Question

Two bodies (binary system) of mass $M$ and $m(M>m)$ revolve in circular orbits of radii $R$ and $r$, respectively about their centre of mass. Their angular speeds are ${\omega }_1$ and ${\omega }_2$. Then :

• A. ${\omega }_{1}R={\omega }_{2}r$
• B. ${\omega }_{1}r={\omega }_{2}R$
• C. ${\omega }_1={\omega }_2$
• D. ${\omega }_1^{2}R={\omega }_2^{2}r$

Answer 1

The correct option is C ${\omega }_1={\omega }_2$

They revolve around their center of mass of system, So mr = MR $\to \frac{m}{R}=\frac{M}{r}$

The gravitational force between them acts as centripetal force.

Thus,

$\frac{GMm}{(R+r{)}^2}=m({\omega }_1{)}^{2}r=M({\omega }_2{)}^{2}R$

Solving this, we get ${\omega }_1=(\frac{GM}{r(R+r{)}^2}{)}^{0.5}=(\frac{Gm}{R(R+r{)}^2}{)}^{0.5}={\omega }_2$