# Homework Help: Newton Law of Universal Gravitation

1. Aug 20, 2004

### scoutfai

we all learn this law. It state that the force between two bodies of mass $$m_1$$ and $$m_2$$ is attractive in nature. The gravitational attraction F is directly proportional to the product of the masses of the two bodies and inversely proportional to the square of the distance between them.
From the definition, the formula
$$F=\frac{Gm_1m_2}{r^{2}}$$
can be formed.
to form this formula, it is not a problem to me since we have the Definition . Now my question is , how Newton realize or found this Definition? What i mean is, how he know that the gravitational force is directly proportional to the product of the masses and inversely to the square of the distance ?
we can form the formula cause we have the definition. But to form the definition we must have something also, how Newton found out this relationship between gravitational force , masses and the distance ? by trial and error ? by experiment ? by assumption ??

2. Aug 20, 2004

### HallsofIvy

Very nice question. Newton, who was one of the main developers of calculus, was able to show, using calculus, that only an "inverse square law" would give orbits of the planets that satisfied Kepler's laws: that the orbit is an ellipse with the sun at one focus, that planets "sweep out" equal areas in equal times, and that the square of the period of an orbit is proportional to the cube of the average radius (hope I got that one right!).
Of course, those laws were due to observation of orbits by Kepler.

3. Aug 20, 2004

### Locrian

Well put Ivy. If you are ever interested scoutfai, Carl Sagan does a fantastic job of going over the history behind this in his infamous series "Cosmos." It is not terribly mathematical, but that's actually something you could do yourself once you see the information.

4. Aug 20, 2004

### BobG

Specifically, the equation of the third law (rearranged) is:

$$\frac{4\pi^2a^3}{\tau^2}= \mu$$

The relationship between the planet's average radius and it's orbital period always came up to the same number - $$\mu$$

Progressing to ellitptical orbit's and Kepler's second law, the satellite has to accelerate towards perigee to keep the area swept out equal. Guess what? $$\mu$$ also works to explain the rate of acceleration.

Apply both of those principles to the Moon orbiting the Earth, and suddenly you get a new value for $$\mu$$. But the new value works just as well for to explain various aspects of the Moon's orbit. In fact, the value for $$\mu$$ for the Moon's orbit work's pretty good on the acceleration of objects on Earth, as well.

One of the values - the one for the Moon's orbit and falling objects on Earth - is called the geocentric gravitational constant. The other - the one for the planets around the Sun - is called the heliocentric gravitational constant.

Years later, after looking at the orbits of Jupiter's moons, etc., it was found that you could break the gravitational constants into two separate terms. One was a universal gravitational constant and the other was dependent on the orbited object's mass.

5. Aug 21, 2004

### scoutfai

Which law came first ?

After i read all of your message, it make me feel that Kepler's Law was introduced to the world first before the Newton Law of Universal Gravitation ,is this true?
So, after Kepler introduced his laws, Newton apply it to find out the relationship between gravitational force, masses and distance, right ?

6. Aug 21, 2004

### arildno

On the basis of Tycho Brahe's extraordinarily precise measurements, Kepler was able, after many years of study, to deduce that all the observations could be fitted into the general patterns suggested by what is now known as Kepler's 3 laws.

That is, Kepler's 3 laws should be regarded as empirical laws, patterns extracted from a mass of data.

Newton formulated an elegant and concise theory of how these empirical laws might be derived from the idea of the universal attraction of matter.

7. Aug 21, 2004

### scoutfai

From the last sentence, it seen like you try to state that Newton Law of gravitation is actually expressing the same knowledge as the Kepler's 3 Laws, the only different is Kepler derive his law by using some data of measurement while Newton is just 'create' it with out any data....... is it ? i know what had state by the Kepler's 3 Laws and Newton Law is different at first, but if you keep working on Newton Law, you will derive the Kepler's Law (ie same knowledge, it just like both of them try to explain how to calculate the value 4 , but one of them state 2+2 while another one state 2*2 , but at the end the answer is the same)
So, after consider all of your replies, it make me feel that after Kepler introduced his Law, Newton try to create his own Law base on Kepler's 3 Laws.....in other word, Newton keep working on the Kepler's Law until he derive that
$$F=\frac{Gm_1m_2}{r^{2}}$$
so, Newton Law of Universal Gravitation is the Subset of Kepler's 3 Laws ( or element ), right ?

Actually, after i read all of your replies, i don't feel that any of you answer my question correctly, that is how Newton find out the relationship between masses, distance and gravitational force.. Newton must base on something to derive that relationship, it is impossible for Newton to use his imagination only to understand the universal law and found the relationship.....i don't believe this world has such clever people. Can you derive a new law by sitting on the chair only ? of course not, you must have some information, just like Kepler, and then you can keep working on that information and finally you may be able to derive a law....... this is the point, i want to know where Newton get the source and the process in deriving the relationship.....

8. Aug 21, 2004

### arildno

Newton's law of gravitation DO contain more "information" than Kepler's 3 laws.
Kepler's 3 laws solely concern a description of how the celestial bodies move.
Newton's law yields Kepler's 3 laws as a special case when applied to movement of celestial bodies.
However, you may also use Newton's law (among other things) to describe the motion of a projectile close to earth, in which Newton's law implies that the projectile's acceleration is (roughly) constant.
You may also use Newton's law to estimate the density of a rotating star, for example.
Nowhere in Kepler's laws can you find evidence for these types of relationships.

9. Aug 21, 2004

### HallsofIvy

By the way, Tycho Brahe did these "extraordinarily precise measurements" with the naked eye without using telescopes which were invented only late in his life.

10. Aug 21, 2004

### arildno

I know, that's why I used the word "extraordinarily"..

Without T.B.'s detailed work, I don't think Kepler would have come upon his own ideas.
It is somewhat unfortunate that Tycho Brahe is mainly remembered for his futile attempt to save the Ptolemean system; he clearly ranks among the finest experimental astronomers of all time, and should be honoured for that.

11. Aug 21, 2004

### krab

Wrong. Newton's law includes far far more than that covered by Kepler. Besides what Arildno mentioned, I should also mention: tides, variation of g at different points on earth, gravitation of irregularly-shaped bodies, ability to deal with more than 2 bodies. The last is very important: perturbation of a planet's orbit due to other planets is a measurable effect, and helped predict existence of yet-unfound planets.

So you see, Newton's one tiny formula is incredibly powerful, much more powerful than Kepler's laws. The way in which Newton arrived at the Law is called induction and is one of the main methods in science that produces progress in understanding.

12. Aug 21, 2004

### HallsofIvy

A crucial point is that Newton recognized that "gravitation" applied to objects on earth AND to planets, moons, etc.

13. Sep 1, 2004

### poolwin2001

Kepler pored over the data given by T.B and emperically derived the laws.

Newton thought let gravity be force between any two objects which has mass
and given by F=M1*M2......

Then with this he was able to derive Kepler's laws!!

With this simple assumption he was able to prove a lot of things.

So Newton's law is not the subset of Kepler's laws
as with Newton's one law ,we can prove kepler's 3 laws.

So Newton's law is More basic than Kepler's.

Cheers
Poolwin2001

14. Sep 1, 2004

### HallsofIvy

By the way, it is a tribute to both Kepler and Tycho Brahe that Kepler dropped his first theory: he had worked out a very "pretty" theory in which Mercury's orbit was inscribed in a tetrahedron, which had a cube circumscribed about it (and Venus' orbit inscribed in that), with a octahedron circumscribed about the cube, etc. In other words, everything was based on the Platonic solids (the stars were on a sphere circumscribed about everything!). Sounds very mystical today (or like some of the stuff we get on this forum!) but very pretty none the less. When Kepler realized that that did not match Tycho Brahe's observations, he chucked the whole thing and eventually came up with "Kepler's Laws". A tribute to his integrity and his respect for Brahe's observations.

15. Sep 1, 2004

### HallsofIvy

One of the reasons I mentioned that was that the textbook "Calculus of One and Several Variables" by Salas, Hille, and Etgen, now in 9th edition(used to be just "Salas and Hille". I suspect Etgen revised the newer editions), in its "biographical" blurb on Brahe, contains the sentence "For more than twenty years he looked through his telescopes and recorded what he saw."

I almost through the book out the window when I saw that!

16. Sep 1, 2004

### arildno

An excellent and (horrifying!) example of the sheer ignorance and thouroughly unjustified contempt (thinly veiled in the example) which so-called scientists of today have of the accomplishments of earlier generations.
It's only their own stupidity which is laid bare by such comments; they are as worms compared to minds like Tycho Brahe's.