a)Using elementary Newtonian mechanics, find the period of amass (m1) in a circu

Question

a)Using elementary Newtonian mechanics, find the period of amass (m1) in a circular orbit of radius r around a fixed mass(m2).b)Using the seperation of variables into Center of Mass andreletive motions, find the corresponding period for the case that(m2) is not fixed and the masses circle each other a constantdistance (r) apart.Discuss the limit of this result if (m2) goes toinfinity.c) What would be the orbital period if the earth werereplaced by a star of mass equal to the solar mass, in a circularorbit, with the distance between the sun and star equal to thepresent earth-sun distance? (The mass of the sun is more that300,000 times that of the earth.)

Explanation / Answer

let w be the angular velocity of the planet m1rw2=Gm1m2/r2 sow=(Gm2/r3)1/2 so T=2/w =2r3/2/(Gm2)1/2
for the motion of both thebodies the distance of the body 1 from the centre ofmass is m2r/(m1+m2) the equivalent mass ism1m2/(m1+m2) Gm1m2/r2  = m1 (m2r/m1+m2)w2
so T = 2r3/2/(G(m1+m2))1/2 if m2 is infinite then thisperiod is 0 as the body revolves at infinite speed
if the earth was replaced by a star of masssame as sun then T =To/(2)1/2 T=258.09 days
sow=(Gm2/r3)1/2 so T=2/w =2r3/2/(Gm2)1/2
for the motion of both thebodies the distance of the body 1 from the centre ofmass is m2r/(m1+m2) the equivalent mass ism1m2/(m1+m2) Gm1m2/r2  = m1 (m2r/m1+m2)w2
so T = 2r3/2/(G(m1+m2))1/2 if m2 is infinite then thisperiod is 0 as the body revolves at infinite speed
if the earth was replaced by a star of masssame as sun then T =To/(2)1/2 T=258.09 days

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