12.72:
a) The radii
2
1
and
R
R
are measured with respect to the center of mass, and so
.
and
,
1
2
2
1
2
2
1
1
M
M
R
R
R
M
R
M
=
=
b) If the periods were different, the stars would move around the circle with respect
to one another, and their separations would not be constant; the orbits would not remain
circular. Employing qualitative physical principles, the forces on each star are equal in
magnitude, and in terms of the periods, the product of the mass and the radial
accelerations are
.
4
4
2
2
2
2
2
2
1
1
1
2
T
R
M
T
R
M
π
=
π
From the result of part (a), the numerators of these expressions are equal, and so the
denominators are equal, and the periods are the same. To find the period in the symmetric
from desired, there are many possible routes. An elegant method, using a bit of hindsight,
is to use the above expressions to relate the periods to the force
,
2
2
1
2
1
)
(
GM
g
R
R
M
F
+
=
so that
equivalent expressions for the period are
G
)
R
(R
R
π
T
M
2
2
1
1
2
2
2
4
+
=
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Mass, Black hole, R1 R2, radii r1

Click to edit the document details