Wheres G (m1*m2)/r^2 come from?

[MAGNIBORO]

hi, I'm not very familiar with physics But I want to know how Newton came to the famous equation
$$F=G \, \frac{m_{1}m_{2}}{r^{2}}$$
He makes a mathematical demonstration and supports it with experimental evidence or
he "creates" an equation that complies with the experimental data?
If the Newton explanation is too long or complex I do not mind seeing an current proof
What I really want to know is why this is true? (in classic physics)
thanks

 

[Dale]

I am not sure I understand the difference between the two things you are asking. In both cases you seem to be describing the scientific method of hypothesizing a theoretical equation and checking for consistency with experimental data.

 

[BvU]

I suppose the inverse square law originated in observations of planet orbits. The proportionality with the masses then is a smaller step.
Google is your friend: here, here and here too

 

[phyzguy]

It's a fascinating story really. Newton was trying to understand what kept the celestial bodies in their orbits. He understood the concept of centripetal force, and realized that if the gravity of the Earth extended out to the moon, then the gravitational pull of the Earth on the moon could be the centripetal force that keeps the moon in its orbit. He then calculated the necessary pull and found that it was about 1/3600 of the pull that the Earth exerts at its surface. Since the moon is about 60 Earth radii away, he hypothesized that if the force fell off as the inverse square, then everything would work out. This wasn't too hard to believe, since it was known that things like the intensity of light fall off as the inverse square, because the area over which the light is spread increases as the square of the radius. Deducing that the same pull of gravity which causes things to fall to the Earth could also explain why the moon doesn't fall to the Earth was, in my opinion, a tremendous stroke of insight and a testimony to Newton's genius. It's worth reading Newton's own words on this topic:

“In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force with wch [a] globe revolving within a sphere presses the surface of the sphere) from Kepler's rule of the periodic times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665-1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then than at any time since.”

 

[MAGNIBORO]

I am not sure I understand the difference between the two things you are asking. In both cases you seem to be describing the scientific method of hypothesizing a theoretical equation and checking for consistency with experimental data.

The first method is to arrive at the equation purely through mathematical demonstrations and then back it with evidence (I do not know if this is done in physics, I am more familiar to seeing mathematical proofs)
The second method would be to search by brute force the equation And then put values and compare them with other experimental ones , supposse:
$F=C_{1} \, \frac{m_{1}+m_{2}}{r^{2}}+C_{2}$
or
$F=C_{1} \, \ \frac{|{m_{1}-m_{2}|}}{r^{2}}+C_{2}$
(but this equations are obviously wrong)

I suppose the inverse square law originated in observations of planet orbits. The proportionality with the masses then is a smaller step.
Google is your friend: here, here and here too

I search in google but I have bad experiences learning physics by myself (Quantum physics)

It's a fascinating story really. Newton was trying to understand what kept the celestial bodies in their orbits. He understood the concept of centripetal force, and realized that if the gravity of the Earth extended out to the moon, then the gravitational pull of the Earth on the moon could be the centripetal force that keeps the moon in its orbit. He then calculated the necessary pull and found that it was about 1/3600 of the pull that the Earth exerts at its surface. Since the moon is about 60 Earth radii away, he hypothesized that if the force fell off as the inverse square, then everything would work out. This wasn't too hard to believe, since it was known that things like the intensity of light fall off as the inverse square, because the area over which the light is spread increases as the square of the radius. Deducing that the same pull of gravity which causes things to fall to the Earth could also explain why the moon doesn't fall to the Earth was, in my opinion, a tremendous stroke of insight and a testimony to Newton's genius. It's worth reading Newton's own words on this topic:

“In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force with wch [a] globe revolving within a sphere presses the surface of the sphere) from Kepler's rule of the periodic times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665-1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then than at any time since.”

wow , Then it was based on experimental data and gave form to the equation Being helped with observations of old physicists. very impressiveThank you for all your replies, they helped me understand the famous equation And to understand how physicists work when they want to demonstrate something =D

 

[Dale]

The first method is to arrive at the equation purely through mathematical demonstrations and then back it with evidence
...
The second method would be to search by brute force the equation And then put values and compare them with other experimental ones

I don't think that either one accurately represents the scientific process. The first method would be closer, but good theories are not just pure mathematics. Instead, they are informed by all previous experimental results and theories. The math of the new theory needs to match the math of previous theories within their domain of experimental validation. In addition, the new theory should predict different experimental results outside the domain of current experimental validation.

I don't think that the second approach is possible, even in principle.