Question

In lecture 3, Prof. V. Balakrishnan, used a technique of scaling to prove kepler's third law from a generalized case where the potential is some power of (r^n), i got confused, how he got to prove the law from scaling factors mu and lamda ?

Answer 1

i tried the following, \[R _{1}=\lambda R_{1}^{'}\] and \[T_{1}=\mu T _{1}^{'}\]
and because we have the dimensionless ratio \[\frac{ \lambda ^{3} }{ \mu ^{2} } \] so we can substitute for the value of lambda and Mu twice, one for \[R^{'}\] and \[T^{'}\] another one for \[R\] and \[T\] and i end up the with following
\[\left( \frac{ R_{1} }{ R_{2} } \right)^{3}\left( \frac{ T_{2} }{ T_{1} } \right)^{3}=\left( \frac{ R _{1}^{'} }{ R _{2}^{'} } \right)^{3}\left( \frac{ T _{2}^{'} }{ T _{1}^{'} } \right)^{3}\]
now i can't do anything more, the only thing that i can do is
\[\left( \frac{ R_{1} }{ R_{2} } \right)^{3}\left( \frac{ T_{2} }{ T_{1} } \right)^{3}= Const\]

is there anyone can help in this

Answer 2

This is how i would prove Keppler's law.. Of course, it is not the way Keppler actually did but it seems Ok!

Let mass of Sun be M and mass of planet be m

Now, to stay in orbit the centripetal force and centrifugal force must be balanced!\[\frac{GMm}{r^2}=mr\omega ^2\]\[\frac{GM}{r^2}=mr(\frac{2\pi}{T})^2\]all other are constants except r and T
From above relation, you have \(T^2\) is directly proportional to \(r^3\)

Answer 3

*\[\frac{GMm}{r^2}\]in second step.. Well you could have chosen to eliminate m too.. Still that doesn't matter as it is a constant!

Answer 4

Sorry about Prof. V. Balakrishnan's technique..

P.S. I don't know who is he..

Answer 5

Equating the centrifugal force to gravity is a little tautologous as Newton used Kepler's Laws and the equation for centrifugal acceleration to deduce the inverse square law.

Answer 6

thank u all, i know all of that staff, but this not my question,
watch this starting min 53
http://www.youtube.com/watch?v=0Zo93VDyacE