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My fantasy novel features a floating city. I wanted to know how much energy (joules/second) it would take to keep a floating city aloft, but the definition of joules has "distance displaced" in its definition, and a floating city does not move, so no matter what I put for the force or the time, the energy comes out to be zero.

I asked a physics-inclined friend how much energy it would take to keep this floating city aloft, and his answer surprised me: The formula was right, and it would take zero energy. He said that it would take a lot of energy to get the city up there, but none to keep it there.

I was incredulous because helicopters very obviously expend energy to stay aloft, but my friend explained to me that in theory, making a city float would be like lifting an apple from the floor and putting it on the table. It takes energy to put it up there, but no energy to keep it there. All you need is some way to prevent the gravitational potential energy from turning into kinetic energy and you've got your floating city. "Magic" seemed a reasonable enough explanation to keep the energy from changing forms.

This confused me as much as it made sense. So my questions:

  • What, from a physics standpoint, is the difference between a helicopter that must expend energy to remain aloft and an apple on a table that does not?
  • How could you tell which of the two paradigms would be correct for a magically floating city?
  • If you did need to spend energy to keep the city aloft, how would you calculate it, given that the displacement is zero?
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  • $\begingroup$ What you choose for your story can be best…. But your friend is right. If you have a magnet floating on another fixed magnet, you might need to do something to stabilize it so it stays there, but the strength of the magnet doesn’t change, it is not expensing energy to float. If you take the magnet away and use it, it will have the same strength it before it floated. $\endgroup$
    – UVphoton
    Aug 1, 2022 at 22:48
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    $\begingroup$ If you want to hover then balance the force of the mass of the city times gravity. Then whatever energy you need to produce that force is your answer. Note you could have problems keeping your city in place, since an easterly wind might cause it to accelerate a small amount but big things are hard to stop moving… same thing if the barometric pressure changes it would change the buoyant force from displacing the air and you might need more or less energy to balance the force…. $\endgroup$
    – UVphoton
    Aug 1, 2022 at 22:59
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    $\begingroup$ A balloon expends no energy to stay aloft; it is an aerostat, meaning that it floats on air. A helicopter expends a lot of energy to stay aloft; it is an aerodyne, meaning that it flies by using aerodynamic forces. $\endgroup$
    – AlexP
    Aug 2, 2022 at 0:38
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    $\begingroup$ If you want it to cost energy, just use some sort of system that converts energy to force. Say a magic artifact that can apply n newtons of force per x Joules of whatever kind of energy you put into it. Then, you apply enough force to counteract gravity, so that the system experiences zero net force. $\endgroup$ Aug 2, 2022 at 15:19
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    $\begingroup$ Joules/second is the unit for power (Wattage), not energy. The unit for energy is simply the Joule. $\endgroup$ Aug 3, 2022 at 3:49

9 Answers 9

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The zero-energy is the correct ideal answer. The act of keeping a city floating takes zero energy, as shown by the fact that the ground beneath our feet expends 0 energy keeping the city out of the Earth's mantle.

If you use magic, this is good enough. However, sometimes you don't have the luxury of doing exactly the physics operation you want. This is where the helicopter comes in. The helicopter expends zero energy "staying aloft," but it expends an immense amount of energy shoving air downwards. The helicopter operates in a non-ideal world where the only operation it can do is apply force to a fluid, which rapidly gets out of the way of its rotors. It does no work to itself (it remains at the same height, thus the same potential), but does a lot of work to the air in order to get the forces needed to combat gravity.

Given that you are doing this with magic, you get to choose. This could be a "force field" like effect which requires no energy input, or it could be more like an updraft of air counteracting the forces of gravity, which would require a great deal of energy. The energy expended will depend 100% on the method you use to keep it aloft. We can say nothing about the energy expenditure without settling on a method and how that method works.

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    $\begingroup$ My preferred 0 energy method is to just stick a stick underneath the city, and let the city rest on it. It's easy to see that no energy is expended by the stick, although there are tremendous forces on it! $\endgroup$
    – Cort Ammon
    Aug 1, 2022 at 22:03
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    $\begingroup$ It doesn't have to be magic, it can also be handwavy science (which is, of course, indistinguishable from magic). E.g. in the world of the game "Bioshock Infinite", the floating city of Columbia is floating because of so-called "Lutece particles", which are atoms (?) that are stuck in space (apparenly relative to Earth) for some reason. See also bioshock.fandom.com/wiki/Lutece_Particle $\endgroup$
    – rob74
    Aug 2, 2022 at 7:06
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    $\begingroup$ @Luaan this is actually already used in Bioshock: infinite to keep a whole city floating. I think in added lore of the Avatar universe the floating rocks are the same, floating because of the unobtainium easily going into superconduction and creating magnetic fields to keep the rocks aloft. $\endgroup$
    – Trioxidane
    Aug 2, 2022 at 13:05
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    $\begingroup$ It's absolutely infuriating that this is correct. Exerting a force over time uses (or more correctly, converts) no energy - that only comes in when exerting force over a distance. Experiences of holding a heavy weight above your head and thinking it was tiring are entirely misleading. $\endgroup$ Aug 3, 2022 at 8:07
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    $\begingroup$ @ZeissIkon Amusingly, having the magic "work" by having a giant hold the city up in the clouds would indeed provide us the information needed to cook up an energy budget required to keep it aloft. Of course, its probably not measured in joules, but rather in fried whole cows per day. $\endgroup$
    – Cort Ammon
    Aug 3, 2022 at 17:39
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The force you're causing may cost energy even if the energy doesn't go into the object it is acting upon.

What you need to consider is "How does the city stay up?"

  1. Is mass being thrown down to counteract the force of gravity on the city? If so, work is being done on that mass being thrown down. Which you could calculate.
  2. Is the force going through some sort of extradimensional space so that effectively the city is actually supported by the ground, but you just can't see the supports because they're in another dimension? If so, then it really does take no energy to keep it up (though it may take some to keep those dimensional spaces open).
  3. Is the force actually a void (vacuum) space with a lot of volume and no density so it really is floating, much like a hot air or helium balloon? Again, no energy required. Though there may be energy needed to evacuate the space.

As you can see "the how" maters a lot in all of these cases.

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  • $\begingroup$ forced are not "caused" and do not "cost energy" $\endgroup$
    – Deipatrous
    Aug 2, 2022 at 9:01
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    $\begingroup$ I agree that they are not always "caused" or "cost energy" but they certainly CAN be "caused", and they CAN "cost energy". This is at the heart of my answer. $\endgroup$
    – Mathaddict
    Aug 2, 2022 at 23:01
  • $\begingroup$ A simple example is an electromagnet. Turning on the current causes a magnetic force in/around the coil, and maintaining that force costs energy because the coil has electrical resistance. $\endgroup$
    – towr
    Aug 3, 2022 at 9:16
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Floating.

https://en.wikipedia.org/wiki/Buoyancy

For this reason, an object whose average density is greater than that of the fluid in which it is submerged tends to sink. If the object is less dense than the liquid, the force can keep the object afloat.

rubber duck

duck

Your city is floating. That means its mass is less than the medium it displaces. This big duck is expending no energy to float. It is supported by the water because it is less dense than the water. .

Your floating city is equal in mass to the atmosphere it displaces. It can thus float there with no energy expenditures. A neutrally buoyant object will stay put where it is in a column of gas or water, not experiencing a force up or down.

A helicopter is not floating. It is flying. Its mass is more than the air it displaces and so it must work to throw air downwards and itself upwards.

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    $\begingroup$ You could create an interesting world, using magic to create a column of hyper-dense air like a bathtub underneath the city. It could be quite a shock to try to walk into that column of air! $\endgroup$
    – Cort Ammon
    Aug 1, 2022 at 22:30
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    $\begingroup$ Hyper-dense air sounds so silly, then you remember that nature takes expired air and removes the oxygen and uses that to make trees the flesh of which we build with. Crazy. $\endgroup$
    – pbhj
    Aug 1, 2022 at 23:56
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    $\begingroup$ It would work in the clouds of Venus... $\endgroup$
    – Deipatrous
    Aug 2, 2022 at 9:02
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    $\begingroup$ I did some reasearch and a 2.500,000 cubic meters vacuum chamber should be able to counteract the equivalent of 3250 Tons of material, let's say you could make that chamber in a sphere shape with some very strong and light material so that it weight only 5 tons, you're left with a good amount of material you could put on top of that sphere, the sphere would be the volume of the pyramid of gyza and given each house weights 30 tons you could build around a hundred houses on that. Doesn't sound like you'd need magic, just some good engineering and technologies $\endgroup$
    – Tofandel
    Aug 3, 2022 at 11:04
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    $\begingroup$ Keep in mind the density of air fluctuates with temperature, so your city will go up and down a lot. Balloons can release hot air or heat more air, and your city will need something similar to control its altitude. This inevitably expends some energy on good days and a lot of energy when you're trying to stay the course through a supercell. $\endgroup$
    – Hene
    Aug 3, 2022 at 12:29
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I don't think any of the other answers are directly answering your questions, so:

  1. A helicopter stays aloft by pushing on the air, and the air annoyingly moves in response. If the air were solid, then it would be difficult to breathe, but it would require 0 energy to keep the helicopter aloft, just like it requires 0 energy to keep it landed on a hill. A hovering helicopter requires energy like walking on a Stairmaster requires energy, because the thing its pushing on is moving in the direction of the push.

  2. Which paradigm is appropriate depends on how your magic works. If nothing has to move, then no energy is expended. If your city needs thrusters of some sort that expel rocket exhaust, or move air, or similar, then it will cost energy just like a helicopter.

  3. Calculations: First, note that joules/sec is power, not energy. The formula for power that applies is force x velocity. The power required to keep the helicopter aloft is the weight of the helicopter times the velocity of the air moving through its rotors. If your city needs thrusters, then the power required to keep it aloft is the weight of the city times the velocity of the thruster exhaust.

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  • $\begingroup$ I'm not sure that's a correct usage of P=Fv, since the thruster force is not constantly applied over a distance as the exhaust is pushed away. It implies that you could use arbitrarily little power to keep the city afloat, simply by using many thrusters with low exhaust speed, rather than few thrusters with high exhaust speed. Suppose we split the city in half - each thruster supports half the weight and can do so with lower exhaust speed. Both combined use less power than one thruster that supports the whole city. Keep splitting to use less and less power - something is wrong here. $\endgroup$ Aug 2, 2022 at 15:17
  • $\begingroup$ @NuclearHoagie It's correct. E = Fx, and P = dE/dt = F * dx/dt for constant force -- no actual distance needs to be measured. And yes, you could indeed keep the city afloat with very little power if you had a whole lot of thrust area... if your thrusters are themselves weightless. $\endgroup$ Aug 2, 2022 at 16:29
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What, from a physics standpoint, is the difference between a helicopter that must expend energy to remain aloft and an apple on a table that does not?

Both are pulled down by gravity. The apple is stopped by the force created by tiny deformation of the table. If you gradually increase the weight of the apple it would eventually break the table and fall. A helicopter uses lift created by the aerodynamic force as rotor blades travel through the air.

How could you tell which of the two paradigms would be correct for a magically floating city?

Well, if your city lies on a table it doesn't really fly. So, it has to combat gravity somehow.

If you did need to spend energy to keep the city aloft, how would you calculate it, given that the displacement is zero?

Let's say your city has mass M. Gravity force acting on it F = M*g. You need to create force that is directly opposite in direction and equal in module.

It likely to be hugely expensive though. Helicopters are not cheap, rockets even more so. The cheapest option would be a lighter-than-air city. It would require tons of helium or hydrogen though.

That's why nobody builds flying cities. Except for mages who don't care about energy.

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  • $\begingroup$ I would suggest magnets, but I seem to recall a theorem about you can't balance something stably on magnets. $\endgroup$
    – Deipatrous
    Aug 2, 2022 at 9:03
  • $\begingroup$ @Deipatrous You could do it on any stable magnetic field. Earth doesn't have one of those, though. $\endgroup$
    – Michael W.
    Aug 2, 2022 at 20:16
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    $\begingroup$ @Deipatrous You can't balance something stably on a house magnet, but you can absolutely make a magnet that is shaped to balance something on it. $\endgroup$
    – Hene
    Aug 3, 2022 at 12:35
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In case your Magic isn't super-powerfull, but you're in a technologically really advanced setting: Use the power of earths orbits! (Think of the International Space Station)

Please note that with different orbits there are different benefits / caveats

  • some are geostationary
  • some need a FAST speed (the ISS travels at an average speed of 28,000 kilometers per hour)
  • some need more energy to counter atmospheric drag than others
  • none provide a human-livable environment

However, some of these caveats might be erasable using magic:

  • use portals to get humans / fuel up there
  • use Force-Shields to shield your city
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  • $\begingroup$ Clever. Let the city orbit the planet at 7,000 meters. $\endgroup$
    – Tony Ennis
    Aug 2, 2022 at 20:03
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Zero energy or fuel

Lets say you have a helicopter with an exact ton of weight. To stay aloft, exactly at the height it is a ton heavy, you need to push exactly a ton of force upwards¹. Less than a ton and gravity pulls it down. More than a ton and the helicopter goes up. That means if there's no acceleration and the helicopter stays on it's place above the Earth it's a zero sum.

This doesn't mean it doesn’t cost energy. We are using energy in this case to keep it at this state. It has a lot of potential energy. If you turn off the helicopter the energy is released. We can see this by the effect of the helicopter crashing down onto the ground. The problem with the blades is that the air is lighter than the helicopter and gaseous. We need to keep moving yhe blades and push air down, or the air will move past the helicopter and not support the weight of the helicopter.

Now we have the exact same situation with the helicopter stationary on it's place high kn yhe air. The difference is that we now made a platform under it, connected all the way yo the ground. Though the helicopter is staying where it is just like in the first scenario we now see only one difference. Instead of the blades pushing a ton of weight up¹, keeping it level, it is the platform pushing up one ton. This keeps the helicopter level. Just like in the first scenario the helicopter has potential energy. If we remove the platform it'll crash down just the same.

You might think the helicopter with spinning blades is then something of an oddity. It is using energy to stay aloft, while the platform is using nothing. But that is a mistake of human perception. Energy isn't created or destroyed. It just becomes something else. That means that even though the helicopter is using fuel no energy leaves the system. The fuel in the helicopter just changes it's potential energy into movement and heat. The platform also experiences some deformation and heating. It might not be as impressive and loud as using the blades, but it does the same thing.

Your city can then choose how it stays aloft. You need to cancel out the amount of power the city pushes down (gravity times mass). Will you try to keep it up by pushing against things, like air, requiring a constant input of energy to keep the city up. Or will you put it on a magical platform, requiring the stationary platform to push back.

¹ Or down, so the helicopter stays up. Matter of perspective. You push other things down to push the helicopter up.

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  • $\begingroup$ I think this answer is quite misleading. For instance: You're measuring force with mass units, and your “The platform also experiences some deformation and heating.” implies that the platform is expending energy to keep the helicopter aloft. (It isn't, and if you actually try this in real life, you won't observe continual stress-heating.) $\endgroup$
    – wizzwizz4
    Aug 3, 2022 at 20:58
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As others have mentionned you would indeed need no energy to keep it there only the initial energy to put it there

I did some calculations, based on the weight of air and buoyancy, I might not be 100% correct though

If you built a very big vacuum chamber, lets say Giza pyramid volume, vacuum weighing nothing except for the enclosure and air weighing 1.3Kg per cubic meter

You'd end up extracting 3250 Tons of air from that chamber, remove maybe 50 Tons to build the enclosure and pump needed to build that vacuum and you'd effectively end up with a balloon capable of supporting 3200 tons of material on top of it, depending on how light you build your houses, let's say 10 tons to 30 tons, you could build 100 to 300 houses on that sphere

Feel free to scale that up (or down) but bigger is always better for the sphere as volume scales exponentially but the enclosure weight scales linearly

So it doesn't look like you'd need magic to build your city, only very good engineering and materials

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  • $\begingroup$ Quite the material challenge, indeed! Building a sphere of that size with 50 tons of material gives you just 2kg per square meter, and that ignores any internal structure. If made of steel, you need an 85m-radius shell with sub-millimeter thickness, which can withstand a few billion kg of pressure. $\endgroup$ Aug 3, 2022 at 19:28
  • $\begingroup$ 50tons was just a random guess, there is a lot of wiggle room so you could make it 200 tons and still have a lot of houses you can build. As for material, you could use something like nanotubes instead of steel, and the advantage of the sphere is that it's very good as withstanding external pressure, its the optimal shape for that. That obviously still leave a lot of technical and Sci-fi feats, but at least not physically impossible $\endgroup$
    – Tofandel
    Aug 3, 2022 at 19:42
  • $\begingroup$ Heck you could even make it a 1500 tons and still build 50 houses $\endgroup$
    – Tofandel
    Aug 3, 2022 at 19:50
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I asked a physics-inclined friend how much energy it would take to keep this floating city aloft, and his answer surprised me: The formula was right, and it would take zero energy. He said that it would take a lot of energy to get the city up there, but none to keep it there.

You might need some better physics-inclined friends. You're focused on gravitational potential energy, but you're ignoring everything else.

The problem here is that energy is not something that acts on things, it's something that things have which can be transformed (or transferred) by performing work. A Newton's Cradle is a perfect example of this. You lift one ball to give it gravitational potential energy, then let it go. The force of gravity acts on the ball causing it to fall, transforming GP energy into kinetic energy. When the ball strikes one of the motionless balls the force of the collission applies to each ball in the collision (almost) equally, transferring the kinetic energy through the collected balls until the final ball rebounds up. The energy transfer and transformation was the result of the forces acting on the balls.

The only way to stop gravity from turning your GP energy into kinetic energy is to provide some force to balance things out. Since force and energy go hand in hand, you're going to be chewing through the power (work done per unit of time) to produce enough force to counter the force of gravity.

Don't believe me? Reach your hand out straight in front of you at eye level. Now hold it there, as motionless as you can. Touch one fingertip to a wall or something, not enough to support it but just so you can feel if you start to move. Now hold it there. I'll be right back.

.

.

.

You still holding? Starting to feel some muscle fatigue yet? Give it a few minutes.

What you're feeling is a combination of things, but the one we're interested in is the chemical potential energy you've just expended to keep your muscles tight enough to provide the force that acted against gravity attempting to drag you down. (Unless you're reading this on the ISS, in which case you're cheating. And also awesome.) You didn't move, your muscles didn't move much, but you consumed (i.e. transformed into bound chemical energy and heat) some energy just by staying perfectly still, and more than if you didn't have your arm up.

So can we please stop with this idea that "the city doesn't move!" means "we don't have to use energy!"? Please?

When you are generating a force - such as the one you're going to need to counter the force of gravity - you are transforming energy in some way. And because Thermodynamics is still a thing, some of that energy is being lost in forms you didn't want: probably at least heat and sound.


Oh, but wait, we have magic! We don't have to worry about conservation of energy and stuff. That's for the poor sods stuck in physics-dominated universes that don't have access to things like extra-planar portals and so on.

So here's how you do it.

First, enslave a bunch of the inhabitants of the nearby planes. Force them to build towers in the place that matches where you want your city to be, just next door in a suitable plane - I hear the Ethereal is relative stable this millenium. Next, enchant some materials to exist in both the material and Ethereal plane, and place them on top of the towers. Back in the material plane, use those as support girders to build the base plate of your city on. Keep the horde of critters working on expanding the forest of support towers - maybe contract an edritch horror or two to keep them in line while you're busy in the real world.

Now we just need to invite some sucker... I mean citizens to come build our amazing floating city. Oh, and maybe get some fae folk to go pretty up the support towers in the ethereal realm. Just in case someone with True Sight or something comes looking, we want to be able to pass them off as epic spell formations or something.

Now all that gravity trying to tear you from the sky is frustrated because the structure is supported by the bedrock of the ethereal.

Just... don't overdo it. Even bedrock will liquify under enough pressure. And you might want to invest in some heavy duty stabilizers to make sure the Ethereal landscape stays where you put it.

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