this episode of real engineering is brought to you by brilliant a problem-solving website 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 that could cross the divide of science fiction to science reality if we somehow improved on existing technologies it's the kind of 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 its seemingly impossible height we can build higher with current materials if we widen the base but this 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 counterbalancing the weight of the structure by pulling upwards we can do this thanks to centrifugal force imagine a tetherball swinging around a pole at a certain angular velocity the string will be held straight and taut against the pole because centrifugal force an apparent force that appears in a rotating reference frame pulls outwards now the problem is the whole point of feather ball is to wrap the string around the pole if the string can't rotate around the center of spin it will simply wrap itself around the pole we are essentially trying to recreate this dynamic but on an astronomical scale and to do that we have to work with the 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're 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 in sync with earth's rotation the structure will curve and break or in some sort of cartoon world wrap around the earth like the tetherball example our orbit will 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 centrifugal force is defined by this equation where ms is the mass of the satellite omega is the angular velocity and r is the distance to the center 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 now we have an equation with all the known values which we can solve for by inputting the values for earth and we find a value of 42 168 kilometers this is the distance from the center of the planet so we will be about 36 000 kilometers above the surface of the planet at the equator okay this gives us a starting point for our construction we are going to put some form of massive satellite with a cable 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 the earth we will shift our center of mass and disrupt our orbit to counter this we're going to need to extend our tether in both directions this keeps our center of mass in geostationary orbit and so maintains our circular orbit if we place a counterweight on the far end we don'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 extending towards 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 the 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 the 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 1 the density of the material we choose if we choose to build this cable out of steel with a density of 7900 kilograms per meter cubed 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 zero at its end points but material at these points have the highest effect on our stress as gravitational force and centrifugal force increases as we move further from our geostationary orbit neutral point so it makes sense to minimize materials at the endpoints and maximize 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 an answer once again this is the equation they derived here as is the area of the tether we choose 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 minimize 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 minimize this value inside this bracket and we only have two values we can control in this equation the density which we want to minimize and the stress value which we are designing for which here is donated by t which we want to maximize normally we wouldn't use the maximum stress the 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 going to be safe and i'm designing it right on the edge of braking 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 analyze the structure with steel is cheap and well understood so let's start there with a high-grade high-strength alloy like 350 margin steel this steel has an ultimate tensile strength that can range from 1.1 gigapascals to 2.4 with a density of 8200 kilograms per meter cubed this paper quotes a steel with an ultimate tensile strength of 5 gigapascals and a density of 7900 kilograms 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 fiber we could easily form into a tether we are going to throw two existing materials into the mix too titanium which we discovered in our last video has excellent specific strength qualities and carbon fiber 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 five millimeters in diameter at the base by multiplying the cross-sectional area at the bottom by the taper ratio we find the cable's widest point for steel this taper ratio is so huge that our cable at its widest point will be this number whatever that is for reference the width of the known universe is 8.8 by 10 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 fiber are looking a lot better they will have a circular diameter of 80 meters and 170 meters respectively still not quite feasible the amount of material required to build something like this would outstrip any cost savings we could possibly supply and that's just assuming the fibers could even be formed into this shape without losing a significant portion of their maximum tensile strength which is a big assumption 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 have to make a material capable of making it feasible especially when you consider we are analyzing this at the ultimate tensile strength and 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 1 300 kilograms per meter cubed at that value the taper ratio is just 1.6 if this material can be manufactured on a massive scale it would revolutionize 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 the 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 overlook i only noticed them because i applied their methods myself and noticed inconsistencies their rounding was so aggressive that the results were off by an inconceivably large number thanks to that exponential function in their equation and i noticed that 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 fundamentally deeper level and this is why i love brilliant and believe they are the perfect complement to my channel i found myself struggling to remember core parts of calculus while following the derivations in the paper and it's been years since i had to integrate anything so i have committed myself to completing brilliance course on calculus and advanced mathematics to brush up on my math skills which have dulled with the 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