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Newton's Law of Universal Gravitation

"God does not care about our mathematical difficulties. He integrates empirically" - Albert Einstein ( 1879 - 1955 )

Proposition: The Kepler Laws allowed Newton to establish his Law of Universal Gravitation without which Newton would have been at a considerable loss!

Johannes Kepler's empirical foundation for Newton's Law of Universal Gravitation


Johannes Kepler

circa 1610 Johannes Kepler portrait - artist unknown


According to Kepler's 1st Law ( Planetary Law of Ellipses ) the planets move in elliptical orbits with the sun at one the foci:


planetary ellipse


Prior to Johannes Kepler ( 1571 - 1630 ), Copernicus ( Polish: Mikolaj Kopernik, 1473 - 1543 ) supposed that planetary orbits were approximate circles which indeed are ellipses of a special kind. In any event, because circles are special cases of generalized ellipses, any derivations based upon Kepler's 1st Law ( Planetary Law of Ellipses ) for ellipses must also hold true for circles.

Kepler's 2nd Law ( Equal Areas in Equal Times ) states that a planet's motion maps out equal areas in equal times for a radius vector, radius_vector.png, drawn from the sun - foci to the planet. This implies that every planet traverses its "assumed circle" with constant velocity, v_velocity_vector.png, but with a centripetal acceleration, acceleration_vector.png, directed inwards towards the center of the assumed circle as follows:


centripetal acceleration


Assuming small incremental changes in s_distance.png,

calculus limit,

we have the following ratios:

centripetal acceleration for circular planetary orbit

Also another derivation:

circular planetary orbit

Kepler's 3rd Law ( Harmonic Law ) mathematically formalized the data painstakingly collected earlier by Tycho Brahe ( 1546 - 1601 ) who was born in Skane, Denmark [ now in Sweden ] by clearly stating the following ratio which has the same value C_constant.png for all planets:

Kepler's 3rd Law

That is, total time period T.png is proportional to the 3/2 power of radius r.png for all planets written as follows:

planetary constant

But we also know from Kepler's 2nd Law ( Equal Areas in Equal Times ) that

centripetal acceleration

This then provides one of the first and most important empirical proofs of Newton's Law of Universal Gravitation since planetary centripetal acceleration solely depends upon the distance of the planet from the sun!!

Derivation of Newton's Universal Law of Gravitation based upon Kepler's equations

Sir Issac Newton

Sir Isaac Newton ( 1643 – 1727 )

Of course, planets do not accelerate and fall into their center bodies due to a countervailing centrifugal force and Newton's Law of Inertia.

Newton's genius shone when he made his 2nd Law of Motion both famous and universal:

Newton's 2nd law of motion

on earth to apply to planetary bodies and the sun as follows:

universal constant

Hence, here is a further interpretation of the force which maintains a planet's orbit about the sun and which is directly dependent upon the planet's distance from this central body.

Moreover, the universality of Newton's concept of forces acting upon bodies did not just allow for the relative mass of the sun to effect a planet's orbital motion, but equally that there exists a mutual force arising from a planet's mass and hence its reciprocal effect upon the sun. This can all be expressed as follows:

sun

Because of these reciprocal and mutual effects of forces involved - i.e., action vs. reaction - we achieve the following:

force

This says that there is some universal value, k.png, for both the sun and any planet and therefore for any body whatsoever in the solar system. This universal factor of proportionality, k.png, is called the gravitational constant and was finally and fairly accurately implicitly determined [ within 1% ] in 1797 by British Henry Cavendish ( 1731 - 1810 ) whose original intent was to calculate earth's density relative to water. Rather, Cavendish misinterpreted his discovery of the gravitational constant as earth's density; nevertheless, the torsion balance apparatus conceived and built earlier by then deceased John Mitchell ( Reverend, geologist and English natural philosopher, 1724 - 1793 ) was crated and sent on to Cavendish which allowed him to complete his 1797 experiment to within 1% accuracy of the modern gravitational constant calculation! Cavendish published his results in Philosophical Transactions of the Royal Society of London, Vol. 88 (1798), pgs. 469-526.

Continuing ...

Newton's law of universal gravitation

Or, in words

mutually attractive force.

Therefore using slightly more explicit words,

celestial suspended bodies

Examples

§ Example 1). Two suspended masses, one large and the other smaller, are separated from each other as follows:

mass

The mutual gravitational force of attraction is therefore:

mutual gravitational force of attraction

§ Example 2). Two suspended celestial bodies, one large and the other smaller, are separated from each other as follows:

celestial bodie

The mutual gravitational force of attraction is therefore:

Newton's mutual gravitational force of attraction

Variation of Gravitational Force of Attraction

§ Example 3). An American astronaut lands on a distant exoplanet whose surface gravity force in terms of acceleration is 3.5 m/sec2 and whose radial distance to its center is 2,300 km.

nasa exoplanet

Hence, find the force of acceleration of gravity at an altitude of 500 km above the exoplanet's surface as follows:

universal gravity constant

This implies

universal gravitational constant

and at the planet's surface

gravity acceleration

Finally, the astronaut will experience gravity acceleration of the following amount:

gravity acceleration at planet surface

Now assume that our American astronaut weighs 150 lbs at earth's surface, he/she will have weight

astronaut weight

on this exoplanet's surface.

At 500 km above the planet surface, our American astronaut will have weight

astronaut mass and weight

note: these examples are used in the future upcoming Relativity Science Calculator Mac application

Newton's Laws Produce Universal Law of Conservation of Energy and Confirm Galileo's Law of Falling Bodies

Newton's 2nd law and Universal Law of Gravitational Force of Attraction

And by applying a bit of integral calculus to Newton's laws, the Universal Law of Conservation of Energy will be clearly evinced:

Universal Law of Conservation of Energy

Notice that in the equations

variation of gravitational force

and

Derivation of acceleration without time

that we have here a mathematical demonstration of Galileo Galilei's Law of Falling Bodies since the final velocities and hence time of any falling body to a certain distance in a vacuum  depends solely upon the gravitational force of the attracting body and has absolutely nothing to do with the mass of the falling body itself!!




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