Question

Consider two particles of masses m₁ and m₂ in orbit about their common center of mass. Let F be defined as the position vector of the particle of mass m, relative to the center of mass.

Part A: Derive the equation of motion for m₁ and show that it can be written in the form. \ddot{\overline{\Gamma_{1}}}+\frac{G m_{2}{ }^{3}}{\left(m_{1}+m_{2}\right)^{2} r_{1}^{3}} \overline{r_{1}}=\overline{0} Part B: Derive the equation of motion for m₂ and demonstrate that it is similar, that is, \ddot{r_{2}}+\frac{G m_{1}^{3}}{\left(m_{1}+m_{2}\right)^{2} r_{2}^{3}} \bar{r}_{2}=\overline{0} Part C: Finally, prove that the relative equation that results from (a) and (b) is \frac{\ddot{r}}{r_{12}}+G\left(m_{1}+m_{2}\right) \frac{\bar{r}_{12}}{r_{12}^{3}}=\overline{0}

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