A few days ago I wrote about xkcd’s Randall Munroe’s ‘What if?’ blog, and how great it is. I thought it might be fun to try and write something similar once, so here it goes! You could say this is a blatant rip-off, but I like to think of it as an homage.
What if I built a tower to space, and I stood at the top. Would I float away?
First off, building a (conventional) tower to space is impossible. There is no material known to man that would be able to withstand the pressure of its own weight within its foundations. There are some alternative designs proposed which might actually work, but these are not likely to be built anytime soon. So for now we’ll just gloss over the structural engineering difficulties as this is a hypothetical question anyways, and get right to the physics.
When you ask a non-physicist why the astronauts in the International Space Station (ISS) float, people tend to respond: “Because there is no gravity in space.” This is a fairly widespread erroneous assumption. The idea that you would float away at the top of a tower to space is rooted in this same idea that “there is no gravity in space”. But why then does the moon orbit the Earth? Or the Earth the sun?
There certainly is gravity in space; it’s what holds our solar system, our galaxy, and every other one in line! Why then do the astronauts up in the ISS float? The answer is simple: They are falling. They seem to be floating relative to the internals of the space station, but relative to the earth they are hurling around at immense speeds. On average about 27,724 km/h in fact. So if they are falling, why do they never crash into the ground? Again, the answer is fairly straightforward; they are falling around the Earth, rather than towards it. This is the definition of a stable orbit. During launch the ISS was accelerated by rockets not just upwards (away from Earth into space) but also laterally (sideways). This caused the spaceship to fall around the Earth.
Perhaps an easier way to explain it (albeit a scientifically outdated one), is that the centrifugal forces of circling the Earth exactly counteract the gravitational forces pulling the ISS back down. If the centrifugal forces were larger, the ISS would be in an escape trajectory and would move further and further away from the Earth. If the centrifugal forces were to be lower than the gravitational forces, the ISS would come crashing back down to Earth.
Okay, so you need high lateral speed for the illusion of zero-gravity. What if the ISS were to halt to a dead stop all of a sudden? Would the astronauts fall on their asses? Yes! But then, as the ISS started plummeting straight down towards Earth due to gravity, the astronauts would once again start to float. Logical, because the spaceship is falling once more; it’s just falling straight down this time. Not for long though, the ISS is not designed for re-entry so it would partially burn up, and then break apart. Sorry…
Back to your tower! It doesn’t move laterally (relative to Earth) and it doesn’t plummet downwards (theoretically). So gravity most certainly is taking a hold of you, and you are not weightless. There is however still the mere gigantic altitude to consider. Gravity changes with altitude, the farther away from the earth you are, the less it pulls on you. Gravitational force is proportional to 1/R², where R is the distance between you and the Earth’s core. The radius of the Earth is 6,378.1km so the distance to the center of the Earth from the top of the tower to space would equal 6,378.1km+the height of the tower.
Before we can calculate this we first need to know the exact height of your tower. “A tower to space” is not very specific. The formal definition of the “edge of space” is at an altitude of 100km. However, there is still a significant atmosphere at that altitude, so let’s build a little higher (we have an infinite amount of hypothetical money anyways). For the purpose of this exercise let’s use the current altitude of the ISS as the height of the tower. The ISS skims the Earth’s atmosphere so its altitude is a good indication of the “edge” of the atmosphere, and therefore a more rigid definition of space. Currently the ISS orbits at around 410km above the Earth.
So your tower is 410km high. The distance to the Earth’s center would then be 6,378.1km+410km=6,788.1km. Gravity at sea-level is 9.81m/s². So now to calculate the gravity at the top of your 410km high tower:
So gravity at the top of the tower would be 8.66m/s² which is 13.3% less gravity than at sea-level. That may perhaps be only faintly noticeable. Therefore standing on the tower’s roof, you would be standing just as firm as you would on the ground.
But what are you doing standing around on the roof?! You just completed building a friggin’ tower to space! I’m going to assume the most logical next step in your entrepreneurial plan (trust me, I have a degree in business) and suggest you open a revolving restaurant at the top.
Now some interesting questions arise as to the customer experience of your brand spanking new restaurant. For instance, how long does the elevator ride up take?
Let’s assume you installed the world’s fastest elevators in your tower. They travel at a dazzling speed of 18 meters per second (That’s 100 floors in a little under 20 seconds!)
It would then be a 6 hour and 20 minute long elevator-ride to the top. One thing is for sure, your clientele is bound to be hungry once they reach the restaurant.
Going back down would be a lot faster, given the right elevator design. We’ll let gravity do all the work, but help it out by vacuum-sucking the elevator-shaft, eliminating air-resistance (and by happenstance significantly simplifying the following calculation). All we really need to calculate is the acceleration due to gravity. Gravity as a function of altitude isn’t linear, but at this size numbers we can use the average acceleration between the gravity at the top and the gravity at the ground floor of the tower as an approximation. We already know both of these from earlier. They were 8.66m/s² at the top, and 9.81m/s² at sea-level. Average between the two is: 9.235m/s². So when we plug that in:So the elevator will plummet for 4 minutes and 58 seconds. Whilst the elevator is falling, everyone in it will experience weightlessness, as explained earlier. Not a good idea right after eating in a restaurant I would think… (How do you think the vomit comet got its name?)
That last calculation does not take into account the actual slowing down as we reach the ground floor. As it stands it takes 4 minutes and 58 seconds of falling at which point you will smash into the ground at a speed of approximately 9,907km/h. Assuming your well-fed customers want to leave the tower alive, we should probably think about installing brakes at some point…
Check back in a few days for part 2, in which we will discuss in more detail the unexpected difficulties of running a restaurant at an altitude of 410 kilometers.
Even though all content in this post was created by me I want to attribute back to Randall Munroe of xkcd.com for the original concept of What If? and the style of the stick-figure illustrations under the Creative Commons Attribution-NonCommercial 2.5 license as stated on his site. I will do the same specific to this post and add the Share-Alike. Thusly this post is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 license.