# What if there was a pole, whose length was the circumference of the Earth?

A slight touch wouldn't do anything. The pole is still extremely massive. And in equilibrium. To move it would require a force to overcome that massive inertia.

Hmm, I suppose that makes sense. But isn't the equilibrium a very very unstable one? I imagine it like balancing a straw on an orange. Once you have it there, even the slightest touch makes it roll off. Of course, a straw isn't really a good example of a very massive object, but still :p

DaveC426913
Gold Member
Hmm, I suppose that makes sense. But isn't the equilibrium a very very unstable one?
The Queen Mary could be pushed by hand, but it would take a fair bit of pushing to get it moving.

DaveC426913
Gold Member
My guess would be that you would have a pole in which the ends surpass each other. It's like, you put down the pole, then it would poke you in the back.

Sorry, what I was getting at is, if the ends came together so that the pole was now a complete ring, what would happen? When slightly longer than the circumference of the Earth, it would actually not be able to lie on the ground - it would be too long.

It would be suspended a short distance above the ground. Assuming it was perfectly rigid, it would also be weightless. You could lift it up and it would stay there. Push it down and it would stay there. It is the inside-out version of Newton's Shell Theorem.

Mapes
Homework Helper
Gold Member
Sorry, what I was getting at is, if the ends came together so that the pole was now a complete ring, what would happen? When slightly longer than the circumference of the Earth, it would actually not be able to lie on the ground - it would be too long.

It's a nifty problem to calculate the compressive stress on this ring. One approach is to equate the gravitational potential energy of the whole ring being a little lower to the strain energy penalty for the ring to compress to make this happen. (Are there other solution approaches?)

I'll leave the details up to the interested reader; I get $\sigma = \sqrt{2r\rho g E}$, where $\sigma$ is the compressive stress, $r$ the distance from the Earth's center, $\rho$ the material density, $g$ the acceleration of gravity, and $E$ the Young's elastic modulus of the ring.

Surprisingly, the answer is independent of cross-sectional area. Not surprisingly, no material in existence could survive this configuration; a yield strength of at least hundreds of GPa would be required, if my calculations are correct.

Note that even if a material's yield strength were high enough, the ring would still surely fail by compressive buckling (i.e., elastic instability), which is a different failure mechanism than simple compressive loading.

diazona
Homework Helper
A slight touch wouldn't do anything. The pole is still extremely massive. And in equilibrium. To move it would require a force to overcome that massive inertia.
$F = ma$, I say - a slight touch (F) would produce a slight acceleration (a). So it would do something, even if the motion would be so small you couldn't measure it.
Hmm, I suppose that makes sense. But isn't the equilibrium a very very unstable one? I imagine it like balancing a straw on an orange. Once you have it there, even the slightest touch makes it roll off. Of course, a straw isn't really a good example of a very massive object, but still :p
Well, the straw-on-an-orange analogy sort of breaks down once the straw moves from the top of the orange: the straw is subject to a uniform force, but the pole would be subject to a radial force. (Point: the straw can fall off the orange, but the pole couldn't fall off the earth)

DaveC426913
Gold Member
$F = ma$, I say - a slight touch (F) would produce a slight acceleration (a). So it would do something, even if the motion would be so small you couldn't measure it.
Well OK. I guess the same thing could be said about the Queen Mary. A slight touch sets it moving imperceptibly?

An other way I thought about looking at this thought experiment is if the pole was infinitely long. Would the pole keep on warping around the Earth? Or will it warp until ends meet (well sorta) and the equal force from each side would cancel any other warping and the rest of the extent will keep on jutting out into space?

Just something I thought would be interesting to add to this already interesting thought experiment.

Edit: Assume that the infinite pole is behaves the same way as the already finite pole.

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DaveC426913
Gold Member
An other way I thought about looking at this thought experiment is if the pole was infinitely long. Would the pole keep on warping around the Earth? Or will it warp until ends meet (well sorta) and the equal force from each side would cancel any other warping and the rest of the extent will keep on jutting out into space?

Just something I thought would be interesting to add to this already interesting thought experiment.

Edit: Assume that the infinite pole is behaves the same way as the already finite pole.

Uh, you lost me. An infinitely-long pole would form a helix of infinite width. I don't see where you're going with this.

Uh, you lost me. An infinitely-long pole would form a helix of infinite width. I don't see where you're going with this.

Well I sorta worded it wrong, I meant a helix when I suggested my first option I just didn't word it properly. Sorry about that.