7. In a double star system, two stars of masses m,
and m, seperated by a distance d rotate about
their centre of mass. Then their common
angular velocity would be
Answers
The centre of mass C will be at distance d/3 and 2d/3 from masses 2m and m, respectively. Both the stars rotate round C in their respective orbits with the same angular velocity ω. The gravitational force acting on each star due to the other supplies the necessary centripetal force.
The gravitational force on either star is G(2m)m/d 2
If we consider the rotation of the smaller star, the centripetal force (mrω
2
) is [m(2d/3)ω
2
] and [(2mdω
2
)/3] for the bigger star, i.e., the same.
∴
d
2
G(2m)m
=m(
3
2d
) ω
2
or Ω=
(
d
3
3Gm
)
Therefore, the period of revolution is given by
T=
ω
2π
=2π
(
3Gm
d
3
)
The ratio of the angular momenta is
(Iω)
small
(Iω)
big
=
I
smell
I
big
=
m(
3
2d
)
2
(2m)(
3
d
)
2
=
2
1
Since ω is the same for both.
The ratio of their kinetic energies is
(
2
1
Iω
2
)
small
(
2
1
Iω
2
)
big
=
I
small
I
big
=
2
1
which is the same as the ratio of their angular momenta.