Artifical gravity is a must for any space station if humans are to live there fo

Question

Artifical gravity is a must for any space station if humans are to live there for any extended length of time. Without artificial gravity, human growth is stunted and biological functions break down. The most effective way to create artificial gravity is through the use of a rotating enclosed cylinder, as shown in the figure. Humans walk on the inside edge of the cylinder, which is sufficiently large that its curvature is not a factor. The space station rotates at a speed designed to apply a radial force on its inhabitants that mimics the normal force they would experience on Earth. A danger of the space station is if an entity (whatever that may be) decides to increase the rotational speed while humans are inside.Humans typically can withstand accelerations up to 40.0g. Suppose alien creatures install a rocket possessing 360650 N of thrust to the outside of the rotating space station, as shown. How long would it take (in days) for the artificial gravity to exceed 40.0g? Assume that the artificial gravity was g before the rocket was installed, and that the inside and outside diameters of the space station are 1.25 km and 1.26 km (respectively), its height is 65.0 m, and that the station is largely aluminum alloy of density 2.60 g/cm3.

Explanation / Answer

Okay, so we need to find the moment of inertia of the space station using the density. First, find the volume of the outer cylinder then subtract the volume of the inner cylinder and multiply this by the density. Remember to change the density to kg/m^3 and to change the diameter given to radius. This will give the mass, next find the moment of inertia using 1/2m(r1^2+r2^2) again remembering to change the diameter given to radius and giving the appropriate units. This should give something on the order of 1.4e15. Now using I*alpha = F*r where r is the radius of the outer circle, you should be able to get alpha to be 1.63-7. Now we can calculate the angular velocity at g using a = w^2*r where r is radius of inner. we can also use this to calculate the needed omega. This should give current omega as .12528367 and needed as .7923635529. Now using w = w0+alpha*t, we get t to be on the order of 4100000 s. In days, this is about 47.3 days

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