All of the questions concern a space station, consisting of a long thin uniform
ID: 2077593 • Letter: A
Question
All of the questions concern a space station, consisting of a long thin uniform rod of mass B. 4.4 x 106 kg and length C. 240 meters, with two identical uniform hollow spheres, each of mass D 1.7 x 106 kg and radius E. 74 meters, attached at the ends of the rod, as shown below. Note that none of the diagrams shown is drawn to scale!
(a) Suppose that the station starts out at rest (not rotating). What we want is to get it spinning about an axis passing through its center of mass, at an angular velocity of F. 0.19 rad/s, which is just what’s needed to produce 1-g of artificial gravity at the end points. To achieve this, we use rocket motors to apply a constant force F = G. 4.7 x 106N to each sphere as shown, directed toward the centers of the spheres. How long must the motors fire in order to bring the station from rest up to an angular velocity of F. 0.19 rad/s? Answer: _________.___ minutes
axISExplanation / Answer
Net torque = I alpha
for Given System:
for hollow sphere, Icm = 2 m r^2 / 3
Icm = 2 x 1.7 x 10^6 x 74^2 / 3 = 6206.13 x 10^6 kg m^2
about given axis, I = Icm + m d^
I = (6206.13 x 10^6) + ((1.7 x 10^6)(240/2 + 74)^2)
I = 70187.33 x 10^6 kg m^2
Inet = [ (4.4 x 10^6) (240^2) / 12] + 2 I
= 1.615 x 10^11 kg m^2
(240/2) (4.7 x 10^6) = (1.615 x 10^11) alpha
alpha = 3.49 x 10^-3 rad/s^2
wf = wi + alpha t
0.19 = 0 + (3.49 x 10^-3)t
t = 54.4 sec
Or 0.91 minutes.