How can we deduce newton's law from Keplr's law?
Answers
Explanation:
Newton’s Law of Gravitation is states that in this universe attracts every other body with a force which is directly proportional to the product of their masses and is inversely proportional to the product of the squares of the distance between them.
Newton’s Law of Gravitation can be easily obtained from Kepler’s Laws of Planetary Motion.
Suppose a planet of mass m is revolving around the sun of mass M in a nearly circular orbit of radius r, with a constant angular velocity ω. Let, T be the time period of revolution of the planet around the sun.
ω=2πT … (1)
The centripetal force acting on the planet for its circular motion is:
F=mrω2=mr(2πT)2=4π2mrR2,
According to Kepler’s Third Law:
T² α r³ (Or) T² = Kr³
Where, K = Constant of Proportionality.
Therefore,
F=4π2mrKr3=4π2K×mr2 … (2)
F∝mr2 (∵ 4π2K is a constant) … (3)
This centripetal force is provided by the gravitational attraction exerted by the sun on the planet. According to Newton, the gravitational attraction between the sun and the planet is mutual. If force F is directly proportional to the mass of the planet (m). It should be directly proportional to the mass of the sun (M).
Hence, the factor 4π2K∝M (Or) 4π2K=GM … (4)
Substitute equation (4) in the equation (2), we get:
F=GMmr2, Which is Newton’s Law of Gravitation.
Answer:
Explanation:
You can prove that Kepler’s 3rd Law (period squared is proportional to semi-major axis cubed) is only true for a 1/r2 force. This can be done by using Kepler’s 2nd law to show that angular momentum is conserved, relating the angular momentum of an elliptical orbit to the semi-major and semi-minor axes of the ellipse, and then showing that only a 1/r2 force produces such an ellipse.
Using the fact that Kepler’s 3rd holds for all planets, you can then show that the force from the Sun must be proportional to the mass of the planet.
This is as far as you can get with just Kepler’s Laws.
From there, you have to argue that the gravitational force must be symmetric to ensure conservations of momentum (Newton’s 3rd law), which means that if the force is proportional to the mass of the planet, it also must be proportional to the mass of the Sun.
And so you have Newton’s Law of Gravity: The force of gravity is proportional to the mass of the first object (Sun) and the second object (Earth), and inversely proportional to the square of the distance.
It’s actually easier to go in the other direction: Start with Newton’s Law of gravity, and show that Kepler’s Laws are necessary outcomes for the case of a bounded (elliptical) orbit. I go through this derivation in my introductory classical mechanics class, it’s fairly standard.