In designing rotating space stations to provide for artificial-gravity environments, one of the constraints that must be considered is motion sickness. Studies have shown that the negative effects of motion sickness begin to appear when the rotational motion is faster than approximately 2 revolutions per minute. On the other hand, the magnitude of the centripetal acceleration at the astronauts' feet should equal the magnitude of the acceleration due to gravity on earth. Thus, to eliminate the difficulties with motion sickness, designers must choose the distance between the astronaut's feet and the axis about which the space station rotates to be greater than a certain minimum value. What is this minimum value?

The maximum angular velocity w is

2 rpm*2 pi(rad/rev)/60 (sec/min) = 0.209 rad/s

R w^2 = g, with the requirement that w < 0.209 rad/s

w^2 = g/R < (0.209 rad/s)^2 = 4.39*10^-2

R > 9.8/4.39*10^-2 = 220 m

I don't think there are currently any plans for a space station that large.

The Russia/USA space station ISS, which is currently up there, does not rotate. The one depicted in the movie 2001 was supposed to produce only 1/6 the acceleration of gravity. For more about rotating space stations real and fictional, see

http://en.wikipedia.org/wiki/Rotating_wheel_space_station

To determine the minimum value for the distance between the astronaut's feet and the axis of rotation in order to eliminate motion sickness, we can use the following formula:

a = ω²r

Where:
a = centripetal acceleration
ω = angular velocity (in radians per second)
r = distance between the astronaut's feet and the axis of rotation

To eliminate motion sickness, the centripetal acceleration (a) should be equal to the acceleration due to gravity on Earth (9.8 m/s²). Let's assume the angular velocity is ω = 2 revolutions per minute.

First, we need to convert ω from revolutions per minute to radians per second. We know that 1 revolution is equal to 2π radians, and there are 60 seconds in a minute:

ω = (2 revolutions/minute) * (2π radians/revolution) * (1 minute/60 seconds)
≈ 0.2094 radians/second

Now we can rearrange the formula:

a = ω²r

And substitute the values:

(9.8 m/s²) = (0.2094 radians/second)² * r

Solving for r:

r = (9.8 m/s²) / (0.2094 radians/second)²
≈ 111.67 meters

Therefore, the minimum distance between the astronaut's feet and the axis of rotation should be approximately 111.67 meters in order to eliminate motion sickness.

To determine the minimum value for the distance between the astronaut's feet and the axis of rotation in order to eliminate motion sickness, we need to consider the relationship between centripetal acceleration, rotational speed, and the distance from the axis of rotation.

Centripetal acceleration (aₙ) is given by the formula:

aₙ = (v²) / r

where:
- v is the linear velocity of the astronaut
- r is the radius or distance from the axis of rotation to the astronaut's feet

The linear velocity (v) can be expressed in terms of the rotational speed (ω) using the formula:

v = ω * r

where:
- ω is the rotational speed in radians per minute

We want the centripetal acceleration (aₙ) at the astronaut's feet to be equal to the acceleration due to gravity (g). Therefore, we can set up the following equation:

aₙ = g

Substituting the expressions for aₙ and v:

(v²) / r = g

((ω * r)²) / r = g

(ω² * r) = g

This equation represents the relationship between the rotational speed (ω) and the radius (r) that satisfies the condition for eliminating motion sickness.

However, we don't have a specific value for either ω or g. We know that motion sickness occurs at rotational speeds above approximately 2 revolutions per minute, but the specific value of g depends on the gravitational field strength. On Earth, the acceleration due to gravity is roughly 9.8 m/s².

To find the minimum value for r, we need to make a few assumptions. Let's assume:

- ω = 2 revolutions per minute (or 2 * 2π radians per minute)
- g = 9.8 m/s²

Substituting these values into the equation:

((2 * 2π)² * r) = 9.8

(4π² * r) ≈ 9.8

r ≈ 9.8 / (4π²)

Using a calculator, the minimum value for r is approximately 0.2466 meters (or 24.66 centimeters).

Therefore, to eliminate motion sickness, designers must choose a distance greater than approximately 0.2466 meters between the astronaut's feet and the axis of rotation.