54. The magnitude of the net gravitational force on one of the smaller stars (of mass m) is

GM m

r

2

+

Gmm

(2r)

2

=

Gm

r

2

M +

m

4



.

This supplies the centripetal force needed for the motion of the star:

Gm

r

2

M +

m

4



= m

v

2

r

where v =

2πr

T

.

Plugging in for speed v, we arrive at an equation for period T :

T =

2πr

3/2



G(M + m/4)

.