Question

In a double star system, the masses of the two stars are M and 3M. When the stars are collinear with the center of mass, the distance between the two stars is R. The time period of each star is

• A. $8\pi [R^3/GM{]}^{1/2}$
• B. $16\pi [R^3/GM{]}^{1/2}$
• C. $\pi [R^3/GM{]}^{1/2}$
• D. None

Answer 1

The correct option is C $\pi [R^3/GM{]}^{1/2}$

Distance between the stars is R

ratio of the distance from center of mass $\frac{r_1}{r_2}\propto \frac{m_2}{m_1}$

therefore,$r_1=\frac{3R}{4}and{r}_2=\frac{R}{4}$

Time period,$T_1=\frac{2\pi r_1}{v}$

$\frac{G3M\times M}{R^2}=\frac{M(v_1{)}^2\times 4}{3R}$

velocity $v_1=\frac{\sqrt{9GM}}{\sqrt{4R}}$

put $v_{1}in{T}_1$

$T_1=\frac{2\pi r_1}{\frac{\sqrt{9GM}}{\sqrt{4R}}}$

$T_1=\pi \sqrt{\left[\frac{R^3}{GM}\right]}$