Space station mathematics question

[Hoichi]

Ok, I am writing a comic book that takes place on a city sized space station. I have planned for somewhere between 500,000- 1 million people on this space station. It's largely a terra forming/farming project. it will be shaped like a giant rolling pin that has the diameter of roughly a mile or so and will be as long as necessary to hold all the people. How would I calculate the spinning speed to create earth-like gravity?

 

[log0]

Hi Hoichi

Centripetal acceleration is
a_z = r * w²

Your parameters are:
a_z = 9.81 m/s²
r = 1/2 mile = 805 m

The angular velocity is:
w = sqrt(a_z/r) = 0.11 rad/s

Rotations per second:
f = w / (2 * pi) = 0.0176 Hz (about 1 rotation per minute)

Tangential velocity:
v_T = w * r = 89.87 m/s

 

[Hoichi]

Thank you very much, is there anything I may not be thinking about that I should keep in mind?

 

[Andy Resnick]

the Coriolis force and (radial) gradient of centripetal force could make everyone sick from disorientation. Think about what will happen when someone jumps 'up', or out of a window.

 

[Hoichi]

I was kind of hoping these issues would possibly have less of an impact because of it's larger scale. Maybe I should just go the Star Wars route and just leave it unexplained why there is gravity.

 

[log0]

The Naval Medical Research Lab did some experiments in 1958 (see Slow Rotating Room). They showed that human can adapt to a rotation of 3-4 rpm within 3-4 days. More recent investigations show fast adaptability(within minutes) up to 10 rpm if the subjects repeat some voluntary movements so that the nervous system can anticipate the coriolis forces (see History of Artificial Gravity).

This point might make your story more immersive as visitors would have motion sickness entering the habitat (would have to make exercises to adapt).

PS:
For smaller space stations(ships) you might consider using a tether to generate centripetal acceleration. A crew module attached to the engine section by a 1-10 km cable rotating around center of mass would look quite spectacular I think.

Gemini 11-12 artificial gravity experiments:

 

[Cleonis]

Thank you very much, is there anything I may not be thinking about that I should keep in mind?

As Andy Resnick remarked, you may need to take Coriolis effect into account.

The largest speeds will be in using motorized transport.
Will there be buildings that are several tens of floors high? Let's say you're in an elevator moving at 5 meters per second. Using the letter 'w' to denote the angular velocity of the space habitat and 'v' to denote the velocity relative to the structure: the magnitude of the Coriolis effect is 2*w*v, so if the elevator travels at 5 m/s the magnitude of the Coriolis effect is an acceleration of about 1 m/s^2, which is about 1/10th G.

If you ride that elevator it's well advised to position yourself against the wall that you'll be pushed towards. If a group of people makes a mistake, positioning themselves at the wrong end of the elevator then when the elevator gets up to speed they will probably tumble over like bowling pins.

The tricky part is that the direction of the effect depends on whether the elevator is going up or down.

 

[ynot]

Get Rendezvous with Rama by Arthur C. Clarke. I think he covered the physics quite well, with changes in apparent gravity caused by the rotation of a spaceship almost exactly like your story.

 

[Hoichi]

I went ahead and sketched out how I saw the Spacestation in my mind and thought I'd post it.

 

[log0]

Looks cool. So the crew modules are attached by tethers to the central spindle. What are the dimensions of the spindle?

Judging from the picture the tether length would be equal to the spindle diameter. Given an overall diameter of 1 mile I guess 1/3 mile spindle diameter and 1 mile spindle length.

 

[Rasalhague]

Theodore W. Hall has a great website with papers on this subject and a calculator for working out, say, angular and tangential velocity for a given radius and centripetal acceleration. You're right that, for a given centripetal acceleration (i.e. strength of artificial gravity), a bigger radius means less extreme Coriolis effect on human scales. The calculator rates a half mile radius and 1g acceleration with in the comfort zone, as judged by five studies from 1962 to 1985.

http://www.artificial-gravity.com/
http://www.artificial-gravity.com/sw/SpinCalc/SpinCalc.htm

 

[Bob S]

Thank you very much, is there anything I may not be thinking about that I should keep in mind?

On Earth, our food plants grow roots downward, and the plant stems grow upward. In your terra-forming project spinning at about 1 RPM, what direction will the roots and stems grow? Does gravitropism dominate roots and phototropism dominate stems? How does phototropism work when the space station is rotating at 1 RPM and the centrifugal force is outward? Describe the corn crop on (in) your farm..

Bob S