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In the future I'm sure there will be a city on the moon. We know how to get there, it's super close, and it could be great for further space exploration. However, there's a problem with living on the moon for any length of time: the weak gravitational field, which is just one-sixth of the value on Earth.
Low gravity is bad for you. It causes your bones and muscles to atrophy and weakens your heart, since, basically, everything is too easy. You need exercise to offset these effects, but that's not so simple. You can't go out and jog around the Sea of Tranquility—you'd just start bouncing and floating.
But a cool new research paper suggests that lunar residents can simulate the effects of gravity by running around in circles on the vertical walls of a cylinder—you know, sideways, like the classic Wall of Death motorcycle show in carnivals. (← Seriously, do not miss that video!)
Would this work? Could humans actually scale the walls and feel like they're pounding the pavement back home? There's some fabulous physics in play here, so let's dive in!
The Gravity of the Situation
A gravitational field is defined as the gravitational force per unit of mass. On Earth, the gravitational field (g) is 9.8. newtons per kilogram. So a 10-kg object on Earth would experience a gravitational force of 98 newtons—which is what we mean when we speak of its “weight.”
Yes, properly speaking, weight is a force, and it's measured in newtons, not kilograms or pounds. (Good to know, but don't try to correct the guy at the grocery meat counter.) So you would have a different weight on other planets, even though your mass is the same.
We can calculate the strength of the gravitational field at the surface of any planet (or moon) as:
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Here, G is the universal gravitational constant, M is the mass of the planet, and R is its radius. Notice that a smaller radius boosts the gravitational field; if the mass were the same, gravity would be stronger on a smaller planet! The moon's radius is about one-quarter that of the Earth, but it has a much lower mass (just 1 percent of Earth’s), so its gravitational field is much weaker.
Apparent Weight
We can't really make artificial gravity, but according to Einstein's equivalence principle, from the theory of general relativity, we can make something that is indistinguishable from gravity. Let me explain with a familiar situation.
Imagine that you step into an elevator and push the button for the 10th floor. The elevator can't get there without moving, and since it's at rest it has to accelerate up to some speed. Acceleration is the rate of change of velocity. Say that three times and hold on to it.
OK, so you're in an upward-accelerating elevator. How do you feel? You have the odd sensation of being heavier, right? You feel the pressure in your feet, just for the short time that the elevator accelerates. Once it reaches a constant traveling speed the feeling goes away.
Were you actually heavier? Obviously not, because the gravitational force was constant. But something must have changed. Here's a force diagram showing a human in an upward-accelerating elevator:
There are only two forces acting on you here: (1) the downward-pulling gravitational force (m·g) and (2) the upward-pushing force from the floor, which we weirdly call the normal force (N). (“Normal” means perpendicular in geometry, and here it means perpendicular to the floor.) The net force is the sum of those two things.
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When the elevator is stopped, the two forces are equal and opposite, and the net force is zero. But if you're accelerating upward, the net force must also be upward. This means the normal force exceeds the gravitational force (shown by the lengths of the two arrows above). So you feel heavier when the normal force increases. We can call the normal force your “apparent weight.”
Get it? You're in this box and it looks like nothing's changing, but you feel yourself being pulled downward by stronger gravity. That's because your frame of reference, the seemingly motionless elevator car, is in fact zooming upward. Basically, we're shifting from how you see it inside the system to how someone outside the system sees it.
Could you build an elevator on the moon and have it accelerate fast enough to regain your earthly weight? Theoretically, yeah. This is what Einstein's equivalence principle states: There is no difference between a gravitational field and an accelerating reference frame.
A Roundabout Solution
But you see the problem: To keep accelerating upward for even a few minutes, the elevator shaft would have to be absurdly tall, and you'd soon reach equally ridiculous speeds. But wait! There's another way to produce an acceleration: move in a circle.
Here's a physics riddle for you: What are the three controls in a car that make it accelerate? Answer: the gas pedal (to speed up), the brake (to slow down), and the steering wheel (to change direction). Yes, all of these are accelerations!
Remember, acceleration is the rate of change of velocity, and here's the key thing: Velocity in physics is a vector. It has a magnitude, which we call its speed, but it also has a specific direction. Turn the car and you're accelerating, even if your speed is unchanged.
So what if you just drove in a circle? Then you'd be constantly accelerating without going anywhere. This is called centripetal acceleration (ac), which means center-pointing: An object moving in a circle is accelerating toward the center, and the magnitude of this acceleration depends on the speed (v) and the radius (R):
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Since this is an acceleration, driving in a circle feels like a sideways gravitational force. But you already know that: If you're riding shotgun in a car and it turns hard to the left, you seem to feel a force pulling you outward, into the door. People call this a “centrifugal force”—which I'm sorry to say is not a real thing. It's really a horizontal normal force from the door pushing you inward, but it feels like g forces pulling you in the opposite direction, just like in the elevator.
Can we use a circular motion to create artificial gravity in space? Yes. Yes we can. You could have a cylindrical spacecraft and make it rotate as it travels, so that passengers stick to the inner walls. In fact they do this in the sci-fi show The Expanse, with a giant ship called the Navoo.
A Wall of Death on the Moon
Back to the Wall of Death! The setup for this stunt is an upright cylinder with a radius of around 5 meters. On the inside, a motorcycle or even a small car can drive around on the vertical wall if it gets going fast enough, and spectators watch from above. It's pretty cool.
By the way, if you're wondering why they don't make the ring bigger, look at that equation for ac again. A larger radius in the denominator would reduce the centripetal acceleration!
On Earth, a vehicle has to drive about 15 meters per second (34 mph) in order to stay on this wall. However, with a weaker gravitational field on the moon, it's possible that a human could run fast enough to accomplish this. Here's a cross-section view of what it would look like:
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Since you're moving in a horizontal plane, the net force in the vertical direction must be zero. That means the gravitational force (m·g2 above) must be balanced by some other, upward-pushing force. That would be the frictional force (Ff) between your shoes and the wall.
The other force in this case is the normal force pushing inward from the wall, causing the runner to accelerate toward the center and thus creating sideways “gravity.” These forces are interdependent: The frictional force is proportional to the normal force—more “weight” means more friction. That makes sense: Think how hard it is to push a car on Earth.
So we have a fun situation here: The faster you run, the greater your inward acceleration, which means a higher normal force, which means more friction to keep you on the wall.
So how fast would you have to go to stay up there? Is it even possible? If we use a frictional constant of µs = 0.8 (for rubber shoes on a rough wall), we can calculate the minimum speed:
The lunar gravitational field (g2) of 1.625 newtons per kilogram gives us a minimum speed of 3.2 meters per second. That's about 7 miles per hour—a bit faster than jogging. You could get by with a lower speed by making the ring smaller or increasing the frictional coefficient (though 0.8 is already pretty high).
One Little Problem
Now, for simplicity I've been treating the runner as a single point in space, so that all forces act on the same point. Unfortunately, if the human is, uh, human-shaped, this doesn't work, because the forces hit at different points on the body. The normal force and frictional force work at the contact point of the foot, but the gravitational force acts at the center of mass, near the waist.
These vertical forces still add up to zero, but because they hit in different spots, they will exert a torque. If you were horizontal, this would cause you to rotate and fall on your head. (You'd fall kind of slow, but still … ) To stay on the wall, you'd actually need to lean upward, like this:
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To analyze this case, it's actually easier if we use that imaginary “centrifugal force” we dismissed earlier. Physicists call this a “fake force,” but sometimes a fake force can help us understand a real situation. To do this we just have to change the reference frame from the stationary cylinder (as seen by a spectator) to the moving runner themself—an accelerating reference frame once again.
To be stable, the torque from the gravitational force and the centrifugal force combined must be zero. We can use this to solve for the angle of the body (θ):
With a running speed of 3.2 meters per second, you'd have to lean upward at an angle of 51.3 degrees (from the vertical). Maybe that's OK, but you'd look pretty silly trying to run like this. I mean, exercise is important and all, but you still want to be cool.
A Better Running Wall
Here's a different idea. What if we make a wall that doesn't require a frictional force and a crazy tilted running style? Imagine replacing the Wall of Death with a Wall of Injury. Instead of vertical walls, it would have walls that slant outward. Now the runner will be going around a banked turn, like a tiny Nascar track.
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If the wall is angled down by 51.3 degrees (same as the tilt angle above), then you could run around the circle without a frictional force. Even better, you'd be oriented perpendicular to the running surface, so it would look normal—in both senses of the word.
There is a trade-off, in that we'd have to design the track for a particular speed. If we go with 3 meters per second, a track with a radius of 5 meters would be banked at 48 degrees. (I'm measuring from the horizontal now.) This would produce an apparent weight (the normal force) for the runner of 2.4 times their weight on Earth (2.4 g's). So if you weighed 150 pounds on Earth, you'd now weigh 360. OK, that's overkill.
If we use a bigger track, we can get that apparent weight down. With a 20-meter radius, the track would be banked at about 15 degrees, so a bigger track would be less steep. That gives us an apparent weight of 1.68 g's. That's not too bad. You'd get your heart rate up for sure.
It would require a much bigger structure. But this banked track might also be easier to build than a vertical Wall of Death. Since it's shaped like a big cereal bowl, maybe we could just excavate it out of the lunar surface and cover it with some grippy surface material.
I know, it's not like running through a park back home, but c'mon, running on the walls is pretty awesome. If humans want to live on the moon, this could be a simple way to simulate gravity and keep them healthy. Sometimes you actually can make progress by going in circles.
