Artificial gravity is a must for any space station if humans are to live there f

Question

Artificial 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 40.0 g. Suppose alien creatures install a rocket possessing 250050 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 45.0 m, and that the station is largely aluminum alloy of density 2.76 g/cm^3.

Explanation / Answer

given

Ri = 1.25 km = 1250 m

Ro = 1.26 km = 1260 m

h = 45 m

volume of cyllinder, V = pi*(Ro^2 - Ri^2)*h

= pi*(1260^2 - 1250^2)*45

= 3.548*10^6 m^3

mass of the cyllinder, m = volume*density

= 3.548*10^6*2760

= 9.792*10^9 kg

moment of Inetrtia of the cyllinder about its center, I = (1/2)*m*(Ri^2 + Ro^2)

= (1/2)*9.792*10^9*(1250^2 + 1260^2)

= 1.54*10^16 kg.m^2

let w is the angular speed required.

a_rad = Ri*w^2

40*g = Ri*w^2

==> w = sqrt(40*g/Ri)

= sqrt(40*9.8/1250)

= 0.56 rad/s

let alfa is the angular acceleration.

now use, torque, T = I*alfa

==> alfa = T/I

= Ro*F/I

= 1260*250050/(1.54*10^16)

= 2.046*10^-8 rad/s^2

let t is the time taken.

use, wf = wi + alfa*t

==> t = (wf - wi)/alfa

= (0.56 - 0 )/(2.046*10^-8)

= 2.737*10^7 s

= 2.737*10^7/(24*60*60)

= 317 days <<<<<<<<<<<<------------------------Answer

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