Question

A double star consists of two stars having masses M and $2$M. The distance between their centres is equal to r. They revolve under their mutual gravitational interaction. Then, which of the following statements are not correct?

• A. Heavier star revolves in orbit of radius $2r/3$
• B. Both the stars revolve with the same speed, period of which is equal to $\left(2\pi /r^3\right)\left(2G{M}^2/3\right)$
• C. Kinetic energy of the heavier star is twice that of the other star
• D. None of the above

Answer 1

Answer: (A), (C)

The correct options are
A Heavier star revolves in orbit of radius $2r/3$
C Kinetic energy of the heavier star is twice that of the other star

The two stars rotate about their centre of mass.

$r_{com}=\frac{M}{3M}.r=\frac{r}{3}$ from heavier mass

Centripetal acceleration is provided by gravitation

$\Rightarrow \frac{G{M}^2}{r^2}=\frac{2}{3}Mr{\omega }^2\Rightarrow {\omega }^2=\frac{3GM}{r^3}$

$\therefore$ Timeperiod $=\frac{2\pi }{\omega }=2\pi \times \sqrt{\frac{r^3}{3GM}}$

Same for both

K.E of stars$=\frac{1}{2}I{\omega }^2{I}_{2}M=2M.(\frac{r}{3}{)}^2=\frac{2}{9}M{r}^2{I}_M=M.(\frac{2r}{3}{)}^2=\frac{4}{9}m{r}^2{I}_M=2I_{2}M\Rightarrow K.E_M=2K.E_{2}M$