How did Newton derive his law of gravity?

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For some reason I have been extremely curious of how Newton derived his mathematical law for gravitation. You know

$$F= \frac{GMm}{r^2}$$

So the first thing I did before i posted this was search Google and I foundhttp://www.relativitycalculator.com/Newton_Universal_Gravity_Law.shtml" . It talks about a possible derivation of how Newton could have derived his law of gravity. It looks pretty good and I understand most of it BUT there is one part I don't understand. There is one part in the derivation that looks like this.

$$\frac{C}{M}=\frac{c}{m}=\frac{k}{4\pi}$$

In this situation k is the gravitational constant and 4pi is... 4pi. My question is WHERE did this k/(4pi) come from? Why set these two equations equal to k/(4pi)? Can someone tell me where this k/(4pi) came from??

Thanks for your responce :D

 

[alxm]

Well, get rid of it and see what happens!

$$\frac{C}{M}=\frac{c}{m}=k$$
then
$$C = kM, c=km$$
$$f^2 = f*f' = m\frac{4\pi^2C}{r^2}M\frac{4\pi^2c}{r^2}$$
$$= m\frac{4\pi^2kM}{r^2}M\frac{4\pi^2km}{r^2}$$
$$= 16\pi^4k^2\frac{M^2m^2}{r^4}$$

$$f = 4\pi^2k\frac{Mm}{r^2}$$

So you get the same result but with the unnecessary constant of $$4\pi^2$$ in front, so they defined k in a way that eliminates that constant..

Of course, if you want to, you could also just say $$G = 4\pi^2k$$.

 

[astrokreng]

Newton's law of universal gravitation is one of the fundamental laws of physics that explains the force of gravity between two objects. It states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. But how did Newton come up with this mathematical equation?

The story of how Newton derived his law of gravity goes back to the 17th century when he was a young student at Cambridge University. He was fascinated by the works of Galileo and Kepler, who had made significant discoveries in the field of astronomy and motion. Newton was particularly interested in Kepler's laws of planetary motion, which described how planets move around the sun in elliptical orbits.

Using Kepler's laws and his own mathematical skills, Newton began to investigate the force that kept the planets in their orbits. He hypothesized that this force must be proportional to the mass of the objects and the distance between them. He also realized that this force must act in a direction towards the center of the objects. This led him to develop the concept of a universal force of gravity.

To further test his hypothesis, Newton turned to experiments and observations. He studied the motion of pendulums and falling objects to understand the effects of gravity on Earth. He also analyzed the motion of the moon around the Earth and the motion of planets around the sun. Through these experiments, Newton collected data and refined his ideas until he finally arrived at his law of universal gravitation.

In terms of the specific equation F= \frac{GMm}{r^2}, the constant G is known as the gravitational constant and was first calculated by Henry Cavendish in the 18th century. It represents the strength of the gravitational force between two objects and is a fundamental constant in physics. The factor of 4\pi in the equation is a result of mathematical simplification and represents the geometry of a three-dimensional space.

In summary, Newton derived his law of gravity through a combination of observation, experimentation, and mathematical analysis. His work revolutionized our understanding of the universe and laid the foundation for modern physics. The k/(4pi) factor in the equation may seem arbitrary, but it is a fundamental part of the equation that represents the strength of the gravitational force and the nature of space.