Gravitation NCERT Solutions Class 11 Physics - Solved Exercise Question 8.4

Question 8.4:
Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 108 m. Show that the mass of Jupiter is about one-thousandth that of the sun.
Solution:
Orbital period of  I0 , TI0 = 1.769 days  =  1.769 X 24 X 60 X 60 s
Orbital radius of  I0 , RI0 = 4.22 X 108 m
Satellite I0 is revolving around the Jupiter
Mass of the latter is given by the relation:
MJ = 4π2RI03 / GTI02    .....(i)
Where,
MJ = Mass of Jupiter
G = Universal gravitational constant
Orbital period of the earth,
Te = 365.25 days = 365.25 X 24 X 60 X 60 s
Orbital radius of the Earth,
Re = 1 AU = 1.496 X 1011 m
Mass of sun is given as:
Ms = 4π2Re3 / GTe      ......(ii)
∴ Ms / MJ  =  (4π2Re3 / GTe2) X (GTI02 /2RI03)   =   (Re3 X TI02) / (RI03 X Te2)
Substituting the values, we get:
= (1.769 X 24 X 60 X 60 / 365.25 X 24 X 60 X 60)2 X (1.496 X 1011 / 4.22 X 108)3
= 1045.04
∴ Ms / MJ  ~ 1000
Ms ~ 1000 X MJ
Hence, it can be inferred that the mass of Jupiter is about one-thousandth that of the Sun.

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