A launch loop can harvest energy and momentum from the rotor to accelerate payloads, but I don't see any such mechanism here.
Which is why I say I “in this design, some of that momentum would be borrowed from the Earth’s rotation via the cable’s coupling to its magnetic field.” The cable is an electrostatic counterweight because we’re using electromagnetism, not the comparably weak gravitation.
Where is your math?
The top of the elevator is travelling at orbital velocity. This is trivial to show in designs with a counterweight. (Here, the magnetic coupling makes it less intuitive.) If you are on an orbiting object, i.e. the top of a space elevator, you’ve achieved orbital velocity.
>The top of the elevator is travelling at orbital velocity. This is trivial to show in designs with a counterweight.
Per the paper this design only reaches 200 km in altitude, therefore it has no counterweight (a counterweight would need to be somewhere above 35,786 km altitude). Speed at the top is far below orbital velocity, so it requires a method of acceleration.
The paper acknowledges this. From the abstract:
"At the top of the loop, vehicles may be accelerated to orbital velocity or higher by rocket motors, electromagnetic propulsion, or hybrid methods."
https://futureboy.us/fsp/frink.fsp?fromVal=2+pi+%28+%28earth...
Depending on the height of the space elevator, the speed of its top will be smaller, equal or greater than the speed required at that height for a stable circular orbit.
The top of a space elevator will have the same angular velocity as the Earth. The angular velocity of an orbiting object is equal to that of the Earth only when it is on a geosynchronous orbit (i.e. an extremely high orbit in comparison with those of most satellites or in comparison with the height of the space elevator from this proposal).
In order to launch a satellite from a space elevator without additional acceleration, it is not necessary for its height to be that of a geosynchronous orbit.
For smaller heights, any object released from the top will fall towards the Earth on an elliptical orbit. If the height is big enough, the elliptical orbit will not intersect the solid Earth or the atmosphere of the Earth. Nevertheless, the minimum height for this is still on the order of a few tens of thousands of km, i.e. at least 100 times the height of the space elevator from this proposal.
Momentum transfer from the cable, which is attached to an orbiting counterweight.
In this design, some of that momentum would be borrowed from the Earth’s rotation via the cable’s coupling to its magnetic field. In general one boosts the counterweight directly or, more practically, by sending things down [1].
[1] https://space.stackexchange.com/questions/22447/how-will-the...