Question

The small dense stars rotate about their common centre of mass as a binary system, each with a period of 1 year. One star has mass double than that of the other, while mass of the lighter star is one-third the mass of the Sun. The distance between the two two stars is $r$ and the distance of the earth from the Sun is $R.$ The relation between $r$ and $R$ is

• A. $r=2R$
• B. $r=6R$
• C. $r=5R$
• D. $r=R$

Answer 1

The correct option is D $r=R$

Binary Star
Considering the heavier star
$\text{Speed of heavier star}=\frac{\text{Circumference of orbit}}{\text{Time taken}}=\frac{2\pi \frac{r}{3}}{1}=\frac{2\pi r}{3}$
$\text{Gravitational force}=\text{Centrifugal force}$
$\Rightarrow \frac{G\frac{m}{3}\frac{2m}{3}}{r^2}=\frac{\frac{2m}{3}\frac{4{\pi }^2{r}^2}{9}}{\frac{r}{3}}$
$\Rightarrow \frac{8m{\pi }^{2}r}{9}=\frac{2G{m}^2}{9r^2}$
$\Rightarrow r^3=\frac{Gm}{4{\pi }^2}........\left(i\right)$
Sun and earth
$\text{Speed of earth}=\frac{2\pi R}{1}=2\pi R$
$\text{Gravitational force}=\text{Centrifugal force}$
$\Rightarrow \frac{GMm}{R^2}=\frac{M\left(4{\pi }^2{R}^2\right)}{R}$
$\Rightarrow R^3=\frac{Gm}{4{\pi }^2}.......\left(ii\right)$
From eq. (i)and (ii), it is clear that $r=R$
Thus (D) is the correct answer.