Practicing Success
A double-star, possessing masses m1 and m2 , are rotating with constant angular speed. When the maximum distance of separation between them is R, what is the angular speed of revolution of the stars: |
$\sqrt{\frac{G\left(m_1+m_2\right)}{R^3}}$ $\sqrt{\frac{G\left(m_1+m_2\right)}{R^2}}$ $\sqrt{\frac{G}{2\left(m_1+m_2\right) r}}$ $\frac{G\left(m_1+m_2\right)}{R^3}$ |
$\sqrt{\frac{G\left(m_1+m_2\right)}{R^3}}$ |
Each star moves under the gravitational force exerted by the other star. Suppose they rotate with constant angular speed ω in the same sense. Gravitational force provides necessary centripetal acceleration to keep the stars in their respective orbits of revolution. $\Rightarrow F_{g_1}=F_{cp_1}=m_1 r_1 \omega^2$ and $F_{g_2}=F_{cp_2}=m_2 r_2 \omega^2$ $F_{gr_1}=F_{gr_2}=G \frac{m_1 m_2}{R^2} \Rightarrow m_1 r_1 \omega^2=m_2 r_2 \omega^2$ $\Rightarrow m_1 r_1=m_2 r_2$ That means, the system rotates about its C.M., O. $\frac{r_1}{r_2}=\frac{m_2}{m_1} \Rightarrow \quad \frac{r_1}{r_1+r_2}=\frac{m_2}{m_1+m_2}$ $\Rightarrow r_1=\frac{m_2}{m_1+m_2}$ R (since $r_1+r_2$ = R) $F_{c p_1}=m_1 r_1 \omega^2$ and $F_{gr_1}=\frac{G m_1 m_2}{R^2}$ Since $F_{cp_1}=F_{g_1}$ $\Rightarrow m_1 r_1 \omega^2=\frac{G m_1 m_2}{R^2} \Rightarrow r_1 \omega^2=\frac{G m_2}{R^2}$ Putting $r_1=\frac{m_2 R}{m_1+m_2}$, we obtain, $\left(\frac{m_2 R}{m_1+m_2}\right) \omega^2=\frac{G m_2}{R^2}$ $\Rightarrow \omega=\sqrt{\frac{G\left(m_1+m_2\right)}{R^3}}$ |