Chapter 35: The Unassailable Fortress and the Missing Key

Lia set down her quill.

The final formula lay serenely on the parchment.

T²=(4π²/(G×M))×a³

It was perfect.

Starting from first principles, she had rigorously derived all three laws governing the motion of celestial bodies, employing only the three laws of motion and universal gravitation.

This treatise, ‘Principles of the Unity of Celestial and Terrestrial Motion,’ now stood as an unassailable theoretical fortress.

Any challenge to its validity would first necessitate the overthrow of the three laws of motion, which served as its very foundation.

Those three laws, in themselves, were so elegantly simple that experimental verification was a straightforward matter.

A long sigh escaped her lips, as a profound sense of satisfaction, born of intense intellectual exertion, suffused her entire being.

She then gathered the entire stack of papers, meticulously reviewing them from beginning to end.

The definitions were lucid, the logic impeccably self-consistent, and the derivations rigorously precise.

From the terrestrial to the celestial, from a falling apple to an orbiting planet, everything was elegantly subsumed within a single, unified framework.

Lia nodded, a contented smile gracing her lips, and prepared to set aside her manuscript.

Then, a sudden thought halted her.

Her hand paused mid-air.

A notion, unbidden and without warning, surfaced from the depths of her consciousness.

‘Was this theoretical system truly complete?’

It undeniably described how forces altered motion.

Yet, it seemed to offer only a single approach to problem-solving—a direct, force-based method.

Lia’s brow furrowed in concentration.

In her previous life, tackling a physics problem, particularly complex dynamics, typically involved two distinct approaches.

One was vector analysis, the very framework she had just completed, based on force and acceleration.

The other was scalar analysis, rooted in the concepts of work and energy.

Oftentimes, the latter method proved considerably simpler.

Her current theoretical framework felt akin to a war chariot equipped solely with a steering wheel.

While it offered directional control, the absence of an accelerator and brakes meant it utterly lacked the most crucial element: an energy system.

‘No rest yet.

The foundation has not been fully laid.’

She unrolled a fresh sheet of parchment, and at the very end of her treatise, inscribed a new heading.

‘Chapter Five: A Novel Analytical Tool — Work and Energy’

“To more conveniently address certain complex problems of motion, I shall introduce a novel conceptual framework.

At the heart of this system lies the concept of ‘energy.'”

“I. The Definition of Work and the Kinetic Energy Theorem”

“First, we must define ‘work’ (symbol W).

When a force F acts upon an object, causing it to undergo a displacement s in the direction of the force, we state that this force has performed work upon the object.

Its magnitude is given by: W = F × s.”

“However, in many instances, the direction of the applied force does not coincide with the direction of the displacement.

In such cases, only the component of the force acting along the direction of displacement performs work.”

“More generally, for an object traversing a curved path, the force F it experiences may be subject to variation.

To calculate the total work, we can employ calculus.

By infinitely subdividing the path into infinitesimal straight segments ds, the work done over each tiny segment is dW = F · ds.

The total work W, therefore, is the line integral over the entire path.”

“Now, let us delve into the relationship between ‘work’ and the alteration in an object’s state of motion.”

“According to the Second Law of Motion, F = ma = m(dv/dt).

Let us substitute this expression into the integral formula for work.”

W = ∫ F·ds = ∫ m(dv/dt)·ds

“By utilizing the relation (ds/dt) = v, we can perform a substitution of variables.”

W = ∫ m·v·dv

“Upon solving this seemingly simple integral, we arrive at a truly astonishing result.”

W = ½mv₂² – ½mv₁²

“The term ½mv² on the right side of the equation represents a physical quantity dependent solely on an object’s mass and velocity.

I shall name this ‘kinetic energy’ (symbol Ek), which signifies the energy an object possesses by virtue of its motion.”

“Thus, we arrive at a profoundly significant theorem, which I shall call the ‘Kinetic Energy Theorem’: The total work performed on an object by the net external force acting upon it is equal to the change in its kinetic energy.”

The emergence of this theorem heralded the birth of an entirely new approach to problem-solving.

No longer would one need to meticulously track changes in acceleration throughout a process; instead, an equation could be established simply by calculating the initial and final states of kinetic energy and the total work done.

“II. Conservative Forces and Gravitational Potential Energy”

“Forces manifest in numerous forms.

Among these, a particular category exists, which I shall term ‘conservative forces.’

When a conservative force performs work, the magnitude of that work is dependent solely upon the object’s initial and final positions, entirely independent of the specific path traversed by the object.”

“Universal gravitation, for instance, stands as the quintessential conservative force.”

Lia sketched several illustrative diagrams alongside her text.

In these, a sphere descending from an identical initial height—whether along an inclined plane, a curved path, or in free fall—demonstrated that the work performed by gravity remained precisely the same in each scenario.

“For conservative forces, we can introduce an exceptionally valuable concept: ‘potential energy’ (symbol Ep).”

“The change in potential energy is defined as the negative of the work performed by a conservative force.

That is: ΔEp = -W_conservative force.”

“Now, let us define ‘gravitational potential energy’ in the vicinity of the Earth’s surface.

For an object of mass m moving from an initial height h₁ to a final height h₂, the work done by gravity is W_grav = mg(h₁-h₂).

Consequently, the change in its gravitational potential energy is ΔEp = -mg(h₁-h₂) = mgh₂ – mgh₁.”

“From this, we can define the gravitational potential energy of an object near the Earth’s surface as: Ep = mgh.

(Here, we designate the ground as the zero potential energy reference point.)”

“III. The Law of Conservation of Mechanical Energy”

“Now, all the pieces of the puzzle are in our possession.”

“According to the Kinetic Energy Theorem, the total work done by the net external force, W_total, is equal to ΔEk.”

“If, within a given system, only conservative forces (such as gravity) perform work, then W_total = W_conservative force.

Combining this with our definition of potential energy, we have W_conservative force = -ΔEp.”

“Consequently, we arrive at the following relation: ΔEk = -ΔEp.”

“Upon rearranging the terms, we unveil one of the most fundamental conservation laws in the entire universe.”

ΔEk + ΔEp = 0

Or, equivalently, it can be expressed as: Ek₁ + Ep₁ = Ek₂ + Ep₂

Lia solemnly penned the textual description of this profound formula.

“In a system where only conservative forces perform work, an object’s kinetic energy and potential energy may transform into one another, yet their sum, termed ‘mechanical energy,’ will perpetually remain a constant value.

I shall refer to this as the Law of Conservation of Mechanical Energy.”

Having penned the final word, Lia at last allowed herself to fully relax.

‘It is done.’

The two towering pillars of classical mechanics—Newton’s Laws of Motion and the Law of Conservation of Energy—had both been successfully transplanted into this world by her hand.

Armed with these two formidable tools, the mechanics of the entire macroscopic, low-velocity world would henceforth harbor no more secrets.

Gazing at the formidable stack of papers before her, a body of work capable of utterly revolutionizing the physics and magical theories of this entire world, a mischievous thought suddenly sparked within her.

This treatise had, without question, resolved the profound mystery of why heaven and earth were unified, and furthermore, how to precisely describe that unity.

Yet, one crucial detail remained, one she had deliberately chosen to conceal.

It was the universal constant—G—that permeated the entire theory, serving as the fundamental link between celestial bodies and terrestrial objects.

Its very existence was unequivocally predicted by the theory.

But what, precisely, was its numerical value?

‘Unknown.’

To ascertain this, experimental measurement was indispensable.

Yet, the challenge of measuring the gravitational force between two ordinary objects—a force so minuscule as to be almost negligible—was a task no less daunting than constructing an entire new theoretical framework from scratch.

Lia picked up her pen once more, inscribing the final paragraph at the conclusion of her monumental work.

“Thus, the theoretical exposition of these principles is now complete.

This framework offers a unified understanding of all motion, from the grand dance of stars to the subtle stirrings of dust.

However, the true perfection of theory invariably demands experimental verification.”

“Among these, the precise determination of the gravitational constant G represents a pivotal step for the entire theory to transition into practical application.

It will empower us to genuinely ‘weigh’ the masses of stars, compute the densities of celestial bodies, and even prognosticate the existence of unseen worlds.”

She paused, a sly smile playing upon her lips.

“Regarding the precise method for determining the value of G through an ingenious experimental apparatus, I already possess a brilliant conception.”

“Alas, the remaining blank space on this parchment is, regrettably, insufficient to fully elaborate upon it.”