Section 1

Introduction

The purpose of this paper is deliberately limited. It does not attempt to establish the complete M4NA theory of mechanics. Rather, it establishes the precise meaning of the central force concepts on which the quantitative structure of M4NA is built.

A recurring difficulty in discussions of mechanics is that the word force may simultaneously refer to a physical interaction, its magnitude or value, and the accumulated quantity associated with its application over time. M4NA therefore separates these aspects without treating them as different physical forces. The starting point is simple: an applied force acts on an object. When that applied force operates the object to travel through an actual distance, M4NA refers to it as an operating force.

Thus, operating force is a functional designation of an applied force, not a new species of force. The subsequent distinction between force value and force quantity provides the quantitative structure required for examining force interval by interval.

Section 2

Concept 1: Applied Force as the Primary Concept

Applied force is the force applied to an object during a specified period of operation. The term applied identifies the force in relation to its action upon the object. It may be considered over a complete period or over successive time intervals. No separate physical entity is introduced by the terminology operating force.

Section 3

Concept 2: Operating Force

An applied force is called an operating force when it acts on an object and operates the object to travel through an actual distance. Accordingly: Operating Force ≡ Applied Force in its operating role

The distinction is therefore terminological and functional:

• Applied force identifies the force by its application to the object.
• Operating force identifies the same applied force by the function it performs in operating the object’s travel.

This distinction is important because it prevents the introduction of an unnecessary new category of force.

Section 4

Concept 3: Value of Applied Force

M4NA distinguishes the value of an applied force from its quantity. The value of an applied force is the quantity of applied force divided by the duration of its application. Therefore:  f = F / t    and consequently    F = f t

For an individual interval i:  F_i = f_i t_i

where:

• f_i = value of applied force during interval i
• t_i = duration of interval i
• F_i = quantity of applied force during interval i

This distinction allows the value of applied force to be considered separately for each interval.

Section 5

Concept 4: Quantity of Applied Force

The quantity of an applied force is the accumulated amount of force applied over its duration. For one interval: F_i = f_i t_i

For several successive intervals, the total quantity is additive: F = Σ_i F_i = Σ_i f_i t_i

Thus, M4NA treats the total quantity of applied force as the sum of the quantities applied during the individual intervals. This formulation is particularly important where the value of applied force changes from one interval to another.

Section 6

Interval-by-Interval Force Structure

Suppose an object’s travel is divided into n successive intervals. For each interval, f₁, f₂, f₃, …, fₙ represent the respective values of applied force. The corresponding force quantities are: F₁ = f₁t₁,   F₂ = f₂t₂,   F₃ = f₃t₃,   …

The total quantity is therefore: F = F₁ + F₂ + F₃ + ⋯ + Fₙ = Σ_i f_it_i

The sequence f₁, f₂, f₃, …, fₙ therefore represents an applied-force pattern. M4NA can examine this pattern in correspondence with the object’s interval-by-interval velocity and travel pattern.

Section 7

Actual Travelling Distance

The force structure is not considered independently of the object’s actual travel. For each interval:  v_i = s_i / t_i    and therefore    s_i = v_it_i

Here, s_i is the actual distance travelled during interval i. The total actual travelling distance is: S = Σ_i s_i = Σ_i v_it_i

If the total duration is T = Σ_it_i, then the time-weighted average velocity is: V = (Σ_i v_it_i) / (Σ_it_i)

Hence: VT = Σ_i v_it_i = S    so    S = VT

This relation identifies the total actual travelling distance through the time-weighted average velocity and total duration.

Section 8

Quantitative Identification of Applied Force

The M4NA proposition is that applied force can be quantitatively identified in relation to the object’s actual travel. Actual travel is represented by S = Σ_iv_it_i = VT, while the applied-force quantity is represented by F = Σ_if_it_i.

Accordingly, the force pattern and travel pattern can be examined on the same interval structure. The central M4NA relationship under consideration is: F ∝ S    or    F = kS

Here, k is the universal proportionality constant adopted by the M4NA framework. Thus, the applied-force quantity can be related directly to the actual quantity of travel rather than requiring velocity change as a prerequisite for identifying the applied force.

Section 9

Identification from the Velocity Pattern

A significant consequence of the M4NA formulation is that actual travel can be resolved interval by interval. Suppose v₁, v₂, v₃, …, vₙ is the velocity pattern associated with successive travelling intervals. The corresponding actual travelling distances are s_i = v_it_i, and therefore: S = Σ_i v_it_i

M4NA can then examine the corresponding applied-force pattern, f₁, f₂, f₃, …, fₙ, and its associated quantities:

The important point is methodological: M4NA does not have to wait for a change in velocity before examining the applied force. It examines the applied force in relation to actual travel interval by interval, including intervals in which velocity changes and intervals in which velocity does not change.

This provides the basis for distinguishing different force patterns that may produce different or identical travel patterns.

Section 10

Constant and Changing Force Values

The framework permits both constant and changing applied-force values.

Constant-value case

If f₁ = f₂ = f₃ = ⋯ = fₙ = f, then the applied-force value is constant across the intervals.

Changing-value case

If the force values differ across the relevant interval sequence, then the applied-force value changes across intervals.

The important point is that the existence of an operating force does not require its value to remain constant throughout the complete travel. Likewise, the existence of actual travel does not by itself require a change of velocity during every interval.

Section 11

Force Pattern and Velocity Pattern

M4NA treats the following as distinct but related quantities:

This structure permits the physical event to be represented simultaneously through the force pattern, velocity pattern, and actual-travel pattern on the same interval basis.

Section 12

A Fundamental M4NA Proposition

The definitions developed above lead to the following proposition:

Section 13

Why the Distinction Matters

The distinction between force value and force quantity is not merely semantic. Consider two operations having the same force value f, but different durations.

If t_A ≠ t_B, then F_A ≠ F_B even though the force value is the same. Conversely, different force values operating for different durations may produce specified force quantities. This is why M4NA does not use the words force, force value, and force quantity interchangeably.

Section 14

The M4NA Identification Procedure

The proposed framework can be expressed as a sequence:

1. Divide the travel into intervals: t₁, t₂, …, tₙ.
2. Identify the actual distance in each interval: s₁, s₂, …, sₙ.
3. Determine the corresponding interval velocity: v_i = s_i/t_i.
4. Construct the actual-travel pattern: S = Σ_iv_it_i.
5. Examine the corresponding applied-force pattern: f₁, f₂, …, fₙ.
6. Determine the interval force quantities: F_i = f_it_i.
7. Obtain the total applied-force quantity: F = Σ_if_it_i.
8. Examine the force–travel relation: F ∝ S or, under the central M4NA proposition, F = kS.

This procedure makes the framework quantitative rather than merely descriptive.

Section 15

Conceptual Significance

The principal conceptual point can be stated very simply: an applied force is not renamed “operating force” because it possesses a different physical nature. It is called operating force because it operates the object’s travel.

From this starting point, M4NA distinguishes applied force into its two quantitative aspects: value of applied force (f) and quantity of applied force (F).

This creates a common interval-by-interval structure through which force, velocity, and actual travel can be quantitatively compared.

Section 16

Conclusion

This paper establishes four foundational terms for the quantitative concepts of M4NA Classical Mechanics:

• Applied Force: the force applied to an object during a specified period of operation.
• Operating Force: an applied force considered in its role of operating the object to travel through an actual distance.
• Value of Applied Force (f): the quantity of applied force divided by the duration of its application.
• Quantity of Applied Force (F): the accumulated applied-force quantity over the relevant duration or successive intervals.

The resulting quantitative structure is:

The significance of this formulation is that force can be examined as an operating quantity associated with actual travel, while its value can be identified interval by interval. Consequently, M4NA provides a framework in which the pattern of applied/operating force can be examined from the corresponding velocity and actual-travel patterns, including both changing-velocity and non-changing-velocity intervals.

This establishes the quantitative vocabulary necessary for the next stage of the M4NA programme: testing the proposed force–distance relationship against specific physical cases and comparing its predictions with conventional mechanics.