Question 53: Gravitation

Question

A system of binary stars of masses $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ are moving in circular orbits of radii $r_{\mathrm{A}}$ and $r_R$, respectively. If $\mathrm{T}_A$ and $\mathrm{T}_B$ are the time periods of masses $m_A$ and $m_B$ respectively, then

Accepted Answer

From the centre of mass, we get
$$ m_{\mathrm{A}} R_{\mathrm{A}}=m_{\mathrm{B}} R_{\mathrm{B}} $$
Force acting on both stars i.e., gravitational force and centripetal force are balanced.
$$ \frac{\mathrm{G} m_{\mathrm{A}} m_{\mathrm{B}}}{\mathrm{R}^2}=\frac{m_{\mathrm{A}} v_{\mathrm{A}}^2}{\mathrm{R}_{\mathrm{A}}}=\frac{m_{\mathrm{B}} v_{\mathrm{B}}^2}{\mathrm{R}_{\mathrm{B}}} $$
As we know, $v_{\mathrm{A}}=\frac{2 \pi \mathrm{R}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{A}}}, v_2=\frac{2 \pi \mathrm{R}_{\mathrm{B}}}{\mathrm{T}_{\mathrm{B}}}$
Substituting equation (iii) in equation (ii) we get
$$ \begin{aligned} \frac{m_{\mathrm{A}} 4 \pi^2 \mathrm{R}_{\mathrm{A}}^2}{\mathrm{R}_{\mathrm{A}}\left(\mathrm{~T}_{\mathrm{A}}ight)^2} & =\frac{m_{\mathrm{B}} 4 \pi^2 \mathrm{R}_{\mathrm{B}}^2}{\mathrm{R}_{\mathrm{B}}\left(\mathrm{~T}_{\mathrm{B}}ight)^2} \ \frac{m_{\mathrm{A}} \mathrm{R}_{\mathrm{A}}}{\mathrm{~T}_{\mathrm{A}}^2} & =\frac{m_{\mathrm{B}} \mathrm{R}_{\mathrm{B}}}{\mathrm{~T}_{\mathrm{B}}^2} \end{aligned} $$
From equation (i), we conclude in equation (iv)
$$ \begin{aligned} \left(\mathrm{T}_{\mathrm{A}}ight)^2 & =\left(\mathrm{T}_{\mathrm{B}}ight)^2 \ \mathrm{~T}_{\mathrm{A}} & =\mathrm{T}_{\mathrm{B}} . \end{aligned} $$

Answer

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

Answer

$\mathrm{T}_{\mathrm{A}}>\mathrm{T}_{\mathrm{B}}$ if $\left(r_{\mathrm{A}}>r_{\mathrm{B}}\right)$

Answer

$\mathrm{T}_{\mathrm{A}}>\mathrm{T}_{\mathrm{B}}$ if $\left(m_{\mathrm{A}}>m_{\mathrm{B}}\right)$

Answer

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

Gravitation

A system of binary stars of masses m_{\mathrm{A}} and m_{\mathrm{B}} are moving in circular orbits of radii r_{\mathrm{A}} and r_R, respectively. If \mathrm{T}_A and \mathrm{T}_B are the time periods of masses m_A and m_B respectively, then

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