Abstract:
This paper traces the conceptual development of the gravitational force law, from Newton's original proportionality (F ~ m1m2/r²) to the modern formulation using the gravitational constant (F = G*m1m2/r²). We argue that the introduction of G, while necessary for standardization and unit consistency, has obscured the underlying simplicity of the law and led to a perception of G as a mysterious fundamental constant. By examining the intermediate step of introducing a system-dependent base force ('f'), we highlight the dimensionless nature of the mass-distance proportion and clarify the role of G as a scaling factor that combines the inherent strength of gravity with unit conversions.
1. Introduction
Newton's law of universal gravitation is a cornerstone of classical physics. However, the modern formulation, with its reliance on the gravitational constant G, can often seem more complex than necessary. This paper aims to demystify G by tracing its historical and conceptual evolution, demonstrating how it emerged from a simpler, more intuitive understanding based on proportions.
2. Newton's Proportional Insight (F ~ m1m2/r²)
Isaac Newton, in his Principia Mathematica (1687), did not express the law of gravity with a constant like G. Instead, he formulated it as a proportionality: the gravitational force (F) between two objects is directly proportional to the product of their masses (m1 and m2) and inversely proportional to the square of the distance (r) between them:
F ~ m1m2/r²
In this formulation, the expression m1m2/r² is treated as a dimensionless proportion. This elegantly captures the fundamental relationships:
Direct Proportionality to Mass: Doubling either mass doubles the force.
Inverse-Square Law: Doubling the distance reduces the force to one-quarter.
Newton could even determine the ratio of masses (e.g., the Sun's mass to the Earth's mass) using orbital data, without needing any constant of proportionality. He did this by equating the proportionalities of centripetal force and gravitational force for orbiting bodies.
3. The Introduction of a Base Force (F = f * m1m2/r²)
As physics progressed and measurements became more precise, the need for a defined unit of force became increasingly important. This led to the introduction of a "base force," denoted here by 'f':
F = f * (m1m2/r²)
In this formulation:
m1m2/r² remains a dimensionless proportion.
'f' represents the base strength of the gravitational interaction within a chosen system of units.
The value of 'f' would depend on the specific units used for mass, distance, and force. This was a crucial step towards standardization, but it still preserved the conceptual clarity of the original proportionality. 'f' was not a universal constant; it was system-dependent.
4. The Rise of G: Standardization and Obscuration (F = G * m1m2/r²)
The 19th and 20th centuries saw the rise of standardized unit systems, most notably the metric system (and later the SI system). This created the need for a universal gravitational constant that would work regardless of the specific (but standard) units used for mass and distance. This is where G enters the picture.
The equation becomes:
F = G * (m1m2/r²)
G's Role: G serves two key purposes:
It reflects the inherent strength of gravity (like 'f').
It incorporates the necessary unit conversions to ensure dimensional consistency when using standard units (kilograms, meters, seconds, and Newtons).
Added Units: The units of G (m³/(kg⋅s²)) are, in essence, "added" to make the equation dimensionally correct within the standard system. These units are not fundamental to gravity itself, but rather a consequence of our measurement conventions.
Cavendish (1798): Measures the force, without setting a value.
G's Introduction: G was introduced to fulfill this role. It's not that G is conceptually different from 'f'; it's that G is specifically designed to work with standard units (kilograms, meters, seconds, and Newtons). G bakes in the unit conversions.
"Added Units": The units of G (m³/(kg⋅s²)) are, in essence, "added" to make the equation dimensionally correct within the standard unit system. These units are not fundamental to the gravitational interaction itself; they are a consequence of our measurement conventions.
Obscuration: The focus on G, with its seemingly arbitrary units and numerical value, obscured the underlying simplicity of the original proportionality and the role of 'f' as a base force. G became perceived as a mysterious fundamental constant, rather than a scaling factor.
1930 date: The value of G was not truly defined until the 1930's, and it is still being refined.
5. Reframing G: A Scaling Factor, Not a Mystery
It's crucial to recognize that G is, at its heart, a scaling factor. It translates the dimensionless proportion m1m2/r² into a force with specific units. The numerical value of G and its units are a consequence of the fundamental proportionality and our chosen unit system, not the cause of gravity. G could just as easily be represented by another variable and given any value, as long as the defined mass and length scales also change to keep the physics the same.
By understanding the historical progression – from Newton's proportions to the base force 'f' and finally to G – we can demystify the gravitational constant. It's not a magical number, but a practical tool that reflects both the strength of gravity and our conventions for measuring physical quantities. G has a defined value.
6. Confusion of units.
If the main thing that the gravitational law is doing is relating force to a proportion then this means that the relationship is F ~ m1m2/r^2 if we changed the unit definition for mass or length that would change as the square of the mass or the inverse square of the length.
But when we rescale G we rescale the units it has and that does not match the units of the proportion. So this is telling me there is some sort of mismatch between how the proportion works and how we think the units of G are working and I think the units of G are not working as we belive they are because they primarily just cancel out the units in the proportion m1m2/r^2 to force it to be unitless. G is still just the simple force formula f that it has always been just with extra units attached.
6. Conclusion
The evolution of the gravitational force law illustrates a broader point about physical constants. While they are essential for making quantitative predictions, they can sometimes obscure the underlying physical relationships. By revisiting the historical development of the law and appreciating the role of proportionality, we can gain a deeper and more intuitive understanding of gravity and the true nature of G.
References. I learned most of this from:
Newton Did Not Invent or Use the So-Called Newton’s Gravitational Constant; G, It Has Mainly Caused Confusion
https://www.scirp.org/journal/paperinformation?paperid=115356
and although I might not agree with a lot of the statements made, the way Newton worked without G or any other constant, just using the proportions is fascinating.
No comments:
Post a Comment