Question

how newton was able to correctly guess.tht gravitational force is inversely proportional to square of displacement...?...though at tht time there was no mean to study it and find its value​

Answer 1

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He did it using the experimental value, which the best available at that time was approximately equal to the inverse proportional to the square of the displacement using the Keplers Laws.

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Kepler had three laws regarding planetary motion.

According to the third law,

Suppose That A Planet revolves around the sun at a velocity V and at a approximate distance r.

The force acting on the planet is directly proportional to V^2 / r.

The force acting on the planet is directly proportional to V^2 / r.T^2 is directly proportional to r^3.......... [ | ]

You can find the derivation of this law in a different chapter.

Let T be the time period of revolution of the planet.

Now, V = D/T.

Now, V = D/T.T = D/V

Here distance is the circumference of the area swept by the planet .

$v \: = \: \frac{2\pi \: r}{t}$

${v}^{2} = \frac{4 {\pi}^{2} {r}^{2} }{ {t}^{2} }$

Therefore,

${v}^{2} \: \alpha \: \frac{4\pi \: {r}^{3} }{ {r}^{3} }$

Hence, he correctly guessed from the above equations that

${v}^{2} \alpha \: \frac{1}{r}$

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Answer 2

Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.[note 1] The publication of the theory has become known as the "first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.[1][2][3]

This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning.[4] It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1687. When Newton presented Book 1 of the unpublished text in April 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him.

In today's language, the law states that every point mass attracts every other point mass by a force acting along the line intersecting the two points. The force is proportional to the product of the two masses, and inversely proportional to the square of the distance between them.[5]

The equation for universal gravitation thus takes the form:

{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},} {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},}

where F is the gravitational force acting between two objects, m1 and m2 are the masses of the objects, r is the distance between the centers of their masses, and G is the gravitational constant.

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