First slide

Gravitational force and laws

Question

Binary stars of comparable masses rotate under the influence of each other's gravity at a distance ${[2 G / ω^{2}]}^{1 / 3}$ where $ω$ is the angular velocity of each of the system. If difference between the masses of two stars is 6 units. Find the ratio of the masses of smaller to bigger star.

Moderate

A

4:10
B

1:7
C

2:8
D

3:9

Solution

$\begin{array}{l} m_{1} r_{1} = m_{2} r_{2} or r_{1} = \frac{m_{2} r}{m_{1} + m_{2}} (i) \\ m_{1} r_{1} ω^{2} = \frac{{Gm}_{1} m_{2}}{r^{2}} \Rightarrow ω^{2} = \frac{G (m_{1} + m_{2})}{r^{3}} \end{array}$
[From (i)]
$or r = {[\frac{G (m_{1} + m_{2})}{ω^{2}}]}^{1 / 3}$
$\begin{array}{l} {(m_{1} + m_{2})}^{1 / 3} = 2 \\ m_{1} + m_{2} = 8 \\ and m_{2} - m_{1} = 6 (given) \end{array}$
which gives $m_{1} = 1 and m_{2} = 7 units$
$\frac{m_{1}}{m_{2}} = \frac{1}{7}$

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