ABSTRACT
In this chapter the data from the main experiment (presented in Chapter 6 ) and a third subsidiary experiment are analyzed for the purpose of rationalizing how amplitude is determined as a function of variations in mass, length and period of the oscillator. The following hypotheses, developed in Chapters 10 and 11, direct the analyses: (i) change in the periodic timing of a local wrist-pendulum system of fixed mass and length is brought about through changes in internal mechanical parameters (such as stiffness, resting length, etc.) that preserve the amount of energy degraded constant over varying frequencies of oscillation—that is, the transformation is adiabatic, preserving the ratio of energy of oscillation to frequency of oscillation constant in the laboratory frame of reference; (ii) the constant of proportionality that relates energy to frequency scales as a function of the moment of inertia of the oscillator.
The following data analyses, bearing on the amplitude-period relation depicted in Figure 10.2, are reported: First, energy by frequency plots are generated for each wrist-pendulum system for each subject. These are characteristically linear (the slope identifying the constant of action) with non-zero frequency and energy intercepts; second, a system's constant of action—as given by the slope of its energy by frequency plot—is used to generate the curve of the system's change in (maximum) 363amplitude against the system's change in timing (both period and frequency), according to the action-based equation for amplitude developed in Chapter 10. These curves capture the qualitative nature of a system's spacing-time relation— the sometimes rising, the sometimes falling and the sometimes stationary trajectory of amplitudes with increases in period. The demonstration of constant action under invariable external boundary conditions (fixed moment of inertia) is taken as the departure point for determining amplitude's scaling to mass and length. A proportionality for scaling in terms of action and inertia follows from the constancy of action: amplitude p∝(Hiτo/Ii)1/2 . The transformation of this action-based proportionality into a scaling relation for spacing in terms of mass and length is brought about by replacing action and periodic timing by their respective moment variables with coordinate space parameters affixed; that is, https://www.w3.org/1998/Math/MathML"> H i = a H ( M L 2 ) c H https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315648460/0d2be172-8107-4727-8706-4ccdaf357555/content/inline-math_17_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> τ o = a τ ( M 1 / 16 L 1 / 2 ) c τ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315648460/0d2be172-8107-4727-8706-4ccdaf357555/content/inline-math_18_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Ignoring coefficients, the scaling relation for spacing then has the form:
The amplitudes of single wrist-pendulum systems, as reported in the main experiment and as reported in a third subsidiary experiment, were used to evaluate the action-derived moment variable for spacing. The fit of predicted and actual mass and length exponents for amplitude was perfected by recognizing that coordinate space inhomogeneities "adjust" the proportioning of dependent quantities (e.g., action, period) to mass and length differentially.
Two major conclusions are drawn highlighting the interplay of laws and coordinate spaces in governing the pendular clocking mode. First, the scaling relation for spacing arises from the nesting of the mechanical law specifying periodic time within the energy degradation law specifying action, with the coordinate spaces of these laws playing the decisive role in determining the signs and magnitudes of amplitude's mass and length exponents. Second, when two local pendular, clocking modes are in isochronous coordination, their common periodic time is specified at the cooperative level by the periodic timing law applied to the virtual mass and length magnitudes and their different amplitudes are specified at the atomisms' level by the locally defined action constants relating energy to frequency.
