Question

The distance between two stars of masses 3MS and 6MS is 9R. Here R is the mean distance between the centres of the Earth and the Sun, and MS is the mass of the Sun. The two stars orbit around their common centre of mass in circular orbits with period nT, where T is the period of Earth’s revolution around the Sun. The value of n is ___.

Answer 1

Centre of mass of system lies at 6R from lighter mass
$\begin{array}{l}\left [ 3M_{s}\omega ^{2}\times 6R \right ]=\frac{G(18M^{2}_{s})}{81R^{2}}\end{array}$
$\begin{array}{l}\omega ^{2}R = \frac{GM}{81R^{2}}\end{array}$
$\begin{array}{l}T^{‘}=\sqrt{\frac{81R^{3}}{GM_{s}}}\end{array}$
T’ = 9T
n = 9