Question

Suppose there existed a planet that went around the sun twice as fast as the earth .What would be its orbital size as compared to that of the earth??​

Answer 1

Let Ms =mass of the sun

Me = mass of the earth

r = radius of orbit of the earth around the sun .

r' = radius of planet around the sun.

w = angular speed of the earth

w' = angular speed of the planet

A/c to question,

w' = 2w { planet around the sun twice fast as earth }

Centripital force balance the Gravitational force in orbital motion.

MeV²/r = GMsMe/r²

V² = GMs/r

We know,

V = wr [use this here,

(wr)² = GMs/r

w² = GMs/r³ -----------(1)

Similarly for planet ,

w'² = GMs/r'³ ---------(2)

w²/w'² = (r'/r)³

r'/r = (w/w')^(2/3)

r' = (w/2w)^(2/3) r

= (1/4)⅓ r

= 0.63r hence, size of planet' s orbital is smaller then the orbital size of the earth .

Answer 2

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Refer the attachment above ..

Hope it Helps !!