Question

A system of binary stars of masses $M_{1}$ and $M_{2}$ are moving in circular orbits of radii $r_{1}$ and $r_{2}$ respectively. If $T_{1}$ and $T_{2}$ are the time periods of revolution of masses $M_{1}$ and $M_{2}$ respectively and if $\frac{r_{1}}{r_{2}}=2,$ then find $\frac{T_{1}}{T_{2}}.$

Answer 1

Centre fo mass of binary star lies at certre of their circular path.
$M_{1}r_{1}=M_{2}r_{2}$
Then, $G\frac{M_{1} M_{2}}{\left(r_{1} + r_{2}\right)^{2}}=M_{1}\omega _{1}^{2}r_{1}=M_{2}\omega _{2}^{2}r_{2}$
$\Rightarrow \omega _{1}=\omega _{2} \rightarrow \frac{T_{1}}{ T_{2}}=1.00$
Same angular velocity and time period of revolution.