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Wednesday, February 21, 2024

How to Simulate Walking on the Moon—Without Leaving the Planet

Let’s say you want to know what it's like to walk on the moon. Is there any way to simulate a moonwalk while staying on Earth? Well, yes. In fact, there are several.

But before we get to them, why is walking on the moon different than walking on Earth? It's all about gravity.

There is an attractive gravitational force between any objects that have mass. Since you have a mass and the Earth has a mass, a gravitational interaction pulls you towards the center of the Earth. The magnitude of this force depends on the mass of the Earth (ME), the distance between you and the Earth (which is essentially the radius of the Earth, R), and your mass (m). There's also a gravitational constant (G).

The formula for the gravitational force pulling down on you looks like this:

People and objects have different masses, which means they have different gravitational forces—also called weight. If you measure the weight of a person or object and divide by its mass, you get the weight per kilogram. (Remember, weight and mass are different.)

We actually have a name for this quantity—it's called the gravitational field. On Earth, it has a value of g = 9.8 newtons per kilogram, and it points towards the center of the Earth. (For humans, that means “down.”)

If you drop an object in this gravitational field, it will have an acceleration in that same direction with a value of 9.8 meters per second per second. Some people call g "the acceleration due to gravity" for this very reason. But if you have any object, falling or at rest, its weight will still be the product of its mass and g. It doesn't have to accelerate to have this weight.

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In general, we can calculate the gravitational field on the surface of a planet (or a moon) as:

In this formula, M is the mass of the planet or moon and R is its radius.

OK, we already know what walking is like on Earth. Now what happens if you move to the moon? The moon is both smaller and less massive than the Earth. This means that the gravitational field on the moon’s surface is different from Earth’s. By itself, a smaller mass would decrease the gravitational field, but a smaller radius would increase the strength of the field. So we need some values for the moon to see which one matters more.

The moon has a mass that is 0.0123 times that of the Earth (about 1 percent of Earth's mass), and a radius that's 0.272 times that of the Earth. We can use these values to find the gravitational field on the moon.

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That puts the gravitational field at about one-sixth (0.166) the value on Earth, or 1.63 N/kg. If you jump or drop something on the moon, it will have a downward acceleration of 1.63 m/s2.

OK, now how do we simulate that gravitational field on Earth?

The Lever Method

First, you would need to do something about that downward-pulling gravitational field. For every 1 kilogram of mass, the Earth pulls down with a force of 9.8 newtons, whereas on the moon, it would just pull down with a force of 1.63 newtons. That means you would need to push up on a person with a force of 8.17 newtons per kilogram to make them feel like they’re walking on the moon.

One way to provide this upward-pushing force would be to use a lever with a counterbalance. (For example, here is French performer Bastien Dausse using a device to mimic the motion of a person on the surface of the moon.) This is the same basic idea behind the seesaw at the local playground. It is essentially a long stick with a pivot point between a large mass and a person, like this:

Even though there's not a straight stick connecting the person to the counter mass, it's still a lever. A lever is one of the classic "simple machines." It's basically some type of beam on a pivot point. If you push with a force on one side (providing the input force), you get some other force on the other side (the output force). The value of the output force depends on the input force, as well as the relative distances of the two forces from the pivot point.

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The magnitude of the output force can be found with the following expression:

So, that's it: You just need to push down on the right side of the lever using some type of weight, and it will push up on the left side with the human.

How much mass would you need? That's a function of the weight of the human (mhg), the length of the two parts of the lever (ro and ri), and the effective vertical acceleration (am). The effective vertical acceleration would be the same as the free-fall acceleration of a human on the moon.

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If I use a human mass of 75 kilograms, and lever arms of 2.0 and 0.5 meters, then the mass on the end would need to be 250 kilograms. But is this really the same as walking on the moon? Well, it's not subjectively the same. The device only supports the person at some attachment point, which means they can only walk in a circle and not go wherever they want.

Is the vertical acceleration the same as on the moon? This device doesn't provide a constant net force. Instead, this force decreases as the angle increases. This creates a small complication. You can see this in the video: When the performer jumps high enough, the lever is mostly vertical. At that point, he just stays there. Clearly, that's not what would happen on the moon.

Let's see if this lever device provides an acceleration similar to that on the moon. I'm going to use Tracker Video Analysis and plot the vertical position of the performer in the video in each frame. This will give me the following plot of position versus time:

This appears to be a quadratic function, as it should be for constant acceleration. An object with a constant acceleration can be modeled with the following kinematic equation:

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The only thing that matters here is that the term in front of t2 is (1/2)a. That means that the fitting parameter in front of t2 for the data must be 1/2 of the acceleration giving this guy a vertical acceleration of 1.96 m/s2. That's pretty close to the acceleration we computed earlier for a jump on the moon, 1.63 m/s2. Nice.

So we can say it's just like walking on the moon—as long as you walk in circles.

The Pendulum Method

There's another way to simulate a reduced gravitational field, one that NASA used in the 1960s to see how astronauts could move about on the moon.

A person lies sideways, supported by slings around their waist and rib cage, which are attached to very long cables connected to a mounting point somewhere above them. Instead of touching the floor, their feet actually touch a wall that is slightly tilted, so it’s not exactly perpendicular to the floor. This gives them a fake “ground” to practice walking, running, and jumping on without feeling the full force of Earth’s gravity.

But how does this work? Suppose there's a person in one of these simulators. Here's what that would look like, along with the forces acting on the person right after jumping off the fake “ground.”

When the person "jumps," there are only two forces to consider. First, there is the downward gravitational force due to the interaction with the Earth. Second, there is the angled force from the tension in the support cables.

The human is also tilted at some angle—but let's pretend like the "vertical" direction is perpendicular to the support cable. I labeled this direction as the y-axis, which then makes the direction of the cable the x-axis. Since the cable prevents motion in the x-direction, the person can only move in the y-direction (which is like the new vertical direction). That means that only a vector component of the gravitational force will pull that way. Using some basic trigonometry and Newton's Second Law, we can solve for the acceleration in this direction.

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If we want a simulated gravitational field (and free-fall acceleration) of 1.63 m/s2, then the person and floor would need to be leaning 9.6 degrees from being completely horizontal.

You might notice a small problem: If a person jumps up off the tilted floor, then the angle between the cable and the real gravitational force (θ in the diagram above) will also increase. This means the component of the real gravitational force that pulls down towards the fake floor will decrease. You can mostly fix this problem with a long cable. If the cable is 10 meters long, a motion in the y-direction won't change the angle too much, and the fake gravitational force will be mostly constant.

OK, but what if you want to practice running on the moon? In that case, the astronaut-in-training needs to move forward on the tilted floor—but the point where the support cable is attached above the person must also move. It's a little tricky, but it can work. Of course, the biggest drawback to this simulation method is that while the human can move up and down or back and forward, motion to the left or right is impossible, since the length of the cable would have to change.

The Robot Method

There's another reduced gravity simulation that's actually quite similar to the pendulum method. NASA calls this the Active Response Gravity Offload System (ARGOS).

This method also uses a cable to pull up on an astronaut—but in this case the person stands on flat ground with the cable pulling them straight up. The tension in the cable is adjusted so that the net downward force (the cable pulling up and gravity pulling down) is the same as the downward-pulling gravitational force on the moon.

But what happens when a person moves? Well, the support point for the cable is some distance above the human and it moves to match the person's motion. That's where the "robot" part comes in. The system is able to measure not just the person’s position but also their horizontal speed, and it matches this motion with the suspension point of the cables above them. This allows the human to move in all three dimensions—just like they would move on the moon—and practice climbing around on objects like ramps and boxes.

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This is the best way to simulate motion on the moon (or any other reduced gravity situation), but it's not as creative as the pendulum method; a system with long cables seems like something you could build in your own backyard.

The Underwater Method

Couldn't you just put a person underwater to simulate the moon? Yes, that is one option—but it too has some limitations. The basic idea is once again to have an upward-pushing force to reduce the net downward force. Instead of a cable pulling up, this upward force is the buoyancy force due to displaced water. The magnitude of this upward pushing buoyancy force is equal to the weight of the water displaced—that's called Archimedes’ principle. So if a person takes up a certain volume of water, and the weight of that water is equal to the weight of the person, the net force on them would be zero and they would "float."

You can modify this simulation so that a person could walk on the seafloor as if it was the moon. Most humans have a weight that's slightly less than the weight of the water they displace, which means they most likely float towards the surface—but you don’t actually want them to do this. You want them to stand upright on the floor. To do this, you need to add extra weight to the person.

But there are some problems with this setup. The first is that humans breathe. Sure, to make sure your test subject survives underwater, you can add a scuba tank so that they can get air—but their breathing is in fact its own problem. When a person inhales, the size of their lungs increases, and this increases the volume of displaced water. One solution to this problem is to just stick the whole human in a pressurized space suit. That will be more like walking on the moon, and it keeps their breathing volume fairly constant.

But there's another problem, and it has to do with the "center of buoyancy." You might have heard of the "center of mass"—it's like that, but different. The center of mass is a single location in an object (or body) on which you can assume gravity acts. Of course, the gravitational force actually pulls on all parts of the body, but if you use this location, calculations for acceleration and motion will work out just fine.

The location of the center of mass for a human depends on how the mass is distributed. Legs are more massive than arms, and the head is at the top of the body. When you factor in all these things, the center of mass is usually just above the waist, although everyone is different.

The center of buoyancy is also a single location inside the body where you could place a buoyancy force and get the same result as the actual buoyancy force acting on a person. But the center of buoyancy only depends on the shape of an object, not the actual mass distribution. When calculating this force on a person, it doesn't matter that their lungs take up space but have very little mass. This means that a person’s center of mass and center of buoyancy can be—and often are—in different locations.

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Even if the magnitude of the gravitational force and the buoyancy force were equal, having a different location for the center of mass and buoyancy will mean the object (or human) won't be in equilibrium. Here's a quick demonstration that you can try. Take a pencil and place it on a table so that it's pointing away from you. Now put your right and left fingers somewhere near the middle of the pencil and push them towards each other. If you push with equal force with both fingers, the pencil just stays there. Now push towards the tip of the pencil with your right hand and towards the eraser with your left hand. Even if the forces are the same, the pencil will rotate.

This is exactly what happens with the gravitational and buoyancy force on an underwater person. If the gravitational and buoyancy forces push with equal and opposite magnitudes, the person could rotate if their center of mass and center of buoyancy are in different locations.

There's another problem with walking underwater: the water. Here's another experiment. Take your hand and wave it back and forth as though you are fanning some air. Now repeat that underwater. You will notice that in water, it's much harder to move your hand. This is because the density of water is around 1,000 kilograms per cubic meter, but air is just 1.2 kg/m3. The water provides a significant drag force whenever you move. That's not what would happen on the moon, since there's no air. So it’s not a perfect simulator.

But still, this underwater method has an advantage: You could build the floor of a pool so that it looks just like the surfaces you want to explore on the moon.

The Einstein Method

Albert Einstein did much more than develop the famous equation E = mc2, which gives a relationship between mass and energy. He also did significant work on the theory of general relativity, describing the gravitational interaction as a result of the bending of spacetime.

Yes, it's complicated. But from that theory, we also get the equivalence principle. This says that you can't tell the difference between a gravitational field and an accelerating reference frame.

Let me give an example: Suppose you get in an elevator. What happens when the door closes and you push the button for a higher floor? Of course, the elevator is at rest and needs to have some velocity in the upward direction to accelerate upward. But what does it feel like when the elevator accelerates upward? It feels like you are heavier.

The reverse happens when the elevator slows down, or accelerates in the downward direction. In this case, you feel lighter.

Einstein said that you can treat that acceleration as a gravitational field in the opposite direction. In fact, he said there is no difference between an accelerating elevator and real gravity. That's the equivalence principle.

OK, let's go for an extreme case: Suppose the elevator was moving with a downward acceleration of 9.8 m/s2, which is the same value as the Earth's gravitational field. In the reference frame of the elevator, you could treat this as a downward gravitational field from the Earth and an upward field in the opposite direction due to the acceleration. Since these two fields have the same magnitude, the net field would be zero. It would be just like having a person in a box without any gravitational field. The person would be weightless.

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You might already know that this works, because some amusement parks use the equivalence principle to build fun rides like the "Tower of Terror," which is basically a set of seats on a vertical track. At some points, the seats are released and they accelerate downward with a value of 9.8 m/s2. This makes the people in the seats feel weightless—at least for some short amount of time before the car turns horizontally to avoid crashing into the ground (which would be bad).

But if you wanted, you could change this ride from the Tower of Terror to the Tower of Just a Little Scary. Instead of letting the car and its chairs fall with an acceleration of 9.8 m/s2, it could move down with an acceleration of 8.17 m/s2. In the accelerated reference frame of the car, this would be the same as having a downward gravitational field of 9.8 m/s2 and an upward field of 8.17 m/s2. Adding these together gives a net field of 1.63 m/s2 in the downward direction—just like on the moon! You've just built a moon simulator.

This, too, has a problem, though. Dropping a car from the height of a tall building yields only a couple of seconds of simulated moon gravity. That's not much fun. What's needed is a method to accelerate downward with a magnitude of 8.17 m/s2 for a longer period of time.

The solution is: an aircraft. This is a real thing—it's called a "reduced gravity aircraft,” and it can achieve a reduced gravity time interval of over 30 seconds. That's at least long enough to get in some practice moonwalks. My favorite example of this reduced gravity aircraft is from the show MythBusters. As part of their series of experiments showing that people really did land on the moon (yes, people really did), they wanted to reproduce the motion of an astronaut walking on a lunar surface. To do this, they put on some space suits and traveled inside one of these planes.

So to review: You can simulate moon-like gravity on Earth, but which method is the best? At this point, I think the NASA ARGOS robot method is going to give you pretty much everything you need. There's no time limit, and you can move around a surface in all directions, as long as you stay underneath the robot.

Of course, this isn't something that you could do at your house. If you want to try this at home, maybe your best option is to go to the park and play on a seesaw. It's both cheap and relatively safe.

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