that teaches you to think like an engineer.

Space elevators are one of those technologies that sci-fi nerds, like me, obsess over. They

straddle the line between outlandish impossibility and genuine engineering potential. It's

a technology which could cross the divide of science fiction to science reality, if

we somehow improved on existing technologies.

It's the kind thought experiment and lofty engineering challenge that could drive the

development of future technologies. Necessity is the mother of invention after all.

Before jumping into the technologies that need to emerge to facilitate space elevators,

let's first explore what a space elevator actually is.

A space elevator is exactly that, a giant elevator shaft that we can climb to reach

space. Eliminating our dependence on rocket fuel to reach orbit and hopefully, in the

process, lower the cost of space travel.

This isn't your typical construction that relies on the compressive strength of a material

to remain standing.

Our buildings are largely restricted in height as a result of the compressive strength of

our building materials. The higher we build the more weight is piled onto the foundations

of the building. We can counteract this by widening the base of the construction, to

spread the weight over a larger area and then taper the building as it rises to reduce the

weight being added as we add more floors. The most obvious examples of this are the

pyramids, but even the burj khalifa uses the same principle, being widest at its base and

gradually narrowing to it's seemingly impossible height.

We can build higher with current materials if we widen the base, but that becomes uneconomic

pretty quickly as the base would take up an unreasonable amount of space.

So how would a space elevator solve this problem? By counter-balancing the weight of the structure

by pulling upwards. We can do this thanks to centrifugal force.

Imagine a tether ball swinging around a poll. At a certain angular velocity the string will

be held straight and taut against the poll, because centrifugal force, an apparent force

that appears in a rotating reference frame, pulls outwards.

Now, the problem is, the whole point of tetherball is to wrap the string around the poll, if

the string can't rotate around the centre of spin it will simply wrap itself around

the poll.

We are essentially trying to recreate this dynamic, but on an astronomical scale and

to do that we have to work with earth's natural rotation.

So, our structure will need to be located on the equator. Let's imagine a base located

in the middle of the Atlantic ocean.

From here we are going to draw a straight line out into space. For now, this is just

a line, no structure exists. But, any structure that is constructed will need to exist along

this line. If it is not insync with earth's rotation the tether will curve and break,

or in some sort of cartoon world wrap around the earth like our tetherball example.

Our orbit will also need to be circular, rather than elliptical, as an elliptical orbit would

require a tether capable of constantly changing length without breaking.

We can find an orbit that will achieve this with some simple algebra. To remain in a steady

circular orbit we need our centrifugal force to equal the gravitational force. [1]

Centrifugal force is defined by this equation. Where ms is the mass of the satellite, w (omega)

is the angular velocity and r is the distance to the centre of the earth.

While the force due to gravity is defined by this equation. Where G is the gravitational

constant and mp is the mass of the planet. The mass of the satellite cancels out while

we manipulate the equations to get a value for r, our orbital radius. [Reference Image

1]

Now we have an equation with all known values, which we can solve for by inputting the values

for earth, and we find a value of 42, 168 kilometres. This is the distance from the

centre of the planet, so this will be about 36 thousand kilometres above the surface of

the planet at the equator.

Okay, so this gives us a starting point for our construction. We are going to put some

form of massive satellite into this orbit and begin the construction process. Building

up from the planet is not an option, we need to build down.

Now this is where things get tricky. If we extend our tether directly down to earth,

we will shift our centre of mass and disrupt our orbit. To counter this we are going to

extend our tether in both directions. This keeps our centre of mass in geostationary

orbit and so maintains our circular orbit. If we place a counter weight on our far end,

we won't have to have equal lengths of tether on either side, to balance our load, and this

counterweight could be a useful platform for operations, so let's do that.

Now, something interesting happens when we start to extend our tethers out. Since this

is our neutral point, where gravitational force and centripetal force equal, any material

extended toward earth will experience more gravitational force, while any material extending

away from earth will experience more centrifugal force. This creates tension in our tether,

which will reach its maximum at our neutral point at geostationary orbit, as everything

below it is pulling towards the earth and everything above it is pulling outwards towards

space.

We can calculate the max tension in a cable with a uniform cross section with this equation.

Where G is the gravitational constant, M is the mass of the earth, rho is the density

of our material of choice, R is earth's radius and Rg is the radius of geostationary

orbit. There is an explanation of how this was derived in this paper, which you can find

by matching the reference number appearing on screen now, to the reference list in the

description.

All of these numbers are fixed, bar one. The density of the material we chose. If we chose

to build this cable out of steel, with a density of 7,900 kg/m^3. Our maximum tensile stress

would be 382 gigapascals. That's 240 times the ultimate tensile strength of steel. In

other words, steel can't do the job.

So can we solve this problem? Steel is one of the strongest materials we have. We certainly

don't have a material 240 times stronger. But we do have less dense materials, which

will reduce the tensile stress we have to endure. On top of this, we don't have to

have a uniform cross section tether.

Our tensile stress approaches 0 at it's endpoints, but material at these points have

the highest effect on our stress as gravitational force and centrifugal forces increase as we

move further from our geostationary orbit neutral point. So it makes sense to minimise

materials at the end points and maximise it where it's needed most.

This will result in an improved design called the tapered tower.

So this brings us to a new question. How can we calculate the area needed at any point

along the tether. Our previous paper has the answer once again. This is the equation they

derived. Here As is the area of the tether we chose at earth's surface. This starting

value will largely depend on design considerations that we can't possibly know right now, but

we are going to want to minimise it, because this right here is an exponential function.

Meaning, our width is going to increase exponentially as we rise. It is imperative that we minimise

this value inside this bracket, and we only have two values we can control in this equation.

The density, which we want to minimise and stress value we are designing for, which he

is donated by T, which we want to maximise.

Normally, we wouldn't use the maximum stress a material can hold as the design stress.

That leaves zero margin for error. We should be designing in a safety factor. But for now,

I'm just gonna go with it and say this thing isn't gonna be safe and I'm designing

it riiiiight on the edge of breaking. So, yeah….bear that in mind.

Remember that strength and density material selection diagram from our last video? Let's

refer to that again to pick a couple of materials to analyse this structure with.

Steel is cheap and well understood, so let's start there with a high grade high strength

alloy like 350 maraging steel.

This steel has an ultimate tensile strength that can range from 1.1 GPa to 2.4 GPa with

a density of 8,200 kilogram per meter cubed.

This paper quotes a steel with a UTS of 5 GPa and a density of 7,900 kilogram per meter

cubed. I don't know what aliens they got their data from, but this is beyond the realm

of reality. We will use steel, but with more realistic material properties.

Then we will pick some better existing materials. They wisely picked Kevlar, which is a widely

available high strength fibre we could easily form into a tether.

We are going to throw two existing materials into the mix too. Titanium, which as we discovered

in our last video has excellent specific strength qualities, and carbon fibre composites, which

have even better specific strength qualities and would be used today if the SR-71 was redesigned.

Using these material properties. We can calculate the taper ratio, which will be the ratio of

the area of the tether at the bottom of our elevator to the area of the tether at its

widest at geostationary orbit.

I'm going to assume a circular area 5 millimeters in diameter at the base. By multiplying the

cross sectional area at the bottom by the taper ratio we find the cables widest point.

For steel, this taper ratio is so huge that our cable at it's widest point will be this

number, whatever that is. For reference the width of the known universe is 8.8 by ten

to the power of 26 meters wide. Even dividing the diameter of this cable by the width of

the known universe yields this number, which I still can't comprehend. Titanium is marginally

better.

Now Kevlar and Carbon Fibre are looking a lot better. They will have a circular diameter

of 80 meters and 170 metres respectively still not quite feasible.

The amount of material required to build something like this would outstrip any cost savings

it could possibly supply. And that's just assuming the fibres could even be formed into

this shape without losing a significant portion of their maximum tensile strength, which is

a big assumption. Footage: I don't know...

So I think it's safe to say that right now space elevators are possible in the sense

the physics of how they work is based in reality, we just don't have a material capable of

making it feasible.

Especially when you consider we are analysing this at ultimate tensile strength in reality

we should be using a value below our yield strength, as above that value our material

will begin necking where the cross sectional area actually decreases as the material elongates.

We aren't even considering strain here.

One future tech that a lot of people are hyping up for future use in space elevators is carbon

nanotubes. Whose strength is off the charts with some studies quoting ultimate tensile

stress values as high as 130 Gigapascals and a low density of 1300 kg per metre cubed.

At that value the taper ratio is just 1.6.

If this material could be manufactured on a massive scale, it would revolutionise life

on earth, but we would still have to solve a huge number of engineering challenges.

Eliminating vibrations and waves propagating through the tether is a huge challenge. Powering

a climber and dealing with the adverse weather of the lower atmosphere and dodging space

debris in orbit are all massive challenges, before even starting on the most fundamental

problem of all. Manufacturing carbon nanotubes.

We will explore these problems and potential solutions in future videos. One on how carbon

nanotubes are made, why they are so strong, and what needs to happen to take them from

the laboratory to regular life. And then, we will revisit this subject with a design

investigation for an actual space elevator using this new theoretical material.

During the research of this video I noticed several mistakes in the paper I referenced,

small mistakes that anyone could make and easily overlooked. I only noticed them because

I applied their methods myself and noticed inconsistencies. Their rounding was so aggressive

that their results were off an inconceivably large number thanks to the exponential function

in their equation and I noticed their material properties for steel was incorrect because

I recreated their calculations for titanium and noticed it was worse, despite the equation

being entirely determined by specific strength.

This is the power of applying knowledge yourself, rather than being a passive observer. You

begin to understand things on a fundamental deeper level and this is why I love Brilliant

and believe they are the perfect compliment to my channel.

I found myself struggling to remember core parts of calculus while following the derivations

in the paper, as it's been years since I had to integrate anything. So, I have committed

myself to completing Brilliants course on calculus and advanced mathematics to brush

up on my maths skills which have dulled with the passage of time. I can complete this course

gradually in my spare time, keeping notes and using their mobile app to steal spare

time to progress through the interactive courses.

This is just one of many courses on Brilliant that will improve my ability to understand

the world around me. You can set a goal to improve yourself, and then work at that goal

a little bit every day.

If you are naturally curious, want to build your problem-solving skills, or need to develop

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As always, thanks for watching and thank you to all my patreon supporters. If you would

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