Kepler's 3rd Law as a result of Newton's Laws

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Discussion Overview

The discussion revolves around demonstrating Kepler's 3rd Law in the context of two gravitating bodies in circular orbits around their center of mass, utilizing Newton's Laws. The focus is on the relationship between the distances of the bodies from the center of mass and their respective masses.

Discussion Character

  • Exploratory, Technical explanation, Homework-related, Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in differentiating between the scenarios of one mass versus two masses in circular orbits.
  • Another participant questions the ratio of distances (r1 to r2) in relation to the masses (m1 to m2).
  • A participant suggests that a larger mass (m1) results in a smaller distance (r1), implying a corresponding relationship with the other mass (m2) and its distance (r2).
  • A further inquiry is made about the specific ratio of r1 to r2 when m1 equals m2, and what occurs to this ratio if the mass of m2 is doubled.

Areas of Agreement / Disagreement

Participants are exploring relationships and ratios but have not reached a consensus on the implications of these ratios or the overall demonstration of Kepler's 3rd Law in this context.

Contextual Notes

The discussion includes assumptions about circular orbits and the application of Newton's Laws, but does not resolve the mathematical steps necessary to demonstrate the law fully.

takbq2
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thanks in advance for any and all help!

The question is to show that Kepler's 3rd Law holds when you assume circular orbit with TWO gravitating bodies both orbiting around the center of mass. (by equating gravitational and centripetal forces and finding the constant of proportionality which includes the masses of the gravitating bodies).

my attempt at a solution:

I already have shown it when assuming just one mass. I'm having a very hard time differentiating between the two situations.

I know I need to start by using Newton's 3rd law to get a relationship between r1 and a (where a is the distance between the two masses) and eliminate r2 or vice versa (r1 is the distance from m1 to the center of mass, and r2 is the distance from m2 to the center of mass)


Please help me if you can, I appreciate it so much.
 
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please.. anyone?
 
What do you think the ratio of r1 to r2 would be as compared to the ratio of m1 to m2?
 
bigger m1 means smaller r1.. which would in turn mean smaller m2 and larger r2
 
Alright, what is the ratio of r1 to r2 if m1=m2.

What happens to the ratio of r1:r2 if you now double the mass of m2?
 

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