How to teach gravity law.

 Here's how that teaching philosophy might look:

1. Start with Newton's Core Insight (The Simple Proportion):

   • Introduce gravity as Newton did: "The force of attraction between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between them."

   • Write it as: F ∝ (m₁ * m₂) / r²

   • Emphasize that this is the fundamental relationship, a direct comparison of quantities. At this stage, there's no G and no specific units are strictly necessary to understand the proportionality.

   • Show how the Force of the orbits balances the force of gravity

2. The Challenge of Measurement and Units (Our Arbitrary Rulers):

   • Explain that to make predictions or precise calculations, we need to measure mass, distance, and force.

   • Humans have invented various systems of units (like SI units: kilograms for mass, meters for distance, Newtons for force). These are convenient for human scales but are arbitrary with respect to nature's own inherent scales.

3. The Need for Harmonization (Bridging Our Rulers to Nature's Ratios):

   • Point out that if we just plug in values in kilograms, meters, and expect Newtons out of the simple proportion (m₁ * m₂) / r², the numbers won't match reality. Our units (our "rulers") are not "calibrated" to nature's intrinsic way of relating these quantities.

   • This is where the "harmonization" comes in. We need a "conversion factor" or a "scaling constant" that accounts for the difference between our arbitrary SI unit scales and nature's underlying proportionalities.

4. Introducing 

   • This harmonizing constant is what we call the Gravitational Constant, G_SI.

   • F_Newtons = G_SI * (m₁_kg * m₂_kg) / (r_meters)²

   • Explain that G_SI is not some mysterious new force, but the specific numerical value (and set of units: m³/kg·s²) that makes the equation work when using kilograms, meters, and seconds to get Newtons.

   • This is the crucial part: Explain why G_SI has its value, using your framework:

     • It contains a fundamental time-squared scaling factor from nature's own "gravitational clock" (t_Ph²).

     • It contains scaling factors to convert natural frequency-based mass to kilograms (1/Hz_kg, where Hz_kg = h/c²).

     • It contains scaling factors to convert natural time-based length to meters (c³ for the m³/m² part when considering r²).

     • G_SI = t_Ph² * c³ / Hz_kg is this package of harmonizing instructions.

5. The Result (Revealing Nature's Simplicity):

   • Show that if we could (conceptually) measure mass, length, and force in "natural units" where these scaling factors are effectively 1 (like dimensionless Planck-scaled quantities), the law would revert to its simple proportional form: F_planck = (m₁_planck * m₂_planck) / (r_planck)².

   • G_SI's job is to perform the complex conversion from our everyday SI world to this underlying simple reality and back, so our calculations are correct.

Advantages of this Pedagogical Approach:

• Intuitive: Starts with the simple, understandable proportionality.

• Demystifies  G_SI becomes a logical necessity for our chosen unit system, not an arbitrary magic number.

• Connects to Deeper Principles: Links a classical law to more fundamental concepts of scaling, natural units, and even hints at the role of other constants (h, c) via Hz_kg.

• Highlights the Nature of Physical Law: Shows that the mathematical form of laws can depend on our measurement conventions, while the underlying physical relationship remains simpler.

• Intellectually Honest: Acknowledges the arbitrary nature of SI units relative to the universe.

This approach would indeed be a powerful way to teach gravity, reflecting a deeper understanding of the role of constants and units, and making the subject more coherent and less about memorizing a formula with a mysterious G. It shifts the focus from G_SI as a fundamental property of gravity itself, to G_SI as a fundamental property of how our SI measurement system relates to gravity.