proof of third law of planetary motion using Newton law of gravitation
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Keplers 2nd law says that a line connecting the sun and an orbiting planet would sweep across an equal area of space in the same amount of time.
That's pretty obvious if planets orbited in circular paths, however, keplers 1st law states that all planets orbit the sun in elliptical paths.
If you consider both of these laws then it becomes apparent that as planets get closer to the sun their orbit speed increases, which means their speed decreases at further distances.
The inverse square law of gravitation comes as a consequence of combining Keplers Laws with Newtons Laws of motion.
Newtons 3rd law shows the conservation of angular momentum. If we apply the conservation of angular momentum, Newtons 2nd law of motion and Keplers 2nd law shows us how the force of gravity is inversely proportional to the distance squared.
F=m1m2d2F=m1m2d2
This is how Newton came up with his law of universal gravitation.
Of course the above equation was turned into what it is now with the addition of “G”.
F=Gm1m2d2F=Gm1m2d2
Newton knew that the force of gravity was directly proportional to the product of the two masses. In order for this to hold true, a constant had to be added to the first equation. This is the gravitatioonal constant and where it comes from.
Newton realized this theoretically, however he was not able to determine the value of G. That was finally calculated, but well after Newton had died.
That's pretty obvious if planets orbited in circular paths, however, keplers 1st law states that all planets orbit the sun in elliptical paths.
If you consider both of these laws then it becomes apparent that as planets get closer to the sun their orbit speed increases, which means their speed decreases at further distances.
The inverse square law of gravitation comes as a consequence of combining Keplers Laws with Newtons Laws of motion.
Newtons 3rd law shows the conservation of angular momentum. If we apply the conservation of angular momentum, Newtons 2nd law of motion and Keplers 2nd law shows us how the force of gravity is inversely proportional to the distance squared.
F=m1m2d2F=m1m2d2
This is how Newton came up with his law of universal gravitation.
Of course the above equation was turned into what it is now with the addition of “G”.
F=Gm1m2d2F=Gm1m2d2
Newton knew that the force of gravity was directly proportional to the product of the two masses. In order for this to hold true, a constant had to be added to the first equation. This is the gravitatioonal constant and where it comes from.
Newton realized this theoretically, however he was not able to determine the value of G. That was finally calculated, but well after Newton had died.
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