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When scientists want to determine empirically the factors that influence a physical phenomenon, they observe and record in a table of data actions and reactions in nature or during laboratory experiments.
In a typical table of data, columns on the left side contain the values of the experiment's setup parameters (cause or actions). Columns in the middle are for recording the values of the observed phenomena when the experiment is performed (effect or reactions). Columns on the right are reserved for data that are calculated from both the setup-parameter and observed-phenomenon values. Each row contains the data for one "run" of the experiment.
Between runs, the value of one of the experiment's setup parameters can be changed to see how it affects the outcome of the next run. After many runs have occurred using various values for the experiment's setup parameters, the experimenter hopes to establish a mathematical correlation between the parameter and phenomenon values. In today's world, of course, computers help correlate the data; however, in Newton's time, this was a manual chore. To see how to create a table of data, we re-enact an inertial-force experiment in an idealized, hypothetical manner. But, before we can examine this experiment, we need to know about inertial force.
Inertial force automatically applies itself to a mass as a reaction to other types of force, which are applied to that mass and cause it to change speed or direction (accelerate).
Newton created the inertial-force proportion in the following form:
a Þ F m-1 (214
which reads:
The acceleration, a, of a mass, m, is proportional to the external forces that are applied to it, which are equal to the generated inertial force, F, that is automatically applied in the opposite direction.
Therefore, if the external forces increase by a particular factor, the acceleration and the reacting inertial force increase in the same proportion. If both the external forces and the mass either increase or decrease by the same factor, the acceleration remains constant.
Conversely, the same external force applied to a larger mass accelerates it to a lesser extent. The total inertial force remains constant, yet the force per mass is less, which makes the mass "feel" less inertial pressure.
Newton's inertial-force proportion came about by repeatedly measuring and correlating the values of various combinations of mass and force and the resultant acceleration. Originally, measuring these values entailed using unrelated, arbitrary units of measure. The dimensions of acceleration, force, and mass were unrelated, as well. Under this state of affairs, formulating a force equation was impossible.
Successors to Newton decided that, because mass times acceleration proved to be proportional to inertial force, the dimension of force could be replaced by those of mass times acceleration. This is unfortunate because inertial force is not equal to mass times acceleration--it is an action that creates a reaction, and, in addition, when the magnitude of the action changes, that of the reaction changes by the same factor.
Thereafter, Newton's formula yielded the following equation:
F = k m a (215
where:
m possesses dimensions of mass,
a possesses dimensions of length divided by time squared,
k is the constant of proportionality, which, when multiplied by the measurements, (ma), must force the numerical value of the measurement, F, to be equal to them.
In a mathematical equation, if the value of a factor is equal to one, you can remove it from the equation without upsetting the equation's equality attribute. Therefore, in order to make k vanish, we change the value of the unit of force so that when it is applied to one unit of mass, it increases its velocity by one length unit per time unit during one time unit. The inertial-force equation now looks like:
F = m a (216
No constant of proportionality appears to be needed because, as far as the numerical values are concerned, the equation is valid. Yet the dimensions and units of measure on either side of the equal sign are not equal. They can never be equal because the action and reaction are different phenomena--the one is a force, and the other is an accelerating mass.
In the SI of unit measures, the units for the force, F, mass, M, length, L, and time, T, dimensions are, respectively, the newton, N, kilogram, kg, meter, m, and second, s. Saying that such-and-such force contains so-many kilogram-meters per second per second is false. Yet, the SI unit of force, the newton, is said to have dimensions of MLT-2 such that N = kg·m·s-2; however, physically, this is false, but, mathematically, it is convenient.
To make the equation mathematically correct, a constant of proportionality, k, should be included, which would make everything equal--dimensions, units of measure, and numerical values. The value of the constant would be:
k = 1 N·s2·kg-1·m-1 (217
In essence, k converts the (m a) units of kg·m·s-2 to N, and the equation's two newton, N, units balance. However, whenever the numerical value of a constant of proportionality is equal to one, we remove it from the equation and try to remember what combination of units are equivalent to what other combination. This practice is a dangerous shortcut in dimensional analysis when manipulating large, complex algebraic expressions.
The simple goal of the inertial-force experiment is to determine what happens when a force is applied to a mass. Evidently, the mass moves, but with what kind of movement? Can it be described mathematically?
The apparatus for the experiment is comprised of the following elements:
1) a series of objects of varying integral mass
2) a flat, level, frictionless surface upon which an object can lie and move
3) a movable arm capable of applying a horizontal force of a predetermined, constant, integral value to an object and of measuring the time of application of that force
The experiment's setup parameters are:
1) the mass of the object
2) the force applied to the object
3) from a standing start, the distance the object moves over the surface while under the influence of the force
When the experiment runs, the phenomenon that is observed and measured is the time during which the arm's force is applied to the object.
Let us assume that we are not using the SI of unit measures. Let each of the units of mass, force, distance, and time be determined arbitrarily and independently of each other. This almost guarantees that any equation that derives from the experiment will contain a constant of proportionality. Also, assume that the force phenomenon has not yet been defined as possessing dimensions of MLT-2, which it does not necessarily have to be.
We run the experiment several times, changing the value of one setup parameter after each run. We record the results in a table of data. For the sake of simplicity, we establish the values of the setup parameters as from one through three.
The resulting table of data is, as follows:
| Setup Parameters | Run Data | Calculations | ||||
| Mass | Force | Distance | Time (T) | T 2 | T -2 | M 1 F -1 D 1 T -2 |
| 1 | 1 | 1 | 0.333 | 0.1111 | 9 | 9 |
| 1 | 1 | 2 | 0.471 | 0.2222 | 4.5 | 9 |
| 1 | 1 | 3 | 0.577 | 0.3333 | 3 | 9 |
| 1 | 2 | 1 | 0.236 | 0.0555 | 18 | 9 |
| 1 | 2 | 2 | 0.333 | 0.1111 | 9 | 9 |
| 1 | 2 | 3 | 0.408 | 0.1667 | 6 | 9 |
| 1 | 3 | 1 | 0.192 | 0.037r | 27 | 9 |
| 1 | 3 | 2 | 0.272 | 0.074r | 13.5 | 9 |
| 1 | 3 | 3 | 0.333 | 0.1111 | 9 | 9 |
| 2 | 1 | 1 | 0.471 | 0.2222 | 4.5 | 9 |
| 2 | 1 | 2 | 0.667 | 0.4444 | 2.25 | 9 |
| 2 | 1 | 3 | 0.816 | 0.6667 | 1.5 | 9 |
| 2 | 2 | 1 | 0.333 | 0.1111 | 9 | 9 |
| 2 | 2 | 2 | 0.471 | 0.2222 | 4.5 | 9 |
| 2 | 2 | 3 | 0.577 | 0.3333 | 3 | 9 |
| 2 | 3 | 1 | 0.272 | 0.074r | 13.5 | 9 |
| 2 | 3 | 2 | 0.385 | 0.148r | 6.75 | 9 |
| 2 | 3 | 3 | 0.471 | 0.2222 | 4.5 | 9 |
| 3 | 1 | 1 | 0.577 | 0.3333 | 3 | 9 |
| 3 | 1 | 2 | 0.816 | 0.6667 | 1.5 | 9 |
| 3 | 1 | 3 | 1.000 | 1.0000 | 1 | 9 |
| 3 | 2 | 1 | 0.408 | 0.1667 | 6 | 9 |
| 3 | 2 | 2 | 0.577 | 0.3333 | 3 | 9 |
| 3 | 2 | 3 | 0.707 | 0.5000 | 2 | 9 |
| 3 | 3 | 1 | 0.333 | 0.1111 | 9 | 9 |
| 3 | 3 | 2 | 0.471 | 0.2222 | 4.5 | 9 |
| 3 | 3 | 3 | 0.577 | 0.3333 | 3 | 9 |
After running the experiment 27 times, we have collected the periods of time that a force was applied to an object and recorded them in the fourth column of the table.
What do we want from this jumble of data? We will create a mathematical term that contains four factors, which represent the four elements for any possible run of the experiment such that the value of the term remains constant. In other words, we want to create a term, which is a mathematical proportion. In this case, the term takes on the following form:
(Mass)e1 (Distance)e2 (Force)e3 (Time)e4 (218
In essence, the problem is to determine the values for the exponents e1, e2, e3, and e4. To do this, we must perform some numerical analysis upon the data in the table, as follows:
1) When the values for the mass and force are equal to each other (1=1, 2=2, 3=3), the distance-time paired values remain respectively the same (1--0.333, 2--0.471, 3--0.577).
Therefore, the mass and force values are directly proportional to each other and form a fraction within the term, as follows:
either: (Mass) (Force)-1 (219
or its reciprocal: (Force) (Mass)-1
2) When the values for the distance and force are equal to each other (1=1, 2=2, 3=3), the mass-time paired values remain respectively the same (1--0.333, 2--0.471, 3--0.577). Therefore, the distance and force values are directly proportional to each other and form a fraction within the term, as follows:
either: (Mass) (Distance) (Force)-1 (220
or its reciprocal: (Force) (Mass)-1 (Distance)-1
3) In either of these analyses, the time-parameter values increase with the increase of either the mass- or distance-parameter values but not in direct proportion. Therefore, the time factor forms a fraction with both the mass and distance factors, yet we still do not know the form of the proportion.
The term, with mass and distance in the numerator, is, now, as follows:
(Mass) (Distance) (Force)-1 (Time)-? (221
The last-remaining unknown to discover is the negative power of the time factor.
4) Let us see if the average speeds can help. By dividing the distance value by the time value for each run, the average velocity of the object increases over time and distance. Therefore, the object is accelerating, which means that for each succeeding unit of time that the object is being pushed by the force, the object travels farther.
5) Acceleration is a change of velocity over time or, more precisely, the velocity at time (t+1) minus the velocity at time (t) all divided by (t+1-t). Mathematically, this is expressed as:
(Distance) (Time)-2 (222
Columns five and six in the table of data contain the values for T2 and T-2.
6. Inserting the proper time factor into our proportion gives:
(Mass) (Distance) (Force)-1 (Time)-2 (223
The right-most column in the table of data contains the values of the proportion, which are the same for every run of the experiment: 9. This value of 9 is the constant of proportionality. We can now convert the proportion into an equation, as follows:
(Mass) (Distance) (Force)-1 (Time)-2 = 9. (224
Traditionally, in force equations, the force factor is isolated to the left of the equal sign. This gives:
(Force) = (Mass) (Distance) (Time)-2 (9)-1 (225
or
F = 9-1 m a (226
This does not look like the inertial-force equation that we all know. How can we get rid of the factor, 9-1? We can change from the arbitrary system of unit measures that we used in the experiment to a system that changes the constant of proportionality from 9-1 to 1. To do this, we can change the magnitude of one or more units of measure. If we change only one unit, we would multiply one of the existing units by a corresponding factor, as follows:
Unit Factor
Force 9
Time 3
Mass 9-1
Length 9-1
In our experiment, we increase the magnitude of the unit of force by a factor of 9. Therefore, instead of needing 9 units of force to balance the equation, we only need 1. The inertial-force equation now looks familiar:
F = m a (227
Notice that the old constant of proportionality, 9-1, no longer appears in the inertial-force equation. This is because it is now equal to 1 and can be removed from the equation. Yet, in an equation, not only must the values on either side of the equal sign be equal to each other, but the dimensions must be equal, as well. In our inertial-force equation, the dimensions are not equal to each other. Force is not equal to mass times length divided by time squared.
However, we can force them to be equal by inserting counter-balancing units of measure into the constant of proportionality, which we give the code, k. In this manner:
F = k m a (228
where:
k = 1 (force unit) (time unit)2 (mass unit)-1 (length unit)-1.
All of the units except the force units cancel each other, and force equals force. The equation is now balanced with regard to units of measure.
The SI unit of force is the newton, N. It is defined as the amount of force that is necessary to apply to a one-kilogram mass such that it accelerates the mass at a rate of one meter per second for each second that the force is applied. This definition means that the value of the constant of proportionality, k, is automatically set to one.
Therefore:
k = 1 N·s2·kg-1·m-1 (229
and, by tradition, is not included in the inertial-force equation.
The SI unit of inertial force, the newton, N, is not based upon the natural attributes of a particular elementary particle and is, therefore, an irrational unit. Physicists use the phenomenon of inertial force to define the SI unit of force, which is used, not only in inertial-force equations but in electromagnetic- and gravitational-force equations, as well.
This practice presents major ramifications for the form of these forces' equations. In particular, it creates the occurrence of an inappropriate FUPCON in them--the fine-structure constant, α, which we see when we eliminate the permittivity, ε0, and permeability, µ0, constants, respectively, from Coulomb's and Ampère's electromagnetic-force equations and the gravitational, G, constant from Newton's gravitational-force equation.
When working with either an electromagnetic- or gravitational-force phenomenon, we should use a unit of force that is appropriate to it. We realize the reason for this when analyzing the two force equations--electromagnetic and gravitational--and comparing them with each other. See Chapters 5 through 11.
| Title Page | Table of Contents | Preface | <<<< | >>>> | Appendixes | References | To Order This Book | WritWord Homepage |