How much would you weigh on the ISS?

If you’re reading this blog then chances are you’ve seen at least one video of an astronaut doing something cool on the International Space Station. In all of these videos, it looks like there is no gravity, right? Let’s plug some numbers into Newton’s universal law of gravitation and see if that is actually the case.

Newton famously discovered* that objects with mass attract each other via a force whose strength is inversely proportional to the distance between the objects squared. That means that if the distance between two objects is doubled, the force of gravity between them will be one-fourth as strong. The complete equation for calculating the gravitational force between two objects is:newtons_universal_law_of_gravitation_equation

Where here m1 and m2 are the masses of the two objects in consideration (in this case, you and the Earth), r is the distance between the objects measured from their centers, and G is a special constant of the universe called the gravitational constant. So some useful values for us to consider are:

Quantity Symbol Value
Mass of the Earth me 5.972 x 1024 kg
Radius of the Earth re 6.371 x 106 m
Gravitational constant G 6.674 x 10−11  N · (m/kg)2
Distance to the ISS yISS 400 km
Mass of a typical person mp 80 kg

So if we plug all of these values into the equation to calculate the force of gravity acting on a person on the ISS, it comes out to about 696 N. By comparison the same person standing on the surface of Earth would experience about 786 N of gravitational force. Given my penchant for making graphs, it may come as no surprise that I plotted the force of gravity versus distance away from the surface of Earth using the numbers from the table above. Here’s the result:

F_gvsy_spacestation

All of this means that there is only about 10% “less gravity” on the ISS compared to standing on Earth  far from zero gravity. So what gives!?

The answer to the question “why do astronauts on the ISS appear weightless?” is not all that complicated, but it defies the intuition that many people have. Check out this image from the Newton’s cannonball Wikipedia page:

Newton_Cannon

This image does a great job of explaining what it means to be in orbit. When something is in orbit (such as the astronauts on the ISS), it is really just falling toward the Earth. However it is also moving sideways fast enough that instead of hitting the Earth it misses and continues to fall. Look at the different paths in the picture above: paths A and B represent an object that wasn’t moving quite fast enough to get into orbit, it was falling and hit the Earth; paths C and D represent objects that are in orbit, they are constantly in free fall towards the Earth but they are moving sideways so fast that they never actually hit it; path E represents an object moving sideways so fast that it just flies away from Earth and therefore doesn’t go into orbit. If you want to play around with this, there are lots of interactive Newton’s cannon demos available such as this one.

So in fact the astronauts on the ISS are just constantly in free fall. This means that they don’t experience any contact forces of objects pushing on them (like we do when we’re standing on the ground and it’s pushing up on us). If you’ve ever been on an amusement park ride where you’re dropped to the ground, you experience the same sensation of weightlessness that astronauts on the ISS do, but the difference is it only lasts for a few seconds before the ground gets in the way.

Kings_Dominion_Drop_Tower_falling

Here is the Python source code to generate the plot in this post. You will need Python 2.7, NumPy, and matplotlib for the script to run.

*Note that the Newtonian theory of gravity is actually an approximation that has since been supplanted by the general theory of relativity. However for the majority of simple calculations such as the one discussed here, the Newtonian theory is a very good approximation and therefore good enough to get the job done.

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5 thoughts on “How much would you weigh on the ISS?”

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