Humans typically can withstand accelerations up to 40.0g. Suppose alien creature

Question

Humans typically can withstand accelerations up to 40.0g. Suppose alien creatures install a rocket possessing 316410 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 2.72 g/cm3.

Explanation / Answer

Here ,

outer radius , r1 = 1260/2

r1 = 630 m

inner radius , r2 = 625 m

mass of the space station , m = volume * density

m = pi * 65 * (630^2 - 625^2) * 2690 Kg

m = 3.446 *10^9 Kg

moment of inertia of station , I = 0.5 * m (r1^2 + r2^2)

I = 0.5 * 3.446 *10^9 *(630^2 + 625^2)

I = 1.357 *10^15 Kg.m^2

Now , let the initial angular speed is wi

a = wi^2 * r

9.8 = wi^2 * 625

wi = 0.125 rad/s

final angular speed is wf

40 * 9.8 = wf^2 * 625

wf = 0.792 rad/s

NOw , let the time taken is t

using second law of motion

change in angular momentum = torque * time

1.357 *10^15 * (0.792 - 0.125) = 537610 * 630 * t

t= 2.672 *10^6 s

t = 30.9 days

the time taken to increase the acceleration to 40g is 30.9 days

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