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 1.26 km 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. 1.25 km 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 338530 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/cm3 Number days

Explanation / Answer

giive thrust of rocket, T = 338,530 N
initial centripital acceleration a1 = g
final centripital acceleraton, a2 = 40g
time taken = t

also, mass of space station = M
M = 2.76*10^3*pi(1.26^2 - 1.25^2)*10^6*45/4
M = 0.2448415942538*10^10 kg

hence, from moment balance
angular accelration = alpha
I*alpha = T*1260/2
M*(1.25 + 1.26)^2 *10^6 *alpha/4 = T*1260/2
alpha = 5.53*10^-8 rad/s/s
hence

w2^2*1.25*1000 = 40g
w1^2*1.25*1000 = g

w2^2 - w1^2 = 39g/1250
also
w2 = w1 + alpha*t
hence
w1^2 + alph^2*t^2 + 2w1*alpha*t - w1^2 = t^2(alpha^2) + t(2*w1*alpha) = 39g/1250
w1 = sqrt(g/1250)
t = 2943963.8454513733659838749568443 s
t = 34.07365561 days

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