Binary stars rotate under mutual gravitational force at separation units, where ω is the angular velocity of each of the star about centre of mass of the system. If difference between the mass of stars is 6 units. Find the ratio of masses of bigger star to smaller star.

Engineering

Physics

Center of Mass

Gravitational Force and Field

Question

Binary stars rotate under mutual gravitational force at separation $2 {(\frac{G}{ω^{2}})}^{1 / 3}$ units, where ω is the angular velocity of each of the star about centre of mass of the system. If difference between the mass of stars is 6 units. Find the ratio of masses of bigger star to smaller star.

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Given that m₁ – m₂ = 6 unit .........(i)

equation of motion

$\frac{{Gm}_{1} m_{2}}{r^{2}} = m_{1} ω_{2} r_{1} = m_{1} ω_{2} \frac{m_{2} r}{m_{1} + m_{2}}$

$∴ m_{1} + m_{2} = \frac{ω^{2} r^{3}}{G} = 8 unit$ ......(ii)

by (ii) and (ii)

$\frac{m_{1}}{m_{2}} = 7$