Physics - Gravitation Question with Solution | TestHub

The center of mass of the two stars is given by $\mathrm{R}=\mathrm{d}/2$

The force of attraction between the two stars is given by $\mathrm{F}={\mathrm{GM}}^2/{\mathrm{d}}^2$

The orbital period T will be given by the centripetal force law $\mathrm{F}=\mathrm{M}\frac{4{\pi }^2\mathrm{R}}{{\mathrm{T}}^2}=\mathrm{G}\frac{{\mathrm{M}}^2}{{\mathrm{d}}^2}$ or ${\mathrm{T}}_{\mathrm{S}}=2\pi \sqrt{\frac{{\mathrm{d}}^3}{2\mathrm{GM}}}=4\pi \sqrt{\frac{{\mathrm{R}}^3}{\mathrm{GM}}}$

The planetoid is a distance z above the plane. The distance to either star is then $\sqrt{{\mathrm{R}}^2+{\mathrm{z}}^2}$. Only the part of the force that is perpendicular to the plane survives the vector addition, so the net force is given by $\mathrm{F}=2\frac{\mathrm{GmM}}{{\mathrm{R}}^2+{\mathrm{z}}^2}\frac{\mathrm{z}}{\sqrt{{\mathrm{R}}^2+{\mathrm{z}}^2}}$

This expression is exact. But if $\mathrm{z}\ll \mathrm{d}$, then we can approximate this as $\mathrm{F}\approx 2\frac{\mathrm{GmM}}{{\mathrm{R}}^3}\mathrm{z}$ This is a restoring force; the planetoid will execute simple harmonic oscillations with period ${\mathrm{T}}_{\mathrm{P}}=2\pi \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$ where k is the effective force constant, or ${\mathrm{T}}_{\mathrm{P}}=2\pi \sqrt{\frac{{\mathrm{R}}^3}{2\mathrm{GM}}}$