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Re: rotating space station



Think about what happens in the limit that the release point
approaches the center of the station. In that limit, the ball
never hits the "ground." Now, unless you think that limit is
approached discontinuously, the implication is that the ball must
fall further and further behind the astronaut's feet as the height
of the release point increases.

Oops, I thought of this limit after I had already posted my message
and left for the day. Quite right, as Skip and John pointed out, I
forgot to reduce the tangential speed to omega*(R-h).

Indeed, good explanations of this fact exist for both the inertial
and rotating observer. In the inertial frame, the distance
traveled by the ball is sqrt(2Rh+h^2) where h is the height of the
release and R is the radius of the space station and the speed of
the ball is omega*(R-h) so the time it takes the ball to fall is

srqt[ 2*(h/R) + (h/R)^2 ]
t_b = -------------------------
omega*(1 - h/R)

(Notice what happens as h/R --> 1.)

To reach the place where the ball will fall, the astronaut's feet
must move through an angular displacement of arccos(1 - h/R), so
the time it takes the feet to get there is

arccos(1 - h/R)
t_f = ---------------
omega

Now it is not *too* difficult to show that t_f < t_b for all h >
0. Perhaps the easiest way to see this is simply to plot

srqt[ 2*(h/R) + (h/R)^2 ]
t_b/t_f = -------------------------
(1 - h/R)*arccos(1 - h/R)

as a function of h/R and note that it is a monotonically
increasing function. Thus, the ball lands behind the feet.

Another form which makes it a little easier to see the difference is:

t_b = tan(theta)/omega
t_f = theta/omega

left as a nice homework problem, where theta is the angle (in rad)
the station has to rotate through, derived under the assumption that
h < R. So we see the ball falls very nearly at the feet of a short
person, but behind a person standing on a tall ladder. Thanks to all
on this marvelous list for the enlightenment as usual! Carl
--
Dr. Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Annapolis, MD 21402-5026 mailto:mungan@usna.edu
http://physics.usna.edu/physics/faculty/mungan/