JEE Main · PhysicsHard

$Two identical stars of mass M orbit around their common center of mass in a circular orbit with a separation distance d . A small planet of mass m is placed at the center of mass of the system and given a small displacement z along the axis perpendicular to the orbital plane. What is the frequency of oscillation of the planet?$

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Correct answer: A — $\frac{8 GM}{d ^{3}}$

Solution

The net force is F = - 2 \frac{GM m}{( d /2 ) ^{2} + z ^{2}} \frac{z}{( d /2 ) ^{2} + z ^{2}} . For z ≪ d, F \approx - \frac{16 GM m}{d ^{3}} z . Comparing with F = - m ω^{2} z, we get ω = \frac{16 GM}{d ^{3}} . The frequency f = \frac{ω}{2 π} is derived from this angular frequency.