Orbital radius vs period question

AI Thread Summary
To determine the new orbital radius when increasing a satellite's period from T to 8T, one must apply Kepler's Third Law of planetary motion, which states that the square of the orbital period is proportional to the cube of the orbital radius. This relationship can be expressed mathematically as T^2 ∝ R^3. By setting up the equation for the initial and new periods, it can be deduced that (8T)^2 = k(R_new)^3, leading to the conclusion that R_new must equal 4R. Thus, to achieve a period of 8T, the orbital radius must be increased to 4 times the original radius. Understanding this relationship is crucial for solving similar orbital mechanics problems.
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Homework Statement


a satellite has period T and orbital radius R. If you wish to increase the period to 8T what must the new orbital radius be? (answer is 4R, I just don't know how to get it)


Homework Equations





The Attempt at a Solution

 
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First think of the relationship between orbital radius and the period.

(Centripetal force = gravitational force of attraction)
 

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