p45_038.pdf

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Chapter 45 - 45.38
38. (a) The mass of the portion of the galaxy within the radius r from its center is given by M =( r/R ) 3 M .
Thus, from GM m/r 2 = mv 2 /r (where m is the mass of the star) we get
v = GM
r
= GM
r
r
R
3
= r GM
R 3
.
(b) In the case where M = M ,wehave
T = 2 πr
v
=2 πr r
GM = 2 πr 3 / 2
√ GM .
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