Question

a planet of mass m moves around the sun of of mass m in a cooler orbit of radius r with an angular velocity so that Omega is independent of mass m of the planet in a circular orbit of radius 4r around the sun the angular velocity decreases to omega/8​

Answer 1

Explanation:

here gravitational force between planet and sun is balanced by centripetal force.

case 1 : mass of planet = m, mass of sun = M, radius of circular orbit = r and angular speed is \omegaω

then, \frac{GMm}{r^2}=m\omega^2rr2GMm=mω2r

or, \omega^2=\frac{GM}{r^3}ω2=r3GM ......(1)

case 2 : mass of planet = 2m

radius of circular orbit = 4r

and angular speed = \omegaω

then, \omega'^2=\frac{GM}{(4r)^3}ω′2=(4r)3GM

or, \omega'^2=\frac{1}{64}\frac{GM}{r^3}ω′2=641r3GM ......(2)

from equations (1) and (2),

\frac{\omega^2}{\omega'^2}=\frac{64}{1}ω′2ω2=164

\frac{\omega}{\omega'}=8ω′ω=8

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Answer 2

Answer:

refer the attachment

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