Let’s say you want to know what it’s like to walk on the moon. Is there a way to simulate a moonwalk while staying on earth? Oh well. In fact there are several.

But before we get to them, why should walking on the moon be any different than walking on earth?

It’s all about gravity.

There is an attractive gravitational force between all objects that have mass. Since you have mass and the earth has mass, a gravitational interaction pulls you towards the center of the earth. The magnitude of this force depends on the mass of the earth (M_{E}), the distance between you and the earth (which is essentially the earth’s radius R) and your mass (m). There is also a gravitational constant (G).

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

gravity

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 it 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 points to the center of the earth. (For humans, this means “below.”)

Dropping an object into this gravitational field will accelerate it in the same direction at a rate of 9.8 meters per second per second. Some people call *G* “the acceleration of gravity” for this very reason. But if you have *any* Object, falling or at rest, its weight is still the product of its mass and *G*. It doesn’t need to accelerate to have that weight.

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

field

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 when you move to the moon? The Moon is both smaller and less massive than Earth. That is, the gravitational field on the moon’s surface is different from that of Earth. In itself would be a smaller mass *reduction* 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 is more important.

The moon has a mass 0.0123 times that of Earth and a radius 0.272 times. We can use these values to find the gravitational field on the moon.

Gmond

This makes the gravitational field about one-sixth (0.166) that of Earth, or 1.63 N/kg. If you jump or drop something on the moon, it has a downward acceleration of 1.63 m/s^{2}.

OK, so how do we simulate this gravitational field on Earth?

The leverage method

First you would have to do something about this downward-pulling gravitational field. For every 1 kilogram of mass, the Earth pulls down with a force of 9.8 Newtons, while on the Moon it would only pull down with a force of 1.63 Newtons. That means you would have to press *high* on a person with a force of 8.17 newtons per kilogram to give them the feeling of walking on the moon.

One way to provide this upward pushing force would be to use a lever with a counterweight. (For example, French performer Bastien Dausse uses a device to mimic a person’s movement on the moon’s surface.) It’s the same basic idea behind the seesaw at the local playground. It’s essentially a long stick with a pivot between a large mass and a person, like this:

moon lever

Even if there is no straight stick connecting the person to the counter mass, it is still a lever. A lever is one of the classic “simple machines”. It’s basically a kind of beam on a pivot. If you push on one side with a force (which provides the input force), you get another force on the other side (the output force). The value of the output force depends on the input force and the relative distances of the two forces from the pivot.

basic lever

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

fexit

So that’s it: you just have to push down the right side of the lever with some kind of weight and it will push up with the human on the left.

How much mass would you need? This is a function of the human weight (m_{H}g), the length of the two parts of the lever (r_{O} and r_{in}) and the effective vertical acceleration (a_{m}). The effective vertical acceleration would be the same as the free fall acceleration of a human being on the moon.

balancing mass

If I use a human mass of 75 kilograms and lever arms of 2.0 and 0.5 meters, the end mass should be 250 kilograms. But is that really the same as walking on the moon? Well, subjectively it’s not *same*. The device only supports the person at one attachment point, which means they can only walk in circles and not go where they want.

Is the vertical acceleration the same as on the moon? This device does not provide a constant net force. Instead, this force decreases with increasing angle. This creates a small complication. You can see this in the video that the lever is mostly vertical if the performer jumps high enough. At this point he just stays there. Of course, that wouldn’t happen on the moon.