A double star is a system of two stars moving around the centre of mass of the system due to gravitation. The distance between the components of the double star is $d = {(\frac{{GMT}^{2}}{x π^{2}})}^{1 / y}$ (given, the total mass is M and time period is T). Find the value $x + y$

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Detailed Solution

$r_{1} = \frac{m_{2} d}{m_{1} + m_{2}} & \frac{{Gm}_{1} m_{2}}{d^{2}} = m_{1} r_{1} ω^{2}$

$\frac{{Gm}_{1} m_{2}}{d^{2}} = \frac{m_{1} m_{2} d}{m_{1} + m_{2}} \cdot \frac{4 π^{2}}{T^{2}}$

$\Rightarrow \frac{G (m_{1} + m_{2})}{d^{3}} = \frac{4 π^{2}}{T^{2}} Given M = m_{1} + m_{2}$

$\Rightarrow \frac{GM}{d^{3}} = \frac{4 π^{2}}{T^{2}} \Rightarrow d = {[\frac{{GMT}^{2}}{4 π^{2}}]}^{1 / 3}$

$compairing d = {(\frac{{GMT}^{2}}{x π^{2}})}^{1 / y} = {(\frac{{GMT}^{2}}{4 π^{2}})}^{1 / 3} x = 4 and y = 3$

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