This tool calculates the g force experienced as a matter of rpm or linear speed, radius from the center of rotation and if you supply a mass, it will tell you the amount of force experienced. http://www.calctool.org/CALC/phys/newtonian/centrifugal I'm using it to calculate the roll rotation of the USS Sagan on it's trip to Jupiter. At 7.5 m radius for a deck, it needs to spin between 10.9 and 11 rpms for the crew to feel 1 g. So the crew actually walks on the hull as it travels.
Don't forget about the Coriolis effect. It can throw people off balance if they are accustomed to planetary 1G, and will cause thrown object to appear as if they are traveling along a curved path. The centripetal force calculation is only a small part of the story (the easiest part). The loner the radius of rotation, the less disturbing the side effects. If the radius is three meters, with 1G at ankle height, the "gravity" at head level will be around .7G (depending on the person's height). Talk about feeling light-headed!
From what I've read, the upper limit that just about any human can handle without ill effect is around 1 rpm. The faster you go above that, the less people can handle it. I believe that was one of the main constraints on the space habitat studies of the 70s, using rotation for artificial gravity. So, at 10 rpm, you'd probably need a special crew who can naturally handle that rate, a lot of time acclimatizing to the oddities of rapid rotation, or genetic engineering that makes them less affected by it.
Ok, found a good nasa article: "The decision to provide 1 g to the colonists means they must reside in a rotating environment; the most feasible way to generate artificial gravity. However, in a rotating system there are forces acting other than the centrifugal force which supplies the pseudogravity. Thus, although the inhabitant at rest in the rotating system feels only the sensation of weight, when he or she moves, another force, called the "Coriolis force," is felt. The Coriolis force depends upon both the speed of motion and its direction relative to the axis of rotation. The direction of the force is perpendicular to both the velocity and the axis of rotation. Thus if the person in figure 3-1 jumps off the mid-deck level of the rotating torus to a height of 0.55 m (21.5 in.), because of Coriolis force he would not come straight down, but would land about 5.3 cm (more than 2 in.) to one side. At low velocities or low rotation rates the effects of the Coriolis force are negligible, as on Earth, but in a habitat rotating at several rpm, there can be disconcerting effects. Simple movements become complex and the eyes play tricks: turning the head can make stationary objects appear to gyrate and continue to move once the head has stopped turning (ref. 6). This is because Coriolis forces not only influence locomotion but also create cross-coupled angular accelerations in the semicircular canals of the ear when the head is turned out of the plane of rotation. Consequently motion sickness can result even at low rotation rates although people can eventually adapt to rates below 3 rpm after prolonged exposure (ref. 6). Again a design parameter must be set in the absence of experimental data on human tolerance of rotation rates. Although there has been considerable investigation (refs. 7-20) of the effects of rotating systems on humans the data gathered on Earth do not seem relevant to living in space. Earth-based experiments are not a good approximation of rotation effects in space because most tests conducted on Earth orient the long axis of the body parallel to the axis of rotation. In space these axes would be mutually perpendicular. Also on Earth a spinning laboratory subject still has Earth-normal gravity acting as a constant reference for the mechanism of the inner ear. Although most people can adapt to rotation rates of about 3 rpm, there is reason to believe that such adaptation will be inhibited by frequent, repeated changes of the rate of rotation. This point is important because colonists living in a rotating system may also have to work in a non-rotating environment at zero g to exploit the potential benefits of weightlessness. For a large general population, many of whom must commute between zero g and a rotating environment, it seems desirable to minimize the rotation rate. There is a lack of consensus in the literature and among experts who have studied the problem on the appropriate upper limit for the rotation rate (refs. 21-28). For the conditions of the space colony a general consensus is that not more than several rpm is acceptable, and for general population rates significantly greater than 1 rpm should be avoided. Therefore, 1 rpm is set as the upper limit of permissible rotation rate for the principal living quarters of the colonists, again reflecting the conservative design criteria." The jury is still out. They haven't conducted tests in the right environment. If I reduce to 3 RPMs then the radii will have to be 100m. I'm projecting the sail to be 15km a side, so the Sagan still won't eclipse it and the backside of the Sagan can be solar panels. Also, the Sagan is doomed to crash into fantasy (aka, a very advanced alien ship), so I'm not going to put too much into it. May eliminate the sail. It's not going to add much due to the mass it's pulling.
Thanks. That's probably the study I was thinking of. Note that, even if you need a 100m radius, it may not need to be a solid 100m cylinder; you could have habitation areas on a boom, or even a tether, particularly if they're stowed during orbit changes so they're not exposed to much force from the engines.
I solved this problem by creating a Torus planet several kilometers in diameter, with the total surface area close to that of the Earth. Also, it is not a natural "rock" planet but instead synthetically engineered with a Tokamak inside. So, uhm, yeah. Sci-Fi. However! The genetically engineered humans living there have no memory of their past, so it's just normal fantasy. P.S. And there are lots of resources like this.
I don't even need to pick up a calculator to know the math won't work. Yeah, you could get that kind of surface area with fractal math, but it still wouldn't come close to that much livable space for 1.5-2.5 meter tall humanoids. Your toroid did remind me of an artifact created by Pak-Brennan in Larry Niven's Protector, although the surface area was nowhere near what you are positing.
I meant the minor diameter not the major one. And also I meant several thousand km. Trust me, the surface area is bigger than that on earth. I tried to do it right. ;-)
