Let’s just make it simple: To achieve an artificial gravity of 0.5 g, you’ll need a radius of 450 meters and a spacecraft-to-counterweight distance of twice that (900 meters).
Just for fun, the Wikipedia page lists the tether distance at 450 meters. This would give a rotational radius of 225 meters. Using the same angular velocity, the astronauts would have an artificial gravity of just 0.25 g’s.
I mean, that’s not terrible. In fact, the gravitational field on Mars is 0.38 g’s, so this would be almost good enough for the astronauts to prepare for work on Mars. But I’m going to stick with my artificial gravity of 0.5 g’s and a tether length of 900 meters.
What Would It Be Like to Slide Down a Tether?
Without going into too much detail, let’s consider what would happen if an astronaut was going to climb one of the cables from the spacecraft to the counterweight on the other side for some reason. Maybe life’s just better on the other side—who knows?
When the astronaut starts up the cable (I’m calling “up” the direction that’s opposite the artificial gravity), physics dictates that they will feel the same apparent weight as the other astronauts on the spacecraft. However, as they get higher on the cable, their circular radius (their distance from the center of rotation) decreases, making the artificial gravity also decrease. They would keep feeling lighter until they got to the center of the tether, where they would feel weightless. As they continued their journey to the other side, their apparent weight would start to increase—but in the opposite direction, pulling them toward the counterweight at the other end of the tether.
But that’s not very exciting for a movie. So here is something very dramatic instead. Suppose an astronaut starts near the center of rotation with very little artificial gravity. Instead of slowly climbing “down” the tether, what if she just lets the fake gravity pull her down? How fast would she be going when she gets to the end of the line? (This would sort of be like falling on Earth, except that as she “falls,” the gravitational force would increase as her distance from the center does. In other words, the further she falls, the greater the force on her.)
Since the force on the astronaut changes as she moves down, this becomes a more challenging problem. But don’t worry, there’s a simple way to get a solution. It might seem like a cheat, but it works. The key is to break the motion into tiny pieces of time.
If we consider her motion during a time interval of just 0.01 seconds, then she doesn’t move very far. This means that the artificial gravity force is mostly constant, since her circular radius is also approximately constant. However, if we assume a constant force during that short time interval, then we can use simpler kinematic equations to find the position and velocity of the astronaut after 0.01 seconds. Then we use her new position to find the new force and repeat the whole process again. This method is called a numerical calculation.
If you want to model the motion after 1 second, you would need 100 of these 0.01 time intervals. You could do this calculation on paper, but it’s easier to make a computer program do it. I will take the easy way out and use Python. You can see my code here, but this is what it would look like. (Note: I made the size of the astronaut larger so you could see her, and this animation is running at 10x speed.)
For this slide down the cable, it takes the astronaut around 44 seconds to slide with a final speed (in the direction of the cable) of 44 meters per second, or 98 miles per hour. So, this is not a safe thing to do.