A system consists of two stars of equal masses that revolve in a circular orbit about a centre of mass midway between them. Orbital speed of each star is $$v$$ and period is $$T.$$ Find the mass $$M$$ of each star ($$G$$ is gravitational constant)
A.
$$\frac{{2G{v^3}}}{{\pi T}}$$
B.
$$\frac{{{v^3}T}}{{\pi G}}$$
C.
$$\frac{{{v^3}T}}{{2\pi G}}$$
D.
$$\frac{{2T{v^3}}}{{\pi G}}$$
Answer :
$$\frac{{2T{v^3}}}{{\pi G}}$$
Solution :
$$\eqalign{
& \frac{{M{v^2}}}{R} = \frac{{G{M^2}}}{{4{R^2}}} \Rightarrow M = \frac{{4R{v^2}}}{G} \cr
& v = \frac{{2\pi R}}{T} \cr
& R = \frac{{vT}}{{2\pi }} \cr
& M = \frac{{{v^3}T2}}{{\pi G}} \cr} $$
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