- #1
Felipe Lincoln
Gold Member
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- 11
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?