A planet of mass M and a single satellite of mass M/10 revolve in circular orbits about their stationary centre of mass, being held together by their gravitational attraction. the distance between their centres is D. the period of this orbital motion is ( IGNORE SPIN OFF PLANET AND SATELLITE)

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The distance between their masses are separated by D

as given we have a two mass system where on mass is M and the other mass is M/10 separated by D distance apart

so the position of the centre of mass , as we know is inversely proportional to the masses and lies on the line joining them

so over here the position of the center of mass from the M kg mass is  M10M+M10D= M1011M10D = D/11while the position of the center of mass from the M/10 mass isD-D/11 = 10D/11now as both are moving about their common center of mass so we have their radius of motion as found aboveFor M radius is = D/11 while for mass M/10 radius is  10D/11now both the masses are moving due to force F = G M×M/10D2so this force provides each of them their required centripetal force so for mass Mwe have G M×M/10D2 = Mv2Rso V = GMR/10D2so the time period of ocillation T = 2πR /Vso T = 2πR/GMR/10D2or T = 2πD10RGMwhere R = D/11 soT = 2πD10D/11GM = 2π10D311GM

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