# Gravitational law and point mass

1. Aug 1, 2013

### jd12345

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

I am seriously confused by "point mass" in this case. I have studied this countless number of times and suddenly today I feel confused about it.
I have two main questions here: -
1) After Newton found this law from Kepler's observations how did he deduce that every "point mass" attracts?
2) What do you mean by "point mass" anyway?

P.S. - I don't know why I always feel gaps in my understanding of physics and math concepts. Once in a while I come up with these crazy questions.

2. Aug 1, 2013

### WannabeNewton

Hi jd! You have all the right to ask about "point masses" because the need to rigorously treat them in various fields of physics is quite deep and in no way trivial.

I don't know the answer to (1), which is ironic given my username, but as for (2) you can intuitively think of a "point mass" as a test particle whose size tends to zero. Often in introductory texts you will see the planets treated as a "point masses" when talking about their orbits around the Sun because their sizes are taken to be negligible compared to their distances from the Sun, the size of the Sun etc.

3. Aug 1, 2013

### technician

when Newton discovered the laws of gravitation he realised that the inverse square law was an indication of spherical symmetry.
He realised (like Faraday in electricity and magnetism) that Force fields could be represented by lines of force originating at a point.
The great insight was to realise that this meant that all masses could be considered as point objects.

PS I dont think Newton made much of the lines of force representation....he was good at maths.
Faraday used it a lot because he was not so good with maths and liked a 'physical' representation....he drew lines of force!!

Last edited: Aug 1, 2013
4. Aug 1, 2013

### Naty1

1/
http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation

Maybe everything he held in his hand and let go moved down to earth? Or maybe because he did not know of any but like charged objects repelling?? Or maybe he copied the idea from Robert Hooke?? [see http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation#Plagiarism_dispute]

2/a 'point mass' in classical gravitational theory means you can approximate gravitational dynamics of external [distant] objects as if their mass was concentrated at a point, the center of mass. A spherically symmetric body, for example, affects external objects gravitationally as if all of its mass were concentrated at its center.

because there are always such'gaps'....the more you think, the more you uncover about what you don't know.
edit: Einstein went from Newton's idea's to the theory of General Relativity!!!!!!!!!!!

5. Aug 2, 2013

### D H

Staff Emeritus
Newton summarizes his theory of gravity in book 1 of his Principia. He details how arrived at this result in book 3. In this book, he starts by looking with the moons of Jupiter and Saturn, then at the planets, then at the Earth's moon. He found that the moons of Jupiter and Saturn are attracted toward Jupiter with a force inversely proportional to the square of the distance to their primary, the planets are attracted toward the Sun with an acceleration inversely proportional to the square of the distance to the Sun, the Moon with an acceleration inversely proportional to the square of the distance to the Earth.

However, Newton found that the Sun is a huge perturber of the orbit of the Moon, and to a lesser extent, the orbits of the moons of Jupiter and Saturn. The planets similarly perturb each other's orbits about the Sun. All of these heavenly bodies apparently were attracted to one another. From that, he deduced that gravity makes *everything* be attracted to everything else.

As an acid test, he looked at why things fall near the Earth. He found that objects fall toward the Earth for the same reason that the Moon orbits the Earth.

Newton didn't talk about point masses in his discussion of gravitation. He talked about "corpuscles" (test particles) and spheres. Newton's theorem XXXI (now called Newton's shell theorem) showed that outside an object with a spherical mass distribution, the gravitational attraction toward that object is proportional to the object's mass and inversely proportional to the square of the distance to the center of the object. This was critical to the development of his universal theory of gravitation because he used the nearly spherical heavenly bodies as a basis for the development of this theory.

Only objects with a spherical mass distribution can be considered as point masses. The Earth is not spherical. It is much better modeled as an oblate spheroid. This oblateness has a marked effect on satellites in low Earth orbit, and to a much lesser extent, on the orbit of the Moon. The oblateness causes the orbital plane of a satellite to precess. Pick just the right orbit and the precession rate will be one revolution per year. These are called sun-synchronous orbits, and a good number of Earth observation satellites are placed in such orbits.

6. Aug 2, 2013

### technician

My post was with reference to what Newton would have known.
If you find that F is proportional to 1/r2 then it is a stroke of genius to realise that this represents what a point mass would produce.
I dont know how much Newton knew about the Earth being an oblate spheroid or how Earth satellites (other than the moon) (!!!!!!Newton!!!) would behave or how the orbit of mercury did not fit with the calculations. But he got something started.
The original post is very clear (I think) in that it asks about Newton's world.
If it needs further discussion that was beyond Newton perhaps we need another post.
I understand that we are supposed to stay 'on topic'?

Last edited: Aug 2, 2013
7. Aug 2, 2013

### technician

Newton summarizes his theory of gravity in book 1 of his Principia. He details how arrived at this result in book 3. In this book, he starts by looking with the moons of Jupiter and Saturn, then at the planets, then at the Earth's moon. He found that the moons of Jupiter and Saturn are attracted toward Jupiter with a force inversely proportional to the square of the distance to their primary, the planets are attracted toward the Sun with an acceleration inversely proportional to the square of the distance to the Sun, the Moon with an acceleration inversely proportional to the square of the distance to the Earth.

However, Newton found that the Sun is a huge perturber of the orbit of the Moon, and to a lesser extent, the orbits of the moons of Jupiter and Saturn. The planets similarly perturb each other's orbits about the Sun. All of these heavenly bodies apparently were attracted to one another. From that, he deduced that gravity makes *everything* be attracted to everything else.

Last edited: Aug 2, 2013
8. Aug 2, 2013

### D H

Staff Emeritus
9. Aug 2, 2013

### Naty1

very interesting...I was unaware of that....

10. Aug 2, 2013

### technician

Take yourself back to 1600 or thereabouts....Kepler made astronomical observations. All he could really measure were times (of orbits) and distances. Go into your back garden with binoculars and you will easily see the moons of jupiter just like Kepler did... it is easy to measure the times of the orbits of Jupiters moons. The distance of the moons from jupiter may not be so easy but you could find a way.
So you have a table of times of orbits and radius of orbits.
Newton looked at this table and 'saw' that T2 was proportional to r3
This suggested to him that a force existed between the planet and the moon !!! amazing
It told him that the force F was inversely proportional to the distance (r) squared. even more amazing.
If you are a student of physics it is a great exercise to use the well known inverse square law to work back
to come up with T2 = const x r3
This is Newton's greatness, it is consistent with 'point masses' and any distractions due to oblate spheroids are for another discussion.

11. Aug 2, 2013

### technician

Last edited: Aug 2, 2013
12. Aug 2, 2013

### Staff: Mentor

13. Aug 2, 2013

### king vitamin

I believe DH's computation is using the moon's observed orbit + Newton's law of gravitation to predict the acceleration of gravity at the Earth's surface. An equivalent computation can be found in Newton's Principia, book 3, proposition IV (in antiquated form). This is the acid test that the force between celestial bodies is "the same as the force which we call gravity" in Newton's words.

14. Aug 2, 2013

### D H

Staff Emeritus
That is exactly what I was trying to show: That an object with a mass of 0.0123 Earth masses that orbits the Earth at 60.36 Earth radii in 27.32 days is consistent with gravitation at the Earth's surface.

(Emphasis mine.)

Newton never said $F=\frac{Gm_1m_2}{r^2}$, or even $F=ma$. We have all kinds of mathematical tools now that simply didn't exist in Newton's time. To make matters worse, Newton didn't even use the mathematical tools that were available in his time in his Principia. He instead used proportionalities instead of equalities, words and geometry instead of algebra and calculus. From a modern perspective, slogging one's way through Newton's Principia is a special kind of torture.

15. Aug 3, 2013

### technician

Newton never said F=Gm1m2r2, or even F=ma. We have all kinds of mathematical tools now that simply didn't exist in Newton's time. To make matters worse, Newton didn't even use the mathematical tools that were available in his time in his Principia. He instead used proportionalities instead of equalities, words and geometry instead of algebra and calculus. From a modern perspective, slogging one's way through Newton's Principia is a special kind of torture.

On page 45 of his book he analyses Centripetal force and consider various possibilities.
The key one as far as gravitaion is concerned is Cor.vi : If the periodic times are as the 3/2 power of the radii and therefore, the velocities are inversely as the square root of the radii then the Centripetal force will be as inversely as the square of the radii
In general he states that if T is proportional to Rn then velocity is proportional to 1/(Rn-1) and F is proportional to 1/(R2n-1)

It is Cor.vi, also known as Kepler's 3rd law that is the basis for F is proportional to 1/R2
Newton analysed motion geometrically by considering small steps. I think he is recognised as one of the major scientists/mathematicians to develop the idea of calculus.