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Satellites

A system of b...

Question

A system of binary stars of masses $m_{A}$ and $m_{B}$ are moving in circular orbits of radii $r_{A}$ and $r_{B}$ respectively. If $T_{A}$ and $T_{B}$ are the time periods of masses $m_{A}$ and $m_{A}$ respectively, then

A

\frac{T_{A}}{T_{B}} = {(\frac{r_{A}}{r_{B}})}^{3 / 2}

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B

T_{A} > T_{B} (i f r_{A} > r_{B})

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C

T_{A} > T_{B} (i f m_{A} > m_{B})

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D

T_{A} = T_{B}

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Solution

The correct option is D $T_{A} = T_{B}$
$\frac{G m_{A} m_{B}}{{(r_{A} + r_{B})}^{2}} = \frac{m_{A} r_{A} 4 π^{2}}{T_{A}^{2}} = \frac{m_{B} r_{B} 4 π^{2}}{T_{B}^{2}}$
$\Rightarrow m_{A} r_{A} = m_{B} r_{B}$
$∴ T_{A} = T_{B}$

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