Physics Question #9011

kyler, a 15 year old from the Internet asks on December 18, 2013,

If I were to tie a rope or something like that to the moon and have it dangle far enough toward the earth so I can reach it on the ground, would I be able to hold on to the rope and travel around the earth very fast because the moon is rotating around the earth?

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The answer

Jess Brewer answered on December 20, 2013

First let's pretend that the Moon and the Earth are both at rest with respect to each other.  Then the rope would feel a net attraction towards the Moon until it was about 1/4 of the way to the Earth; then it would feel no net gravity; then as it got closer to the Earth it would be attracted more and more in that direction.  If it were stretched all the way from the Moon to the Earth, the net force on it would be towards the Earth (because the Earth is heavier) so you'd better make sure it's tied tightly to the Moon.
 
Now, no ordinary rope would be able to support its own weight; in fact I doubt that even carbon nanotubes (the strongest stuff we know, if I remember correctly) would be strong enough for THAT application.  But even if it could, there's another problem.
 
The Moon and the Earth are NOT at rest, either by themselves or with respect to each other.  The Moon is in orbit around the Earth; it goes around once every 28 days or so.  It's moving pretty fast 'way out there on the end of the rope, just exactly fast enough that it "falls" toward the Earth at just the right acceleration to keep going in a circle so it never gets much closer or further away.  The Moon end of the rope would also be in "free fall" with respect to the Earth, so the Earth's share of the tension on the rope would be zero there.
 
Note that it would be pretty hard to get the rope stretched between the Earth and the Moon in the first place, because of that motion.  Perhaps if you stretched the rope out straight by itself in a place where the Earth and Moon would "run into it" and then grabbed the ends... but you'd put an enormous strain on the rope to get it moving at the speed of the Moon around the Earth, and it would probably just break.  But let's imagine that you've solved that problem.  There is an even worse one.
 
The Earth is spinning on its axis, once a day.  Where would you attach the rope?  Maybe at the North pole?  Be sure to have a good bearing!  In the end, the rope would not be moving with respect to the Earth, except to rotate about the Earth's axis.  (Actually the rope would rotate 1/28 as fast as the Earth, which would be rotating under it.)  So you wouldn't move with respect to the Earth's surface until you climbed up the rope towards the Moon.  At that point you'd feel a small pull sideways (called a Coriolis force) depending on how fast you climbed, and your apparent weight would decrease gradually as you got higher up, until you got up to the altitude of "geosynchronous orbit" ('way above the atmosphere, so be sure to wear a space suit), at which point you'd feel weightless, because you'd be in orbit!
 
That part would be pretty cool.  Forget the Moon; you can just attach a big rock to the rope a little above geosynchronous orbit and use this as a "space elevator" to get into orbit without rockets!  You should read Arthur C. Clarke's "Fountains of Paradise" to learn more about this possibility.

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