtracting 2 * pi * R gives the differences in the circumferences of the two spheres:

[2 * pi * R + (2 * pi)] - [2 * pi * R] = 2 pi

Thus, for any given R, adding 1 foot to the R adds 2 * pi to the circumference, or, 6.283 feet.

Circumference of a circular is

C = 2𝛑R

where R is some unit of length (doesn't matter what unit of length).

The proof is in the formula for circumference of a circle: increase R by 1 unit and the circumference increases by 2𝛑 units.

But we can prove it using simple algebra: The assertion is that

2𝛑(R+1) - 2𝛑R = 2𝛑

For the purposes of this exercise, we can treat 2𝛑 as a unitary symbol. So:

Distributing 2𝛑 across (R+1) gives us, because of the identity a(b+c) = ab + bc,

2𝛑R + 2𝛑 - 2𝛑R = 2𝛑

Reordering the terms (order doesn't matter for addition and subtraction, the identities a-b = a + -b, and a+b = b+a) gives us

2𝛑R - 2𝛑R + 2𝛑 = 2𝛑

Reduction (the identity x - x = 0) gives us

0 + 2𝛑 = 2𝛑

Or

2𝛑 = 2𝛑

Q.E.D.

[P.I. Stazzer-Newt]

How far, north or south, would you need to move the equatorial rope to generate enough slack for the one foot of clearance?

[ShorelineJohn]

But how do you get the pi symbol? Alt + O does not work on this machine.

[P.I. Stazzer-Newt]

Easy as π on an android phone.

Press the ?123 button then the =\< button and there it is.

[epoxyboy]

Make this simple. Swap in 3 in place of pi. It's close enough, the math still holds true, and its easier to see what is going on.

For r = 10 and r = 11
2x3x10 = 60
2x3x11 = 66


For r = 1,000,000 and r = 1,000,001
2x3x1000000 = 6000000
2x3x1000001 = 6000006

[oldcodger]

oops!

[oldcodger]

How far, north or south, would you need to move the equatorial rope to generate enough slack for the one foot of clearance?

For simplicity using the metric system.


How far, north or south, ( H ) would you need to move the equatorial rope to generate enough slack for the one metre of clearance?


Possibly a slight approximation but use Pythagorus theorem.


Diameter of earth 12756 km radius 6378 km


R x R = (R – .001)x (R – .001) + H x H


6378 x 6378 =6377.999 x 6377.999 + H x H


40678884 =40678871.24 + H x H


12.76 = H x H


H = 3.572 km



Being a nerd, some years ago I was bored watching a middle distance race and wondered how much further the athlete in the first lane out had to run than the one in the innermost lane.

Say for a 5000 m race. 12.5 laps x width of lane 4 feet x 6.28 = 314 feet.

I know there are tactical reasons for running wider but that is a huge penalty.

I am surprised there isn't more of a fight to hug the kerb.

[bluedog225]

But the earth isn’t round. Pie are round.

[Paul Pless]

I am surprised there isn't more of a fight to hug the kerb.

[Gerarddm]

I am having a hard time grokking this.

[L.W. Baxter]

ohhh, i get it now. you guys are assuming the earth is a sphere.

the rope has to go all the way around the turtles.

[WI-Tom]

Being a nerd, some years ago I was bored watching a middle distance race and wondered how much further the athlete in the first lane out had to run than the one in the innermost lane.

Say for a 5000 m race. 12.5 laps x width of lane 4 feet x 6.28 = 314 feet.

The starting line is staggered to account for that. Each runner runs the same distance, no? Or am I confused?

Tom

[Kevin T]

I think it is really hard to put a rope across an ocean. And dangerous to put one across a road.

But that is outside the scope of the problem or solution.

I'm with TLT on this one, but if any of you math wizards want to test out your theories in real world applications can I at least get the contract to supply the rope?

Will you be looking for sisal, manila hemp, hemp. cotton, coir, perhaps low stretch polypropylene. Just let me know the actual length of cordage you think you'll need and I can get you a price*.

*Special discounts available to WBF members.

[Nicholas Carey]

The starting line is staggered to account for that. Each runner runs the same distance, no? Or am I confused?

Tom

Not confused.

That delta in distance is also why speed skaters switch lanes during a heat.

[CWSmith]

Not confused.

That delta in distance is also why speed skaters switch lanes during a heat.

Now you have me wondering if there is a speed difference between the inside and outside tracks. I don't mean a distance difference, but a better ability to get speed on a tighter or less tight turn?

[Breakaway]

[Dan McCosh]

FWIW, a friend set a goal of bicycling the distance around the world. Just turned 80, and is quite close to making her goal.

[Too Little Time]

Now you have me wondering if there is a speed difference between the inside and outside tracks. I don't mean a distance difference, but a better ability to get speed on a tighter or less tight turn?

The races are based on distance. So a speed difference is unimportant. It is certainly not enough to make up for the distance difference.

radius of the inner curve is not less than 25 metres and not more than 26 metres

That would suggest that there is a speed difference.

[CWSmith]

The races are based on distance. So a speed difference is unimportant. It is certainly not enough to make up for the distance difference.

I believe the point is that the tracks are run so that the distance is the same. Therefore, the speed difference is the only thing that is important.

[oldcodger]

The starting line is staggered to account for that. Each runner runs the same distance, no? Or am I confused?

Tom

Yes and no.

In the 200m and 400m you have to stay in your lane all the time and the starts are staggered to give an identical distance.

In the 800m you have to stay in your lane for the first bend and the stagger allows for this, after that it is free for all.

Above that there are no lanes, the starting line is curved to allow those on the outside a theoretical equal chance to get to a point on the kerb.

After that for every foot from the kerb, you run an extra 6.28 ft.

[WI-Tom]

Yes and no.

In the 200m and 400m you have to stay in your lane all the time and the starts are staggered to give an identical distance.

In the 800m you have to stay in your lane for the first bend and the stagger allows for this, after that it is free for all.

Above that there are no lanes, the starting line is curved to allow those on the outside a theoretical equal chance to get to a point on the kerb.

After that for every foot from the kerb, you run an extra 6.28 ft.

Interesting--I didn't know that. I coached cross country. "Lanes? We don't need no stinkin' lanes."

I think running more than 800m on a track would bore me to the point of just walking away. Even pavement is boring compared to good dirt trails through the woods and over the hills.

Tom

[John Smith]

Maybe I'm overly simplifying things, but it seems to me as the diameter of a circle increases, so does the circumference. I may also be missing something, but.

If one has a phonograph album and cuts a string long enough to encircle the innermost groove, that string will not encircle the outermost groove. And the further from the center you go, the greater the shortfall would be.

No?

[jdmcintyre]

Its turtles all the way to the bottom so the correct answer is 3.14 m + 2 x infinity