All of these questions concern a space station, consisting of a long thin unifor

Question

All of these questions concern a space station, consisting of a long thin uniform rod of mass 3.7x 10^6 kg and length 621 meters, with two identical uniform hollow spheres, each of mass 2.8 x 10^6 kg and radius 188 meters, attached at the ends of the rod, as shown below. Note that none of the diagrams shown is drawn to scale!

Now, another feature of this station is that the rod shaped section can change its length (kind of like an old fashioned telescope), without changing its overall mass and remaining uniform in its density. Suppose that, however it was accomplished, the station is now rotating at a constant angular velocity of 0.12rad/s. If the length of the rod is reduced to 383 meters, what will be the new angular velocity of the space station?

(c) Now, another feature of this station is that the rod-shaped section can change its length (kind of like an old-fashioned telescope, without changing its overall mass and remaining uniform in its density. Suppose that, however it was accomplished, the station is now rotating at a constant angular velocity of F rad/s, If the length of the rod is reduced to K. meters, what will be the new angular velocity of the space station? Answer: rad/s

Explanation / Answer

initial angular velcity= w1=0.12 rad/s

initial moment of inertia=i1= ml2/12 + 2 [(2/3)mr2+ m(l/2 +r)2]

i1=(3.7x106 (621)2)/12 + 2 [(2/3)(2.8x106)(1882)+ (2.8x106)(621/2+188)2]

i1=1.189x1011+1.524x1012

i1=1.643x1012kgm2.

final mment of inertia= ml2/12 + 2 [(2/3)mr2+ m(l/2 +r)2]

i2=(3.7x106 (383)2)/12 + 2 [(2/3)(2.8x106)(1882)+ (2.8x106)(383/2+188)2]

i2=4.52x1010+9.38x1011

i2=9.83x1011kgm2.

using law of conservation of angular momentum,

i1 w1=i2 w2

1.643x1012 (0.12)=9.83x1011 (w2)

w2=0.21 rad/s

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