(Thus, the total force on the tether would be 100 tons.)
Forces in cables don't work like that. If the cable has negligible mass (for this application, that's not a bad approximation), the tension at every point is equal, so the tension is 50 tons everywhere. If the weight's not negligible, the tension is highest in the center, because the mass of the cable adds more tension, but this could be greater than, less than, or equal to 100 tons depending on the cable (in fact for reasonable cables it will be much closer to 50 tons than 100 tons).
To get acceptable speed/radius, I used Theodore Hall's SpinCalc, and set the rotation rate to 3 RPM. This yields a radius of 100m. (Most sources suggest 3RPM is ok, one source suggests 2 RPM, which would not about double the material requirements.)
So for actual numbers for actual materials (sorry, I can't be arsed to get numbers for carbon nanotube, because we can't make serious cables of that at present), we need a steel cable* that can support 50 tons. Actually, we want more like 50 steel cables that can support over (let's say 10% over) 1 ton each, so we can make a cross-linked structure, to reduce vulnerability to meteoroid strikes. (I'd think something like a hyperboloid tower would work nicely, where you have 25 lines slanting to the left and 25 to the right, and they're fastened everywhere a left and a right line cross.)
Casting about the internet for a suitable cable, we find this, with 2240 pound working load and weight of 0.18 pounds/ft. We need 50x200m = 10km = 33000 ft., so 6000 pounds. Adding in something for the fasteners to make the cross-linked structure, maybe 4 tons.
Given 4 tons of cable etc., it should be obvious the extra tension due to the cable structure's own weight is less than 2 tons (2 instead of 4 because half of it is on each side of center) -- after all, the centripetal acceleration is a maximum of 1g at the outside, but drops off to zero at the center. In fact, since acceleration varies linearly with radius, the average acceleration over the cable is 0.5g, and tension varies from 50 tons at each end to 51 tons at the center.
(Of course one would need to analyze the actual layout of the cable structure proposed, to ensure that cutting any one segment would in fact redistribute tension among the intact cable segments such that none exceeds the working limit -- the 10% (now diminished by the 2% increase in cable tension) margin was just a guess. If we need more strength, use more strands and/or heavier cable.)
*Why a steel cable? Of course NASA can do much better, with a cable of Kevlar or some such fiber, and save valuable mass. But boring old steel cable is what I'm most confident estimating with, so it's what I ran the actual numbers with as well.