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Consider different planets revolving in ...

Consider different planets revolving in different circular orbits around a star of very large mass. If the gravitatonal force between the planet and the star varies as `r^((-5)/2), r` = distance between them. How does the square of the orbital period T depend on the distance r? 

Text Solution

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`implies` Since the star is very large, its orbital radius is R, hence distance between planet and star is r = R
`:.` Centripetal force of star = Gravitational force
`:.(mv^2)/R=(GMm)/R^(5//2)`
Now ` v = Romega = (2piR)/T`
`:. (4pi^2R)/T^2 = (GM)/(R^(5//2))" "[ :. omega(2pi)/T]`
`:. R^(7//2)=((GM)/(4pi^2))T^2`
`:.T^2 =((4pi^2)/(GM))R^(7//2)`
`:. T^(2) prop R^(7//2) " "( :. (4pi^2)/(GM) ` = constant )
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