Two planets A and B of equal mass are having their period of revolutions T A and T B such that T A =2 T B . These planets are revolving in the circular orbits of radii rA and rB respectively. Which out of the following would be the correct relationship of their orbits?

Question

Two planets A and B of equal mass are having their period of revolutions T A and T B such that T A =2 T B . These planets are revolving in the circular orbits of radii rA and rB respectively. Which out of the following would be the correct relationship of their orbits? (A) 2 r A 2= r B 2 (B) r A 3=2 r B 3 (C) r A 3=4 r B 3 (D) T A 2- T B 2=(π2/ GM )( r B 3-4 r A 3)

Tardigrade Admin · Accepted Answer

beginarrayc T 2 propto r 3 ( T A 2/ r A 3)=( T B 2/ r B 3) r A 3=4 r B 3 endarray

$2 r_{A}^{2} = r_{B}^{2}$

$r_{A}^{3} = 2 r_{B}^{3}$

$r_{A}^{3} = 4 r_{B}^{3}$

$T_{A}^{2} - T_{B}^{2} = \frac{π ^{2}}{GM} (r_{B}^{3} - 4 r_{A}^{3})$

$T^{2} \propto r^{3} \frac{T _{A}^{2}}{r _{A}^{3}} = \frac{T _{B} ^{2}}{r _{B}^{3}} r_{A}^{3} = 4 r_{B}^{3}$

Q. Two planets A and B of equal mass are having their period of revolutions TA​ and TB​ such that TA​=2TB​. These planets are revolving in the circular orbits of radii rA​ and rB​ respectively. Which out of the following would be the correct relationship of their orbits?

2rA2​=rB2​

rA3​=2rB3​

rA3​=4rB3​

TA2​−TB2​=GMπ2​(rB3​−4rA3​)

T2∝r3rA3​TA2​​=rB3​TB​2​rA​3=4rB3​​

$2 r_{A}^{2} = r_{B}^{2}$

$r_{A}^{3} = 2 r_{B}^{3}$

$r_{A}^{3} = 4 r_{B}^{3}$

$T_{A}^{2} - T_{B}^{2} = \frac{π ^{2}}{GM} (r_{B}^{3} - 4 r_{A}^{3})$

$T^{2} \propto r^{3} \frac{T _{A}^{2}}{r _{A}^{3}} = \frac{T _{B} ^{2}}{r _{B}^{3}} r_{A}^{3} = 4 r_{B}^{3}$