Derivation of Newton's Law of Universal Gravitation

In summary: For a square pyramid, would the lines at the base be pointing into the center of the base? Because the base is a square and the four sides can be seen as flat surfaces. And at the top, would they be pointing horizontally?For irregularly shaped objects, is there a way to visualize the lines of force?The lines of force for a gravitational field are not necessarily straight lines, like they are for an electric or magnetic field. They represent the direction and magnitude of the force at each point in the field. For a point mass, the lines would radiate outward from the mass, getting further apart as you move away from it. For a cylinder, the lines would be perpendicular to the flat surfaces and pointing towards the
  • #1
WK95
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##F = G \frac{ m_{1} m_{2}}{ r^{2} } ##

Where does the formula come from? And why does it work that way?

How would it relate to Newton's Second Law?
##F = ma##
Using Newton's Second Law, is it possible to get the Law of Universal Gravitation?
 
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  • #2
WK95 said:
##F = G \frac{ m_{1} m_{2}}{ r^{2} } ##

Where does the formula come from? And why does it work that way?
It comes from Issac Newton. No one knows "why" it works.

WK95 said:
How would it relate to Newton's Second Law?
##F = ma##
Using Newton's Second Law, is it possible to get the Law of Universal Gravitation?

No, I don't think so. Newton's second law and the law of universal gravitation involve two different properties of mass (inertial and gravitational).
 
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  • #3
WK95 said:
##F = G \frac{ m_{1} m_{2}}{ r^{2} } ##

Where does the formula come from? And why does it work that way?

How would it relate to Newton's Second Law?
##F = ma##
Using Newton's Second Law, is it possible to get the Law of Universal Gravitation?

Newton's law of universal gravitation relates to ##\vec{F} = m \vec{a}## in the following way:

$$\vec{F}=\frac{GM_1 m_2}{r^2}\hat{r} = \frac{GM_1}{r^2}m_2 \hat{r}=m_2 \vec{a}$$ where ##a=g=\frac{GM_1}{r^2}##

Why? it is what it is.
 
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  • #4
WK95 said:
##F = G \frac{ _{}m_{1} m_{2}}{ r^{2} } ##

Where does the formula come from? And why does it work that way?

[itex]G[/itex] is a constant of proportionality, and can therefore be made to be 1 under the correct choice of units.

[itex]F = m{1}m{2}/r{2}[/itex]

The force of gravity falling off as the inverse of the distance squared is due the generally spherical shape of the source (usually planets and stars and such). If the gravitational source is a cylinder, then the strength of gravity would fall off as the inverse of the distance, not the inverse of the distance squared. If the gravitational source were an infinite plane, then (perhaps surprisingly) the strength of the force of gravitation does not fall off with distance.

[itex]F = m{1}m{2}[/itex]

That leaves the force of gravity as simply the product of the two masses. As to the exact nature of why matter attracts gravitationally, scientists are still working on that.
 
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  • #5
MikeGomez said:
If the gravitational source is a cylinder, then the strength of gravity would fall off as the inverse of the distance, not the inverse of the distance squared. If the gravitational source were an infinite plane, then (perhaps surprisingly) the strength of the force of gravitation does not fall off with distance.

Since everything with mass has gravity, how come if the gravitational source is a cylinder, the gravity's strength would fall of as the inverse of the distance? So let's say I have a cylinder. At two points one distance x and another distance 2x from the cylinder, the strength at the latter point is only half the strength at the former?

What if the object in question were another shape such as a square pyramid? How could one find the gravitational strength of such an object? Or what about irregularly shaped objects?

With a infinite plane, I get how if one were to move parallel to the plane, the gravitational strength would not fall off but I don't get why it wouldn't fall if a given point were to move away in a directional perpendicular to the plane.
 
  • #6
It might help if you can picture gravitational lines of force. Lines of force are commonly used to illustrate electric and magnetic fields.
The strength of the field is indicated by the number of lines per unit area.
This should help you to 'see' the inverse square law for a point mass.
Can you now 'see' the cylindrical situation?
 
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  • #7
Superposition of vector-fields.
Use Gauss' Law (as one does in electromagnetism) for highly-symmetrical situations.
 
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  • #8
WK95 said:
At two points one distance x and another distance 2x from the cylinder, the strength at the latter point is only half the strength at the former?

When you're "distance x from the cylinder", you're at distance x from only one small part of the cylinder, the part that's directly "opposite" you. When you look "up" or "down" at other parts of the cylinder, they're further away from you, at different distances. Each part of the cylinder produces a gravitational field at your location, whose strength depends on how far that part is from you. To get the total gravitational field at your location, you have to add up the contributions from all the pieces, taking into account that they're in different directions. This is an exercise in vector integration.

What if the object in question were another shape such as a square pyramid? How could one find the gravitational strength of such an object? Or what about irregularly shaped objects?

Again you add up the contributions to the field from each small part of the object, by vector integration.
With a infinite plane, I get how if one were to move parallel to the plane, the gravitational strength would not fall off but I don't get why it wouldn't fall if a given point were to move away in a directional perpendicular to the plane.

One way to look at this is that a featureless infinite plane "looks the same" to you no matter how far you are from it.
 
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  • #9
technician said:
It might help if you can picture gravitational lines of force. Lines of force are commonly used to illustrate electric and magnetic fields.
The strength of the field is indicated by the number of lines per unit area.
This should help you to 'see' the inverse square law for a point mass.
Can you now 'see' the cylindrical situation?

How would unit area work when the distance from a given point to a gravitational source is simply an imaginary straight line?

With a point mass, lines of force would just be lines pointing towards the point.

With a cylinder at the flat surfaces, the liens of force would be point in perpendicular to the flat face and at the rounded part, the lines would be pointed into towards the cylinder's center line.
 
  • #10
WK95 said:
how come if the gravitational source is a cylinder, the gravity's strength would fall of as the inverse of the distance?
I think you meant inverse of the square of the distance, and it doesn't, at least not close to the cylinder. At very great distances from the cylinder it does, but that's because every compact mass distribution looks like a point mass at a large enough distance from the object in question.
 
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  • #11
WK95 said:
How would unit area work when the distance from a given point to a gravitational source is simply an imaginary straight line?

With a point mass, lines of force would just be lines pointing towards the point.

With a cylinder at the flat surfaces, the liens of force would be point in perpendicular to the flat face and at the rounded part, the lines would be pointed into towards the cylinder's center line.

Have a look at electric fields and magnetic fields to see how lines of force are used.
Faraday came up with the idea of lines of force to represent force fields (magnetism in his case) because he was not so good at mathematics.
 
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  • #12
D H said:
I think you meant inverse of the square of the distance, and it doesn't, at least not close to the cylinder. At very great distances from the cylinder it does, but that's because every compact mass distribution looks like a point mass at a large enough distance from the object in question.

I was quoting Mikegomez on that part.
" If the gravitational source is a cylinder, then the strength of gravity would fall off as the inverse of the distance, not the inverse of the distance squared." - MikeGomez
 
  • #13
There is an implicit assumption that the cylinder is infinitely-long.
 
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  • #14
technician said:
Have a look at electric fields and magnetic fields to see how lines of force are used.
Faraday came up with the idea of lines of force to represent force fields (magnetism in his case) because he was not so good at mathematics.

A genius scientist who isn't that good at mathematics? Is that even possible?!
 
  • #15
WK95 said:
A genius scientist who isn't that good at mathematics? Is that even possible?!

This is a different question !...start a new thread if you want to raise this question and you may get appropriate responses.
The genius is in the idea of lines of force. They link nicely with the mathematics so you have 2 techniques to get to grips with force fields.
 
  • #16
WK95 said:
I was quoting Mikegomez on that part.
" If the gravitational source is a cylinder, then the strength of gravity would fall off as the inverse of the distance, not the inverse of the distance squared." - MikeGomez

I like Dr. Ramachandran's explanation in this video. He is discussing the electric field, but it applies to the strength of the gravitational field as well. At 55:10 he summarizes the three cases for a sphere, line (cylinder), and infinite plane. Look a little earlier for his proofs (infinite plane is at 47:00), or better yet watch the entire video.

 
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  • #18

1. What is Newton's Law of Universal Gravitation?

Newton's Law of Universal Gravitation is a physical law that describes the force of gravity between two objects. It states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

2. How did Newton derive this law?

Newton derived his law of universal gravitation using mathematical equations and observations of the motion of planets and objects on Earth. He combined his laws of motion and his law of inertia with Kepler's laws of planetary motion to come up with a universal law of gravitation.

3. What are the key components of this law?

The key components of Newton's law of universal gravitation are mass, distance, and force. The law states that the force of gravity between two objects is directly proportional to their masses and inversely proportional to the square of the distance between them.

4. How does this law explain the motion of objects in the universe?

This law explains the motion of objects in the universe by showing how the force of gravity between two objects affects their motion. It can be used to calculate the gravitational force between any two objects and predict their motion, such as the orbit of planets around the sun.

5. Are there any limitations to Newton's law of universal gravitation?

Yes, there are some limitations to this law. It does not account for the effects of relativity or quantum mechanics, and it only applies to objects with mass. It also assumes that the mass of an object is concentrated at its center, which may not always be the case. However, for most everyday situations, Newton's law of universal gravitation is a very accurate and useful tool for understanding the motion of objects in the universe.

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