In circular orbit as angular momentum is conserved so we can write $v$ is inversely proportional to $r$ but by equating gravitational force and centripetal force we get that v is inversely proportional to root of r. how is this possible?
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$\begingroup$ Have you tried calculating the total mechanical energy of the satellite (knowing orbital velocity) and checking the dependency on $v$? $\endgroup$– controlgroupCommented 2 days ago
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$\begingroup$ why is this a problem? $\endgroup$– JEBCommented 2 days ago
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Your statements are consistent because they are answering two different questions.
The first case, where you keep the angular momentum fixed, $v \propto r^{-1}$. Consider a planet orbiting in circular path of radius R. If you move the planet to radius 2R without applying any additional torque, its velocity will be halved. However, note that in this case, the final orbit of the planet is not necessarily circular. In fact, its final orbit has to be elliptical.
In the second case, you equate centripetal force and gravitational force. This is only true for circular orbits. So when this says $v \propto 1/\sqrt{r}$, the question it answers is “What is the velocity of the planet if I move it to a circular orbit with double the radius?” This doesn’t have to, and will not conserve angular momentum. To move the planet to a circular orbit of twice the radius, you must apply additional torque.
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$\begingroup$ Why the orbit with velocity halved has to be ellipse. $\endgroup$ Commented 2 days ago
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$\begingroup$ Easiest way to see is in that case, $mv^2/r \neq GMm/r^2$, so you have radial acceleration. $\endgroup$ Commented yesterday