From: spike66 (spike66@attbi.com)
Date: Mon Feb 17 2003 - 21:45:28 MST
Damien Broderick wrote:
> spike66 says before departing over the hill on his hog with a red rose
> clenched between his teeth:
OK Im back. Weather didn't cooperate, had to drive
the Detroit. Had a good time anyway.
>>Think of a long
>>thin rod as two rods attached end to end, with a closer
>>rod and a farther rod. The Earth attracts the closer
>>rod stronger....a gravitational torque results that tends to
>>pull the rod to vertical.
>
> Yeah, yeah, but... The long lower rod in this case is not just *sitting*
> there to begin with. The higher end of the lower rod is moving like a bat
> out of hell (compared with what its lengthening tip wants to do), and
> therefore (to start with) so is that lower end. Grav torque drags it
> backward, but still the brute is roaring upward toward the central mass's
> orbit, not just sitting there placidly at 45 degrees.
Yes yes, my good man, but please draw up a free body diagram.
First let me review my original argument. Imagine an
asteroid in GEO, tidelocked so that one face stays
always toward the earth. A space elevator is created
by the mechanism Damien has suggested in his latest
story, wherein nanobots on the surface of a GEO
asteroid convert the material into a continuous
cable building it downward toward the surface of
the planet and upward toward space to counterbalance
the elevator.
The earth's sidereal rotation rate is one revolution
per day (or rather 23 hrs 56 minutes) and so is the
asteroid. The moment of inertia of a sphere is 2/5MR^2
and the moment of inertia of a long thin rod is 1/12ML^2.
If one assumes that all the mass of the Broderick-esque
asteroid (Broderic-inspired? Damienian?) ends up in a
rod 70,000 km in length, starting with a 1 km diameter
sphere, the moment of inertia is increased by a factor
of right at a billion. Since angular momentum is
conserved, one would need to start with a rotation
rate of a billion revolutions per day to end up with
a synchronous cable, which presents its own engineering
challenges.
Robert has suggested spinning up the cable as it is
built using propulsion, which would require a great
deal of propellant.
Alternatively, if the cable is 45 degrees from vertical,
the earth's gravity will apply a torque, which will
supply angular momentum.
Let theta (t) be the angle from vertical and let the
X axis be along the length of the rod with the origin
at the center of mass of the rod. Let p be the mass
density and A be the cross sectional area. Let Rg
be the distance to geosynchronous orbit and let w be
my best approximation of an omega with this keyboard.
The mass of the differential is then pA(dX) so the
expression for the torque on the rod as a function
of theta is
Torque = integral{MGpAdX/(Rg + Xcos(t))^2*Xsin(t) +
pAdx(w^2)(Rg + Xcos(t))(Xsin(t)dX}
The integral is from -L/2 to L/2.
Some of you math guys check my work while I go
off and integrate this function while watching
the finale of Joe Millionaire.
spike
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