# Space elevator problem: how long is the cable?

• raving_lunatic
In summary, the problem involves finding the value of η, the distance from the center of the Earth at which the tension in a cable, used as part of a space elevator, reaches its maximum value. Using a differential force balance on a section of the rope, and considering the gravitational force and the centrifugal force, a differential equation is obtained for the tension at different points along the rope. Integrating this equation from the surface of the Earth to the point where the tension is zero gives the value of η.
raving_lunatic

## Homework Statement

"Consider a cable, of fixed mass per unit length, which extends radially from just above the surface to a distance ηR measured radially from the center of the Earth. By determining the total force on the cable, or otherwise, find an expression and numerical value for η. At what point in the cable does the tension reach the maximum value?"

The cable is to be used as part of a space elevator, so it must rotate with the Earth.
This is pretty much all the information that the question gives.

## Homework Equations

g = GM/r2

Radius of geostationary orbit obeys r3 = GM/ω2

## The Attempt at a Solution

I think I'm approaching this problem in a way that's far too simplistic because of how many marks it's worth on the exam paper it was part of. I thought that to satisfy the condition of rotating with the earth, the CM of the cable must be at a geostationary orbit height, and given that the cable's CM will be halfway along its length, we can simply double that geostationary orbit height and add on Earth's radius to get the total difference from the Earth's center. Formally I guess I just equated the centrifugal force to the gravitational force, solved for when they were equal, and said that this had to be the center of mass for the cable.

Then, to evaluate the tension, it seemed like we can find the maximum just by considering where the magnitude of the resultant of gravity and the centrifugal force is maximised, i.e differentiating rw^2 - GM/r^2 with respect to r -- and this again just gave the result that tension is maximised at the cable's CM, which again I'm not sure is correct. Any help would be greatly appreciated as I feel like I've missed the point of the problem

Try doing a differential force balance on the section of rope between r and r + dr. Let T(r) be the tension at r, and let T(r+dr) be the tension are r + dr. If the linear density of the rope is ρ, how much mass is there over the length dr? Besides the difference in tension, the other force on the differential mass is gravitational. This analysis will lead you to a differential equation for dT/dr. (Don't forget to include the ma term for the differential section of rope). Integrate it from r = R to the location where the tension is again zero. This will give you the value of η. I guess this analysis is supposed to be for a rope that's over the equator.

Chet

## 1. How long would the cable of a space elevator need to be?

The length of the cable for a space elevator would need to reach approximately 36,000 kilometers, which is the distance from the Earth's surface to geostationary orbit.

## 2. What material would the cable be made of?

The most commonly proposed material for a space elevator cable is carbon nanotubes, which have a high strength-to-weight ratio and can withstand the stresses of supporting the weight of the elevator and its cargo.

## 3. How would the cable be anchored to the Earth and the counterweight in space?

The cable would need to be anchored to a base station on the Earth's surface and a counterweight in space. One proposed method is to use a floating platform at sea for the Earth base and a captured asteroid for the counterweight.

## 4. How would the cable be able to support the weight of the elevator and its cargo?

The cable would be under immense tension, with the weight of the elevator and its cargo pulling downward and the centrifugal force from the Earth's rotation pulling upward. The cable would need to be strong enough to withstand these forces without breaking.

## 5. What are the potential benefits of a space elevator?

A space elevator could greatly reduce the cost of launching objects into space, as it would use electricity instead of costly rocket fuel. It could also make space travel more accessible and open up opportunities for space-based industries, such as solar power generation and asteroid mining.

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