Gravitation

JEE Main level, single correct, medium

17 of 223

Question 17 of 223

Consider a binary star system of star AA and star BB with masses mA{m}_{A} and mB{m}_{B} revolving in a circular orbit of radii rA{r}_{A} and rB,{r}_{B}, respectively. If TA{T}_{A} and TB{T}_{B} are the time period of star AA and star B,B, respectively, then:

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Answer: (B) TA=TB{T}_{A}={T}_{B}

1) From the centre of mass, we get
M1R1=M2R2−−−−−−1{M}_{1}{R}_{1}={M}_{2}{R}_{2}------1
2) Force acting on both starts i.e. gravitational force and centripetal force are balanced.
GM1M2R2=M1V12R1=M1V12R1−−−−−−2\dfrac{{GM}_{1}{M}_{2}}{{R}^{2}}=\dfrac{{M}_{1}{V}_{1}^{2}}{{R}_{1}}=\dfrac{{M}_{1}{V}_{1}^{2}}{{R}_{1}}------2
As we know, {V}_{1}=\dfrac{2{\pi R}_{1}}{{T}_{1}}&{V}_{2}=\dfrac{2{\pi R}_{2}}{{T}_{2}}
Substituting equation 3 in equation 2, we get
M1R1(T1)2=M1R2(T2)2−−−−−−4\dfrac{{M}_{1}{R}_{1}}{{\left({T}_{1}\right)}^{2}}=\dfrac{{M}_{1}{R}_{2}}{{\left({T}_{2}\right)}^{2}}------4
From equation 1, we conclude in eqution 4
(T1)2=(T2)2orT1=T2{\left({T}_{1}\right)}^{2}={\left({T}_{2}\right)}^{2}or{T}_{1}={T}_{2}

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