Table of Contents
Big G aka Newton’s constant of gravitation
This not a constant, it is invented to be a unit conversion factor to make Newton’s force equations work.
Units and dimensions
[chatgpt]
Explain units and dimensions of Newton's gravitational constant
The Newtonian gravitational constant \( G \) appears in Newton’s law of universal gravitation:
\[
F = G \frac{m_1 m_2}{r^2}
\]
where:
- \( F \) is the gravitational force between two masses (\( m_1 \) and \( m_2 \)),
- \( r \) is the distance between their centers of mass,
- \( G \) is the gravitational constant.
Units of \( G \):
From the formula, the units of \( G \) can be determined by isolating \( G \):
\[
G = \frac{F r^2}{m_1 m_2}
\]
- \( F \) (force) is measured in newtons (N).
- \( r \) (distance) is measured in meters (m).
- \( m_1 \) and \( m_2 \) (masses) are measured in kilograms (kg).
Substituting the SI unit for force (\( 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 \)):
\[
G \, \text{has units of} \, \frac{\text{N} \cdot \text{m}^2}{\text{kg}^2} = \frac{\text{kg} \cdot \text{m/s}^2 \cdot \text{m}^2}{\text{kg}^2} = \text{m}^3 \cdot \text{kg}^{-1} \cdot \text{s}^{-2}.
\]
Dimensions of \( G \):
The dimensions of \( G \) can be expressed using the base dimensions of mass (\( M \)), length (\( L \)), and time (\( T \)):
\[
[G] = \left[ \text{L}^3 \cdot \text{M}^{-1} \cdot \text{T}^{-2} \right].
\]
Numerical Value (SI Units):
The experimentally determined value of \( G \) is approximately:
\[
G = 6.67430 \times 10^{-11} \, \text{m}^3 \cdot \text{kg}^{-1} \cdot \text{s}^{-2}.
\]
This value is universal and applies to all gravitational interactions.
Force definition with G exposed
\[F= \left [ \frac{R^3}{T^2}\cdot \frac{1}{M} \right]\frac{1}{r^2}\cdot\frac{Mm}{1}\]
Here we see that Kepler’s Rule is written or is hidden inside \(G\) but with \(1/M\)
Multiplication of two masses
Can we associate a physical interpretation to multiplication of 2 masses
\[(\text{Mass of Sun} \times (\text{Mass of Earth})\]
\(M^2\), what does this mean?
If these mass terms simplify at least one of it does. If so, why are we writing it?
There is a deceptive use of mathematics here.
Numerical computations.
\(G\) is \[\frac{R^3}{T^2}\cdot \frac{1}{M}\] or more correctly \[\frac{R_0^3}{T_0^2}\cdot \frac{1}{M_0}\]
This is the constant term in Kepler’s Rule with an additional term called “mass”. How is this mass computed?
If so, why do we call this \(G\) “Newton’s Universal Constant of Gravitation”? It looks more like the constant term of Kepler’s Rule which does not have force term, or gravitation term.
The two terms \(R\) ant \(T\) are not Newtonian terms. In what sense \(M\) is a Newtoniam quantity?
We’ll calculate the numerical value also.