- #1

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## Homework Statement

The two components of a double star are observed to move in circles of radii ##r_1## and ##r_2##. What is the ratio of their masses? (Hint: Write down their accelerations in terms of the angular velocity of rotation ##\omega##)

## Homework Equations

##m\ddot{\vec{x}}= \vec{F}_i##

Being ##\vec{F}_i## all the interaction forces acting on body ##i##

##v=\omega r##

## The Attempt at a Solution

##m_i\ddot{\vec{x}_i}= \vec{F}_{ij}##

##m_j\ddot{\vec{x}_j}= \vec{F}_{ji}##

##\vec{F}_{ij}=-\vec{F}_{ji}\implies m_i\ddot{\vec{x}_i}+m_j\ddot{\vec{x}_j}=0##

##m_i\dot{\vec{\omega_i}}r_i+m_j\dot{\vec{\omega_j}}r_j=0##

I guess ##\dot{\vec{\omega_j}} = \dot{\vec{\omega_i}}## but can't argue properly why. It just feels that since their movement depend on each other and there's no reason to have different angular acceleration, I don't know..

But this way we conclude that ##\dfrac{m_i}{m_j}=-\dfrac{r_j}{r_i}##

I don't think these scalars ##r_i## and ##r_j## can be negative. What's wrong with my resolution?