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Home / Doubts and Solutions/NEET/Physics

A system of binary stars of masses M1 and M2 are moving in circular orbits of radii R1 and R2 respectively. If T1 and T2 are the same time periods of masses M1 and M2 respectively then:

(a) T1/T2=(R1/R2)^1/2

(b)T1>T2 (if R1>R2)

(c) T1>T2 (if M1>M2)

(d)T1=T2

Asked by nasir.mirza 12th June 2018, 9:10 PM
Answered by Expert
Answer:
Figure shows two binary stars of respective mass M1 and M2 rotating in circular orbits of radius R1 and R2 respectively.
The common center of both orbits is also centre of mass of both the stars.
 
From the definition of centre of mass, we have  M1×R1 = M2×R2   ........................... (1)
 
Forces acting on both the stars, i.e., gravitational force and centripetal forces are balanced.
 
begin mathsize 12px style G fraction numerator M subscript 1 space M subscript 2 over denominator R squared end fraction space equals space fraction numerator M subscript 1 v subscript 1 superscript 2 over denominator R subscript 1 end fraction space equals fraction numerator M subscript 2 space v subscript 2 superscript 2 over denominator R subscript 2 end fraction end style...............................(2)
where R = R1+R2 .Let T1 and T2 are time periods of revolution of respective stars

let us substitute begin mathsize 12px style v subscript 1 space equals space fraction numerator 2 pi space R subscript 1 over denominator T subscript 1 end fraction space a n d space v subscript 2 space equals space fraction numerator 2 pi space R subscript 2 over denominator T subscript 2 end fraction end style in eqn.(2) and after simplification we get
begin mathsize 12px style fraction numerator M subscript 1 space R subscript 1 over denominator T subscript 1 superscript 2 end fraction space equals space fraction numerator begin display style M subscript 2 space R subscript 2 end style over denominator begin display style T subscript 2 superscript 2 end style end fraction space end style..................................(3)
using eqn.(1) in eqn.(3) we get the condition T1 = T2
 
Answered by Expert 16th June 2018, 9:47 AM
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