Chapter 13, Problem 84CP

Show that the period of orbit for two masses, $m_1$ and $m_2$ , in circular orbits of radii $r_1$ and $r_2$ , respectively, about their common center-of −mass, is given by $T=2\pi \sqrt{\frac{r^3}{G\left(m_1+m_2\right)}}$ where $r=r_1+r_2$ . (Hint: The masses orbit at radii $r_1$ and $r_2$ , respectively where $r=r_1+r_2$ . Use the expression for the center-of-mass to relate the two radii and note that the two masses must have equal but opposite momenta. Start with the relationship of the period to the circumference and speed of orbit for one of the masses. Use the result of the previous problem using momenta in the expression for the kinetic energy.)