Consider a space habitat that consists of a rotating cylinder. If the variation in g between one's head and feet is to be less than 1/105 g, then compared to one's height, h, what should be the minimum radius of the space habitat? (in terms of h)
The centripetal acceleration is proportional to the distance from the spin axis, r. If the feet are at r, then the feet of a person of height h would be at at 104/105 r = .9905 r
(r-h)/r = 0.99048
h/r = 1 - .99048 = .00952
R = 105 h
To find the minimum radius of the space habitat in terms of height (h), we can use the principle that the variation in gravitational acceleration (g) between one's head and feet should be less than 1/105 g.
In a rotating cylinder space habitat, the centripetal acceleration is proportional to the distance from the spin axis, represented by the radius (r). If we assume the feet are at a distance r, then the feet of a person with height h would be at a distance of 104/105 r = 0.9905 r.
Using the fact that the gravitational acceleration (g) is inversely proportional to the square of the distance from the center of mass, we can define the ratio of gravitational acceleration at the head to that at the feet as (r-h)/r.
Setting this ratio to be less than 1/105, we can equate the expression to 0.99048:
(r-h)/r = 0.99048
Simplifying this equation, we have:
h/r = 1 - 0.99048 = 0.00952
Now, let's solve for the minimum radius (R) in terms of h:
R = 105 h
Therefore, the minimum radius of the space habitat is 105 times the height (h).