Question on gravitation law derivation

AI Thread Summary
The discussion centers on understanding the derivation of Newton's Law of Universal Gravitation, specifically the initial mathematical steps involving ratios of angular velocity and linear distance. The user seeks clarification on the relationships between angular velocity (ω), linear velocity (ν), and their respective distances (s and r) as well as the time period (T) of an orbit. It is explained that as time changes, the distance traveled also changes proportionally, reinforcing the concept that doubling one variable affects the others similarly. The user expresses gratitude for the clarification, confirming their intuitive understanding of the ratios. This exchange highlights the importance of grasping the fundamental relationships in orbital mechanics to comprehend gravitational laws.
Thiafon
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Hi, guys.
I was wondering on Newton's Gravity Law derivation, and I found this page: http://www.relativitycalculator.com/Newton_Universal_Gravity_Law.shtml
Everything seems clear, but the first step is just killing me, because I can't get it.
Assuming small incremental changes in s; <br /> \lim_{t\rightarrow 0} {s} \rightarrow 0<br />
we have the following ratios
\frac{\omega}{\nu}=\frac{s}{r}, and \frac{t}{T}=\frac{s}{2πr}
Could someone help me out? Explain, or just say, which part of math do I have to cover in order to understand that?
(btw, I did pre-calculus, and calculus, so concept of limits is familiar to me)
Thanks in advance.
 
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It's just telling you the ratios of everything. The vector changes at the same rate that the distance traveled does since v and r are equal. If you double w you double s as well.

In the 2nd ratio, T is the total time of one orbital period and 2*Pi*R is the total distance of the orbit. As t changes, which is the time it takes to transverse the incremental time period s, s changes as well. If you double t you double s. Does that make sense?
 
Yes, thank you a lot! I had kind of intuitive feeling about it, but I wasn't sure if it is correct.
 
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