Stanford Torus vs Bernal Sphere

On Mon, 18 Oct 2004 09:09:39 -0500, Combs, Mike wrote:
> In a Stanford Torus, one might possibly have a spiral staircase or
> ladder in the spokes adjacent to the elevator shafts. But that would be
> a ladder half a mile long, and lord knows how much longer the spiral
> staircase would be. People stopping from exhaustion would be creating
> traffic jams (people ascending to the spin axis of a Bernal Sphere would
> be coming in radially from all directions). Maybe if you were losing
> weight with each step, it might not be as unmanageable as it would seem
> from our Earthly perspective. But walking up a much shorter hill sure
> sounds a lot easier.

It's fairly easy to work out how much effort it is to climb.

A climbers 'g' is directly proportional to the distance from the axis.

The energy the climber must expend varies as the g-force varies
linearly with distance from the axis.

Let distance from the central axis be 'r' and the rim radius of the
habitat be 'h', m is the climbers mass, w is the rotation rate of the
habitat in radians per second (i.e. 2 Pi times the rotation rate per
second)

The force on the climber at any point in the habitat is f=m r w^2
_ _
energy = force times distance = _| f dr = _| mrw^2 dr

= [0.5 m r^2 w^2] (0 to h) = 0.5 m h^2 w^2 - 0

Separating out 'g' (which happens to be the acceleration at rim
divided by m = m h w^2/m = hw^2)

= 0.5 m g h

This compares with the earth-based equation over the same distance
which is just mgh.

In other words, the energy you need to expend to climb from the rim to
the axis is half that of climbing the same distance at 1g. So,
climbing 1km vertically to the axis in a habitat is the same as
climbing 500 m on earth. Tiring, but fit people should have no great
problem.

A 70kg man climbing up a 1km radius habitat exerts as much as climbing
500m on earth = mgh = 70 x 9.81 x 500 = 343 kJ = 82 kCals

However peoples muscles aren't particularly efficient so you'd
actually need to eat several times that to regain the energy you lost
from the climb. I forget the inefficiency ratio. Multiply by a few.

This 'half the height' rule only works from the rim to the axis.
Obviously walking upstairs one floor isn't twice as easy, since the
gravity hardly varies.

The general rule if you climb one floor, or from say 3/4 of the rim
radius, to 1/3 way up or whatever, is to use the gravity at the
halfway point between the two points and multiply by the distance
climbed. (Proof left as an exercise to the reader.)

> Regards,
>
> Mike Combs

--
-Ian Woollard

"In theory there is no difference between theory and practice, but in
practice there is."