We know that \({F}_{ob}\) acts on the system of total mass \({M}_{e} + {M}_{s}\), the force per unit mass is
\({Fp}_{ob} = \frac{1}{{M}_{e} + {M}_{s}}{F}_{ob}\),
Let the forces on the masses be,
\({F}_{e}\), \({F}_{m}\)
These forces act on both ends of the system keeping Orbs constant, the force per unit mass must be equal for the acceleration to be equal (and opposite).
\(\frac{{F}_{e}} {{M}_{e}} = \frac{{F}_{m}}{ {M}_{m}} \)
and are equal to \({Fp}_{ob} \); because that is the force per unit mass restraining the system as a whole. Therefore,
\(\frac{{F}_{e}} {{M}_{e}} = \frac{1}{{M}_{e} + {M}_{s}}{F}_{ob}\)
And so,
\({F}_{e}= \frac{ {M}_{e}}{{M}_{e} + {M}_{s}}{F}_{ob}\)
\({F}_{m}= \frac{ {M}_{m}}{{M}_{e} + {M}_{s}}{F}_{ob}\)
The direction of these forces are such that they oppose the increase in Orbs, the orbital distance, acting on the masses at both ends of the system.
