Consider a binary star system of star A and star B

Question:

Consider a binary star system of star A and star B with masses $\mathrm{m}_{\mathrm{A}}$ and $\mathrm{m}_{\mathrm{B}}$ revolving in a circular orbit of radii $r_{A}$ and $r_{B}$, respectively. If $T_{A}$ and $T_{B}$ are the time period of star $A$ and star $B$, respectively, then:

  1. $\frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}=\left(\frac{\mathrm{r}_{\mathrm{A}}}{\mathrm{r}_{\mathrm{B}}}\right)^{\frac{3}{2}}$

  2. $\mathrm{T}_{\mathrm{A}}=\mathrm{T}_{\mathrm{B}}$

  3. $\mathrm{T}_{\mathrm{A}}>\mathrm{T}_{\mathrm{B}}\left(\right.$ if $\left.\mathrm{m}_{\mathrm{A}}>\mathrm{m}_{\mathrm{B}}\right)$

  4. $\mathrm{T}_{\mathrm{A}}>\mathrm{T}_{\mathrm{B}}\left(\right.$ if $\mathrm{r}_{\mathrm{A}}>\mathrm{r}_{\mathrm{B}}$ )

Correct Option: , 2

Solution:

$\mathrm{T}_{\mathrm{A}}=\mathrm{T}_{\mathrm{B}}\left(\right.$ since $\left.\omega_{\mathrm{A}}=\omega_{\mathrm{B}}\right)$

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