Rope around the earth problem

[WaxLyrical]

A variation of this problem was discussed during an episode of S1 of House of Lies. Found it quite interesting.

Suppose you tie a rope around the earth at the equator (circumference approx. 25,000 miles).
Let's say you pull the rope as tight as it will go and then add back 6 feet of slack before tying the knot.
If the extra rope is distributed evenly around the globe will there be enough
space between the rope and the surface of the earth for a worm to crawl under?

Assume the earth is a perfect sphere and the rope does not stretch.

 

[LazyLion]

No, a rope that strong to go around the earth would be fairly heavy, so a worm would not fit under it.

 

[vij]

Surely so, since you are not assuming the earth's surface is perfectly round. Also is the rope of uniform circular cross section?

 

[satanboy]

No, a rope that strong to go around the earth would be fairly heavy, so a worm would not fit under it.

earthworm


doh

 

[ScottulusMaximus]

The rope would not suddenly levitate... It would lay flat on the surface(would just be a bit more "wiggily") as the gravity pulls it inwards towards the centre. If the rope was solid, perfectly balanced, and the entire earth was uniform the gravitational forces would cancel each other out and the ring would then "float"

 

[LazyLion]

earthworm
doh

OP said "worm" not "earthworm".
doh

 

[Leonidas]

earthworm


doh

But in some places the soil will be of a softer consistency meaning the rope will dig into the soil, and thus an earthworm would not be so able to dig in underneath the rope.

 

[LazyLion]

The rope would not suddenly levitate... It would lay flat on the surface(would just be a bit more "wiggily") as the gravity pulls it inwards towards the centre. If the rope was solid, perfectly balanced, and the entire earth was uniform the gravitational forces would cancel each other out and the ring would then "float"

 

[satanboy]

OP said "worm" not "earthworm".
doh

so an earthworm is not a worm

interesting

 

[ScottulusMaximus]

View attachment 307952

Need a diagram?

 

[Hamish McPanji]

When the earth was created 6000 years ago, it was done in 2 halves as it is easier to make like that.Those 2 halves were joined together at the equator. By pulling on the rope like that, you put strain an already weak point and the earth will split in half.

To answer the question, the worm would be dead because of no oxygen and wouldn't be able to crawl

 

[azbob]

If the worm is less than 30cm round, yes.

 

[rpm]

This is a very interesting Mathematical "problem".

The height the rope lifts is independent of the radius of the object it goes around. It will lift the same height, whether it goes around the earth, a table tennis ball, or the universe. It feels counter-intuitive, but it is fairly simple to prove.

 

[Lager]

No, impossible

 

[Hamish McPanji]

.

 

[Lager]

there's a no-impact negative variance factor of 0,0000000454544 on the tightness of the rope
Only the earth worm will manage

 

[Hamish McPanji]

/snip

Thanks you bastid, now I won't be able to sleep

 

[Hamish McPanji]

This is a very interesting Mathematical "problem".

The height the rope lifts is independent of the radius of the object it goes around. It will lift the same height, whether it goes around the earth, a table tennis ball, or the universe. It feels counter-intuitive, but it is fairly simple to prove.

As much as the maths works, I just can't get my head around this. Am going to test it with a football, a tennis ball using some strapping band

 

[wombling]

An equal distribution of 6 feet across 25 000 miles is not going to be enough slack for a worm to fit under unless its a really really small worm; or an earthworm

 

[battletoad]

As much as the maths works, I just can't get my head around this. Am going to test it with a football, a tennis ball using some strapping band

let R be the radius of the sphere in question in metres. Suppose you tie a rope tautly around the (equator of the) sphere. Its length is 2 x pi x R metres.

Now, if you want the rope to be 1 metre off the surface of the sphere all round, the rope needs to be 2 x pi x (R+1) metres.

So the difference in the lengths of rope is 2 x pi x (R+1) - 2 x pi x R = 2 x pi metres. Note that the initial radius R magically disappeared, meaning that irrespective of the initial radius of the sphere (marble, tennis ball, jupiter, sun, etc.), it only needs an additional 2pi metres of slack to lift the rope 1 metre off the surface all around. Weird, yes