The reason behind the inverse proportionality r^2 to the force of attraction

[parsa418]

Does anyone know how Newton knew that the force of attraction between two objects is inversely proportional to the distance between the two objects.

 

[russ_watters]

He derived it mathematically from the knowledge (from Kepler) that planets' orbits are elliptical.

fter the exchanges with Hooke, Newton worked out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector...

http://en.wikipedia.org/wiki/Isaac_Newton

 

[parsa418]

Thanks.
Could you tell me what the mathematical proof was or how to prove it exactly.

 

[Bill_K]

You only need Kepler's Third Law, "The period squared goes as the radius cubed."

T² = (2πr/v)² ~ r³ shows that v ~ r^-½. Then the centripetal force required is F = mv²/r ~ 1/r²

 

[parsa418]

Thank you this is making a lot more sense now. I only have one remaining question:
How is the centripetal force related to the attracting forces between the two objects. I mean these objects aren't rotating each other or anything? are they?

 

[Andrew Mason]

Thank you this is making a lot more sense now. I only have one remaining question:
How is the centripetal force related to the attracting forces between the two objects. I mean these objects aren't rotating each other or anything? are they?

If the force is only gravity from the other mass, then the acceleration of the mass is simply the gravity force/mass. The acceleration is the vector sum of the centripetal acceleration and the tangential acceleration. If it is circular motion with no tangential acceleration and M1>>M2:

\vec{F_g} = \vec{F_c} so:

GM_1/R^2 = v^2/R \implies v = \sqrt{GM_1/R}

AM

 

[parsa418]

Thank you very much

 

[parsa418]

Two Questions:
Could you prove why the period squared equals to the radius cubed
and do all objects rotate around each other?

 

[RedX]

Two Questions:
Could you prove why the period squared equals to the radius cubed
and do all objects rotate around each other?

I think it's fairly obvious to prove that if the potential U(r) satisfies:

U(\kappa r)=\kappa^\nu U(r)

then the period 'T' is related to the size of the orbit 'a' by:

T \mbox{ } = \mbox{ } a^{-\frac{\nu}{2}+1}

In particular for the Kepler orbit \nu=-1 which reproduces his 3rd law, and for the harmonic oscillator \nu=2 you get isochronicity (that is period doesn't depend on size of the orbit which is true of a spring). For a practically free particle, \nu=0, you get the the period 'T' is proportional to the size 'a', which makes sense.

 

[ank160]

You only need Kepler's Third Law, "The period squared goes as the radius cubed."

T² = (2πr/v)² ~ r³ shows that v ~ r^-½. Then the centripetal force required is F = mv²/r ~ 1/r²

But how did Kepler derived that T^2 is proportional to r^3.

 

[Nabeshin]

But how did Kepler derived that T^2 is proportional to r^3.

I believe he just "derived" this empirically.

 

[Vagn]

But how did Kepler derived that T^2 is proportional to r^3.

He didn't, he observed them.He was Tycho Brahe's student and had access to Brahe's catalogues which where extremely accurate with over a decades worth of data. This allowed him to use the data to empirically show his laws to be true. He didn't know why they were true though, and it was only with Newton's Universal Law of Gravitation that they were derived mathematically

 

[parsa418]

Reason behind inverse proportionality of R^2 to force of attraction

Hi
Could anyone please tell me how did Newton prove that the force of attraction between two objects is inversely proportional to the distance between them squared. I know that if you use Kepler's law that the T^2=r^3 you could prove it. However, I still don't get how kepler proved that T^2=R^3. A lot of people told me it was based on observations. But I know there has to be some proof for this observation by now. Please give me a full proof of this matter.
Thank you

 

[Andrew Mason]

Hi
Could anyone please tell me how did Newton prove that the force of attraction between two objects is inversely proportional to the distance between them squared. I know that if you use Kepler's law that the T^2=r^3 you could prove it. However, I still don't get how kepler proved that T^2=R^3. A lot of people told me it was based on observations. But I know there has to be some proof for this observation by now. Please give me a full proof of this matter.
Thank you

There is no "proof". One does not prove things in science like one does in math. Science is only as good as the evidence that supports it.

Kepler developed his laws from observational data accumulated by Tyco Brahe. Newton observed that if the force between two masses was proportional to their masses and inversely proportional to the distance between them squared, this fit Kepler's laws. The "proof" of Newtons law of universal gravitation is simply that it works.

AM

 

[pete20r2]

He didn't, he observed them.He was Tycho Brahe's student and had access to Brahe's catalogues which where extremely accurate with over a decades worth of data. This allowed him to use the data to empirically show his laws to be true.

Indeed,
I believe at the end of his paper with this subject he wrote:

“If thou [dear reader] art bored with this wearisome method of calculation, take pity on me who had to go through with at least seventy repetitions of it, at a very great loss of time; nor wilst thou be surprised that by now the fifth year is nearly past since I took on Mars. . . .”

 

[parsa418]

Then how did Newton exactly observe this. I mean there weren't any force sensors at the time or was there something like it?

 

[pete20r2]

The equation is in parts: eg 1/r^2 because if the radius of a circle doubles, the circumference quadruples. This means the same force is distributed over 4 times the area and is 4 times as weak.

Someone else may enlighten you about the other parts of the equation.

 

[Drakkith]

Then how did Newton exactly observe this. I mean there weren't any force sensors at the time or was there something like it?

The observations and data from Tycho Brahe and Kepler provided huge amounts of data for him to use in his calculations. It took him YEARS to do these calculations by hand.

 

[Andrew Mason]

Then how did Newton exactly observe this. I mean there weren't any force sensors at the time or was there something like it?

Newton postulated that the force of gravity provided the centripetal acceleration that could be calculated from the observed data.

It is fairly simple for a circular orbit:

This acceleration is a_c = v^2/r = (2\pi r/T)^2/r =4\pi r/T^2

If the force of gravity on a planet was proportional to m/r^2 where m is the planet mass and r is the radius of orbit, (ie. F_g = km/r^2 where k is the constant of proportionality) then the acceleration would be:

a_g = F_g/m = k/r^2 = 4\pi r/T^2

which means that:

r^3/T^2 = k/4\pi = constant

This is just Kepler's third law of planetary motion.

(It is much more complicated to work it out for elliptical orbits).

AM