# Gravity around the Earth (or other large spherical object)

Shimo
In a nutshell: Two people, standing a few meters apart facing each other, are holding a rope taut. Assuming the rope is infinitely long, the people are 'infinitely strong', so they may hold any length of rope, and that they could walk all the way around the earth...

If the two people walked backwards, all the while holding the rope taut, until they were standing back to back on the other side of the Earth, would the rope touch the Earth at any point, or would it be taut and above the surface of the Earth at every point?

A friend and I have been debating this for about a week now, and neither of us, nor any of our teachers, can come up with a foolproof way of giving an answer.

I've been looking for a proper answer for a while now, and Google brought me here. Is there an actual answer, or will it remain a hypothetical?

If the problem isn't clear, I'll be more than happy to try to clarify any misunderstandings. Thanks for your time!

Mentor
The rope touches the earth. Draw a diagram and figure out how far apart they will be when the rope touches.

dst
Simple enough - it depends on the length of the rope. An infinitely long rope will NEVER have tension in this manner. The question is meaningless in that sense.

Regardless, if you had a rope that long and in that scenario, it would touch the ground in the same manner as a rope on a pulley.

Shimo
berkeman: The distance between the people when the rope touches the Earth should be proportional to the height the people are holding the rope at.

dst: I used an infinitely long rope only because I don't know the circumference of Earth. Assume the rope is just slightly longer than the circumference of the Earth, so that the two people may stand back to back with the rope taut in their hands. I don't understand the comparison between this and a rope and pulley.

dst
berkeman: The distance between the people when the rope touches the Earth should be proportional to the height the people are holding the rope at.

dst: I used an infinitely long rope only because I don't know the circumference of Earth. Assume the rope is just slightly longer than the circumference of the Earth, so that the two people may stand back to back with the rope taut in their hands. I don't understand the comparison between this and a rope and pulley.

The Earth is round, is it not? Hence, no matter what you try, it will have to contact the ground - pulling only helps this, it won't make it go against gravity on the opposite side of the Earth.

Here's a simple way to verify this - find a cylinder such as a bottle. Using your two hands, stretch a piece of string taut over it - it doesn't contact, does it? Now loop it all the way round the bottle and try again. Same principle.

Mentor
berkeman: The distance between the people when the rope touches the Earth should be proportional to the height the people are holding the rope at.
"proportional" isn't the right word, but you have the right idea. So pick a height and run a calculation for us...

Shimo
To be honest, I don't actually know how to do a calculation like this. I have to run now, but I'll work on it and edit this post later if I've made any progress.

Gold Member
If the rope is unbreakable, and they're pulling with infiite strength, wouldn't the rope just cut through the Earth and remain straight?

atom888
I know what you saying. In an enlarge point of view, the rope will touch the Earth's surface. In the rope puller point of view, the rope is parallel to the Earth's surface, but as he get further and further, he will see the rope not parallel, not sagging, but a straight line slanting down. Just like u tie the rope to the ground and try to pull to make it parallel.

Gold Member
I know what you saying. In an enlarge point of view, the rope will touch the Earth's surface. In the rope puller point of view, the rope is parallel to the Earth's surface, but as he get further and further, he will see the rope not parallel, not sagging, but a straight line slanting down. Just like u tie the rope to the ground and try to pull to make it parallel.
Well, what he'll see is :
1] a rope that vanishes to the horizon. (At that distance, it is for all intents and purposes, visually parallel to the ground.)

2] his partner-in-crime, very tiny and mostly below the horizon, only his outstretched arms and head visible.

atom888
The final answer is the rope touch the ground. The only problem left is which country will buy the big rope now and hire the guys for power generation. lol

Shimo
I've been working on the calculation, but I can't seem to make any progress. According to Google, the radius of the Earth is 6,378.1 km, but needless to say that's an estimate. If we approximate pi to 3.141592, the arc length of 1 degree is 111.318822 km. This leads to the question, not of how far we can see into the horizon, but of how many meters, (or km?) a 'rope' above Earth will 'drop' per unit arc length (presumably 1 degree, for simplicity.) If anyone has any other suggestions or calculations that can help, they would be much appreciated. This is driving me nuts right now!

Thanks again.

Gold Member
All you need to know is the distance to the horizon for a 5 foot tall person (i.e. a 6 foot tall person at their shoulder height).

According to this http://boatsafe.com/tools/horizon.htm the horison is about 3 miles away, mking the rope 6 miles in total.

S1nG
Ok i understand what you mean. Does this involve the flatness problem of the universe? Where a body (Eg, a huge planet ) tends to curve as the observer nears it. We have to take into consideration that the rope doesn't bend and that the 2 person holding the rope makes sure that its taut.

Lets say, this 2 guys stood just opposite of Mount Everest, and that they start to taut the rope opposite of the globe. I think, when the 2 guys reaches halfway the circumference of the Earth, the rope would touch the ground.

atom888
I've been working on the calculation, but I can't seem to make any progress. According to Google, the radius of the Earth is 6,378.1 km, but needless to say that's an estimate. If we approximate pi to 3.141592, the arc length of 1 degree is 111.318822 km. This leads to the question, not of how far we can see into the horizon, but of how many meters, (or km?) a 'rope' above Earth will 'drop' per unit arc length (presumably 1 degree, for simplicity.) If anyone has any other suggestions or calculations that can help, they would be much appreciated. This is driving me nuts right now!

Thanks again.

I don't see why u have to go drop/unit arc length. If u just want to know how far u go to flush your head (meaning what is the arc length to have a certain drop) with horizontal surface, then subtract the radius from the height, use it as a leg and the radius as the hypotenus to find the angle by cos(theta)=(radius-ft drop)/radius . From theta u can find the arc length using r x theta.

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Gold Member
Ok i understand what you mean. Does this involve the flatness problem of the universe? Where a body (Eg, a huge planet ) tends to curve as the observer nears it. We have to take into consideration that the rope doesn't bend and that the 2 person holding the rope makes sure that its taut.

Lets say, this 2 guys stood just opposite of Mount Everest, and that they start to taut the rope opposite of the globe. I think, when the 2 guys reaches halfway the circumference of the Earth, the rope would touch the ground.
What?? No!