Newton's Universal Law of Gravitation?

[ender7x77]

So, I've been wroking a lot with graphs this week, which seems to be leaning towards Newton's Universal Law of Gravitation. We have done many proportionality, but derving the formula has proven to be very difficult. I'm not sure of this means anything to anyone, but its work the try anways. After a series of three intense self-hand made graphs I was able to develop two proportionalities: one which related Force versus Distance, the other relationg Mass versus Distance. My proportionality statements are the following in order of how I just listed the two: F is directly proportional to 1/Δd^2 and M is directly proportional to Δd^2. Eventually, I found my constank (k), and was able to develop a formula for the two proportionalities which were F = k/Δd^2 eventually becoming (3.989 x 10^14)/Δd^2, and M = kxΔd^2which became M = (2.455x10^{-14})xΔd^2.

 

After all these calculations and graph manipulations were are asked to write a proportionality statement relating the two other statements with all three variables (Δd, F, M). In doing so I'm completely bewildered due to the fact of there being two different constants (one of mass and one of ....?). I believe this to be newtons universal law of graviation, but I could be wrong. Does anyone have any ideas? I manipulated the statements so that they both beging with Δd, but then I dont know how to plug it in.

 

Anyways, I appreciate anyone who can shed some light on this but reflecting upon the amount I just wrote, I can see that many will feel reluctant to do so. Thanks.

[ydoaPs]

Mass is not proportional to distance. Mass is a constant.

 

[math]F=G{\frac{Mm}{d^2}}[/math]

 

Where M is the larger mass, m is the smaller mass, d is the distance between their centers, and G is the gravitational constant.

[hotcommodity]

Mass is not proportional to distance. Mass is a constant.

 

[math]F=G{\frac{Mm}{d^2}}[/math]

 

Where M is the larger mass, m is the smaller mass, d is the distance between their centers, and G is the gravitational constant.

 

I thought it was over r squared instead of d squared...or are they the same thing?

[ydoaPs]

I thought it was over r squared instead of d squared...or are they the same thing?

 

They're the same thing.

[ender7x77]

I must have the wrong theory then. We had to do graphs, which showed the proportionality. From one of these graphs which was relationg to Mass and Distance, I discovered that Mass was directly proportional to (Delta Distance)^2. I, then had to combine the two proportionalities that I had acquired and try to relate all variables.

[Klaynos]

I must have the wrong theory then. We had to do graphs, which showed the proportionality. From one of these graphs which was relationg to Mass and Distance, I discovered that Mass was directly proportional to (Delta Distance)^2. I, then had to combine the two proportionalities that I had acquired and try to relate all variables.

 

Mass is conserved (well energy but I doubt we're converting mass-energy into anything else here so lets stick with mass), what was the formula of the graph?

[ender7x77]

I stated the formulas I created for each different graph other than that there is no others because he just give us points and told us to graph...eventually, patterns emerged.

[timo]

F = G m M / r² => r² = C * m with C = GM/F. So distance squared is proportional to mass m when C is constant. Not sure what sense that makes but the statement technically can be considered correct.

 

EDIT: ^^ Forget that, I think I missed your point. You are looking to derive Newton's gravitational law from proportionality relations, are you? Given enough relations, that should be possible with a little combinatory work, leaving a single proportionality constant that Newton called G.

[Klaynos]

I stated the formulas I created for each different graph other than that there is no others because he just give us points and told us to graph...eventually, patterns emerged.

 

Sorry I missed them in your original post, I'll re post them in latex

F is directly proportional to 1/Δd^2

[math]F \propto \frac {1} {\Delta d^2} [/math]

and M is directly proportional to Δd^2

[math]M \propto \Delta d^2 [/math]

 

. Eventually, I found my constank (k), and was able to develop a formula for the two proportionalities which were F = k/Δd^2

[math]F = \frac {k} {\Delta d^2} [/math]

 

eventually becoming (3.989 x 10^14)/Δd^2, and M = kxΔd^2which became M = (2.455x10^{-14})xΔd^2.

 

These aren't really the full formula for the lines. How did you get from the Force proportional equation to the Mass one? I assume [math]\Delta d[/math] is the change in radius? In that case don't you think it's odd that the force is directly related to the change in distance and not the total distance, as that would imply that if you stay at the same altitude you have no force action upon you... I'm not trying to be rude, I'm trying to understand where you're coming from...

[ender7x77]

So, since I cannot seem to explain the setup of the question I thought I'd just show you it so you can visually interpret it. I do not expect you to graph it, but I'm hoping it gives you a better idea of what I'm trying to say..

 

Data Measured and Recorded Bu NASA (verify important law of phyiscs)

_____________________________________________________________

| Mass (Kg) | 1..0 | 5.0 | 10.0 | 15.0 |

| Force (N) | Distance (m) |

| 9.800 | 6.380 x 10^6 | 1.427 x 10^7 | 2.017 x 10^7 | 2.417 x 10^7 |

| 2.450 | 1.276 x 10^7 | 2.853 x 10^7 | 4.035 x 10^7 | 4.942 x 10^7 |

| 1.089 | 1.914 x 10^7 | 4.279 x 10^7 | 6.053 x 10^7 | 7.413 x 10^7 |

| 0.612 | 2.552 x 10^7 | 5.706 x 10^7 | 8.070 x 10^7 | 9.884 x 10^7 |

 

 

Now, with this set of data I'm hoping you can visualize how each proportionality may come about. I did the first graph based on Force versus Distance when the mass was 1 kg, which eventually gave me:

[math]

F \propto \frac {1} {\Delta d^2}

[/math]

I then, to find my constant, I substituted points from my graph into my equation, which I found k to be 3.989 x 10^14. So...

[math]

F \propto \frac {3.989 X 10^{14}} {\Delta d^2}

[/math]

On my second graph I did Mass versus Distance, which generated:

[math]

M \propto \Delta d^2

[/math]

Repeating the same steps as Force versus Distance graph, I found k and placed it into my equation: M = (2.455x10^{-14})xΔd^2.

 

Eventaully, we are asked to combine the two proportionalities so that we create one statement that relates all the variables. The problem is when I'm doing this the different constances confuses me... I know I have to set both in terms of Δd, but I'm not sure after that. I think I got Δd = [k(m)/F]^(1/2). Anyways, that is all I know and I'm stuck.. I understand this a lot throw on you, but I'd really appreciate it. Thanks.

 

PS- The problem is the relationship between the masss of an object, the distance from the centre of the earth and the force of gravity experienced by the object.