Lesson Plan: The Constant Mythology

J. Rogers, SE Ohio

Course: Introductory Physics / Advanced High School Physics
Time Allotment: 50 minutes
Objective: By the end of this class, students will be able to derive any dimensional constant in physics and explain that these constants are composite conversion factors whose values are determined entirely by our choice of SI units.

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Materials:

• Whiteboard or Projector

• Handout with "Dimensional Imbalance Table" and "Practice Problems"

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Lesson Structure & Instructor Notes

(0:00 - 5:00) | I. The Setup: The Illusion of Mystery (5 mins)

What the Teacher Does:

1. Write F = G · m₁m₂ / r² and G ≈ 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻² on the board.

2. Ask the class: "We're told G is a fundamental constant of nature. But look at it. Does that number look fundamental? Do those units look like they were designed by nature, or by a committee of humans in France?"

3. Let the absurdity sink in. "If G is fundamental, why does its value depend on a lump of platinum in a vault outside Paris?"

4. State the lesson's premise: "Today, I'm going to prove to you that G, and every other dimensional constant, is not fundamental at all. It's a conversion factor, just like 2.54 cm/inch. Its entire existence is an artifact of our chosen rulers. We're going to learn the algorithm that reveals this."

Teacher's Notes (The Goal):

• This opening directly attacks the reification of SI units. You are immediately framing the kilogram, meter, and second as arbitrary, local conventions, not a privileged window into reality.

• The 2.54 cm/inch analogy is introduced immediately. It is the central metaphor of the entire lesson. G is not a mystery; G is a 2.54.

(5:00 - 15:00) | II. The Algorithm: The Bookkeeping Trick (10 mins)

What the Teacher Does:

1. Erase G. "Let's start with the real physics: a simple proportionality." Write F ~ m₁m₂ / r².

2. "This is the invariant law. The problem is, we measure F in Newtons, m in kilograms, and r in meters. Our units are mismatched. We need a conversion factor to make the books balance."

3. Introduce "The 4-Step Algorithm for Finding the Conversion Factor."

   • Step 1: Write the proportional law. (F ~ m₁m₂ / r²).

   • Step 2: Check the dimensional imbalance in our chosen units (SI). Left: [M L T⁻²]. Right: [M² L⁻²]. They don't match.

   • Step 3: Define the required 'fudge factor' to fix the imbalance. "We need a factor with dimensions [L³ M⁻¹ T⁻²] to make our SI numbers work in this equation."

   • Step 4: Build this factor from the DEFINITIONS of our unit system. "Now for the big reveal. The SI system was officially redefined in 2019 by fixing the numerical values of constants like h and c. This means our units (kg, m, s) are now defined in terms of these constants. We can use these definitions, which we'll call non reduced Planck units for convenience (l_P, m_P, t_P), as the building blocks for our conversion factor."

   • Assemble the factor: G = l_P³ / (m_P · t_P²).

4. State the conclusion clearly: "G is not a property of gravity. G is a composite conversion factor built from the definitions of our own rulers. Its job is to correct for the dimensional mismatch created by our choice to measure mass, length, and time with separate, arbitrary units. G is the 2.54 of gravity."

Teacher's Notes (The Goal):

• The language is critical. Avoid "natural rulers." The Planck units are now correctly framed as the defining components of the SI system itself. They are not "nature's units"; they are the formal basis vectors of our chart.

• The "fudge factor" framing is powerful. It presents the constant as a patch we invented, not a property we discovered.

• The 2019 SI redefinition is mentioned to ground this in modern, factual metrology. This isn't a speculative theory; it's an explanation of how our measurement system actually works.

(15:00 - 30:00) | III. The Pattern: Exposing All the Conversion Factors (15 mins)

What the Teacher Does:

1. "This isn't just about G. Every dimensional 'constant' is a conversion factor."

2. Rapid-fire, derive c, h, and k_B.

   • For c: "The conversion factor between our meter-stick and our clock." Imbalance [L T⁻¹]. Build it: c = l_P / t_P.

   • For h: "The conversion factor between our clock's frequency and our energy meter." Imbalance [M L² T⁻¹]. Build it: h = m_P · l_P² / t_P.

   • For k_B: "The conversion factor between our thermometer and our energy meter." Imbalance [M L² T⁻² K⁻¹]. Build it: k_B = m_P · l_P² / (t_P² · T_P).

3. Introduce the Fine Structure Constant, α.

   • "What's the conversion factor needed for a dimensionless ratio?"

   • Imbalance is [1].

   • "You don't need a conversion factor if there's no imbalance. This is why α is different. It is not a 2.54. It's a real number that describes the physics itself, independent of our rulers."

Teacher's Notes (The Goal):

• Consistently label each constant as a "conversion factor" between two specific measurement axes. c converts between the length axis and the time axis on our chart. h converts between the frequency axis and the energy axis.

• The α example remains the crucial contrast. It distinguishes between the map's grid lines (the Jacobians) and the actual features on the map (the dimensionless physics).

(30:00 - 45:00) | IV. The Practice: Becoming a Metrologist (15 mins)

What the Teacher Does:

1. Hand out the worksheet. "Your turn. Stop being a physicist and start being a metrologist. Find the conversion factors needed for these formulas."

2. Have students work in pairs on the problems.

3. Walk the room, reinforcing the language: "What's the imbalance in your units? Okay, now build the conversion factor that fixes it."

Teacher's Notes (The Goal):

• The framing "Stop being a physicist and start being a metrologist" is a deliberate act of re-identification. It encourages students to adopt the critical, "outsider" perspective required to see the truth.

(45:00 - 50:00) | V. The Conclusion: The World After the Red Pill (5 mins)

What the Teacher Does:

1. Display the "One-Slide Summary."

2. "The 'fundamental constants' are not fundamental properties of the universe. They are the conversion factors that arise from our historical choice to measure a unified reality with a fragmented and arbitrary set of rulers."

3. Hold up a meter stick and a kilogram weight. "The value of G is not a mystery of gravity. It is a statement about the relationship between this stick and this weight, according to the rules we invented."

4. Deliver the final takeaway: "Don't be fooled by the mythology. The physics is simple. The complexity comes from our tools. Your job is to learn to see through the tools to the simple reality they are measuring. Never again let anyone tell you that 2.54 is a mystery of geometry."

Teacher's Notes (The Goal):

• This conclusion is direct, physical, and visceral. Holding up the props makes the abstract concept concrete. G is about the relationship between these specific objects.

• The final line brings the entire lesson back to the central 2.54 analogy, making it unforgettable. The students don't just leave with a new algorithm; they leave with a new, fundamentally more honest, understanding of what science is actually doing.