A Quantitative Evaluation of Natural Systems, Volume I A Quantitative Evaluation of Natural Systems

Volume I

by Dr Profidious Vile

Prologue

Since the foundation of the universe, every plane of existence has always been given its own set of physical laws, with effects which can be measured and predicted with perfect accuracy, given a knowledge of all contributing factors.

These laws are superceded only by the act of magic and the gods on our world. It is my belief that magic was the founder of all physical law, thoughout every physical plane in the universe. Everything we know and feel about our world is the product of this mysterious, chaotic force, which in contrast to physical law, appears volatile, unpredictable and ever-changing - a menace to the good, ordered world which we have all come to love. While we must love magic for giving birth to everything around us, we must despise it also, for it threatens the fragile integrity of our joy-giving planar laws.

How do we define magic? Magic is any change which is impossible to explain with physical laws. Nature encompasses everything else. I believe that even life itself, the most complex property of our plane's physical law which we have perceived, can be reduced to a series of fundamental principles which assert causality in every thought and action we make.

But this is not a book for the discussion of magic, or the founding of our universe. Rather, it is a yearning and a searching for a full knowledge of the physical processes and phenomena in natural systems, justified by mathematics and higher reasoning. Come with me, Dr Profidious Vile, and I'll share the fruits of my own search for knowledge, in the hopes of opening a new door to society. Behind that door? A world where we are no longer enslaved by magic, but walk hand-in-hand with science. Mankind will no longer be controlled by a minority of mad and reckless mages and devout worshipers of selfish and destructive gods with questionable agendas, but will be placed in the hands of every man and woman, clad in the bulwark of scientific research and invention.

Take my hand, and we'll walk through the door together.

Energy and two basic laws

I have coined the term "Energy" to explain one of the most important properties which defines our plane's natural physical laws. Energy, in this sense, is a very difficult concept to describe. To assist the reader in understanding what Energy is, I will refer to an example.

A ball made from the coagulated juice of the rubber tree is stationery on the ground. It is in a state of low energy. To lift the ball to a height, we do work to resist a force called 'gravity' which I will describe in a later section. To lift the ball, we have applied energy to it. It now has latent energy which will be released when the ball is dropped, in a form of energy which expresses itself as motion. When the ball reaches the ground, the energy we have put into the ball by lifting it into the air does not allow it to immediately return to its low energy state. The ball bounces back up into the air, but not before making sound, and generating a bit of heat in the collision with the ground. The sound and heat produced steal a bit of energy from the ball. The remainder of the energy is consumed in lifting the ball back into the air, where it reaches the pinnacle of its height, beneath the height to which the ball was originally lifted, showing that some energy has been lost. This continues until all of the energy in the ball has in fact been applied to heat and sound.

I'm sure that you are now beginning to understand my definition of Energy. Energy is a fundamental entity of nature that is transferred between parts of a system in order to produce change in the system.

Let's assign a unit of measurement to Energy and call it the 'Schnitzel', abbreviated Sz. Energy can be represented in mathematical equations with the symbol E.

I have extrapolated two laws which apply to all instances of energy I have observed in the absence of magic:

LAW I: In any system which is closed and does not receive interaction from magic or entities outside of it, the total energy in the system remains constant.

Mathematically:

∑E(initial) = ∑E(final)

or

dE/dt = 0

This means that all energy is conserved - without magic, it can't be destroyed or created. Rather, energy can only shift from one form to another. This also proposes that the net sum of energy in our plane is constant, unless there is a change in energy imposed by magic, or if the isolation between our plane and another is broken.

This prevents the bouncing ball from the example from rising higher in the air than it was dropped. This means that if we spend all day lifting rubber balls and dropping them on the ground, we'll eventually get exhausted as we spend our own internal energy. Our internal energy comes from what we eat and drink, which in turn receives its energy from some other source, and so on.

LAW II: Systems without magic tend to a state of greater entropy, approaching a maximum energy. Just as a glass figurine which drops and shatters on the ground does not naturally reassemble itself, energy continually shifts to a state of higher chaos. In all observed cases, this is heat. Mathematically:

∑H(initial) < ∑H(final)

or

dH/dt > 0

Where H is the amount of heat energy within a system, measured in Schnitzels (Sz).

These laws have powerful implications on the destiny of our plane. They suppose that all capacity for change in energy must gradually and eventually run down to a state of low or no non-heat energy, prohibiting the existence of further motion or life. All indications suggest that this will not be occurring in the perceiveable future.

Also to consider, with the massive amount of energy-producing magic which is utilized on our plane, it is also puzzling that the net sum of heat in our plane hasn't increased to the point of making our plane uninhabitable. There are a few ways of explaining this:

a) Energy produced (or removed) by magic is fleeting. Even permanent magic (continual light sources, for instance) is continually balanced by a subsequent loss of the resulting entropy (at some stage). b) Energy 'produced' somewhere on the plane by magic is the consequence of a magical reduction of energy somewhere else on the same plane. c) The entropy produced is not yet sufficient to cause a noticeable overall increase in temperature d) There is an unseen outlet for heat which prevents the buildup of heat on our plane. This makes the most sense, and would place the outlet for heat as the surface sky, which is extremely extensive. This would provide a form of buffer against temperature changes.

The same problem applies to the constant source of heat we receive from the sun. This can also be explained by points c) and d), or c) and d) with a combination of a) and/or b) making c) and d) the most persuasive arguments.

Force and Motion

What is force? In the simplest terms, a force is a push which is exerted to cause a change in motion. I will give my definition of motion later. For now, persist in the classical understanding of the term as I state the three laws which I have experimentally found motion and force to fully abide by:

LAW I: All bodies in nonmagical systems persevere in their state of rest or of moving uniformly straight forward, unless they are compelled to change their state by an applied force.

Simply, zero net force implies zero acceleration, where acceleration is a change in velocity.

I'm sure that most intelligent people would be in an outcry by now. Objects released in mid-air accelerate towards the ground, with no apparent application of force.

This occurs because there is a force which pulls all bodies towards the ground, which I have called "Gravity". While the actual source of gravity is a great mystery, it has been determined that gravity applies the same acceleration to all objects, meaning that the force gravity applies is proportionate to the mass of the object it is applied to. This is why two objects with different masses, ignoring air resistance, dropped from the same height will hit the ground at the same time.

LAW II: The alteration of motion of an object in a nonmagical system is directly proportionate to the force which is applied to it, and is always in the same direction as the force. If we double the force, we double the alteration of motion. If we triple the force, we triple the alteration of motion. If we apply a force for twice as long, the net alteration of motion will double. If two or more forces are applied to the same object, antiparallel, obliquely or otherwise, the net force, and thus net alteration of motion will be the resultant of the sum of the force vectors.

We'll assign "Force" (abbreviated F) a unit, and give it the unit Lederhosen (abbreviated Lh), which is a composite unit.

Consider that the acceleration of a mass, by the second law, is:

a α F/m

Therefore:

a = F/m

And:

F = ma

Where F fully resolves the constant in this relationship.

We will also consider a new term for motion, Momentum, p, which will be proportionate to both the velocity and the mass of the object to which it is attributed. Thus:

p α mv p = mv

Where p is measured in kgm/s

And the change in momentum with respect to time, obtained by differentiation of momentum, p with respect to t:

-dp/dt = ms/t^2 = mv/t = ma = F

Hence, change in momentum, impulse:

∆p = Ft

This also supports the second law, which states that a change in motion is proportionate to the time for which force is applied.

To add forces, both direction and magnitude must be considered. This is why it is necessary to use vectors when evaluating multiple Forces. For instance, if an arbitrary mass is being pushed towards the north with four Lederhosens of force, and another force is pushing the arbitrary mass towards the east with three Lederhosens of force, the net force acting on the arbitrary mass is:

∑F = (4^2 + 3^2)^-2 = 5 Lh tan(θ) = 3/4 θ = arctan(3/4) = 036º

5 Lh at 036º

LAW III: All forces occur in pairs, which are directly opposite to one another and equal in magnitude.

e.g. As a man jumps, he applies a force to the ground, down. This third law dictates the existence of a second force - the ground pushing up against the jumper with just as much force as the jumper is exerting against the ground. This opposing force accelerates the jumper into the air.

The Kinematic Equations

It is very simple to derive equations of velocity, displacement, acceleration and time, as doing so only requires a bit of logical thinking, and an understanding of these concepts.

Let's begin with our simple equation for acceleration. Acceleration is the change in velocity with respect to time, so:

a = ∆v/t a = (v - u)/t where u is the initial velocity, and v is the final velocity. at = v - u v = at + u ...(3) v(average) = (v + u)/2 ... (1)

Now:

v(average) = s/t s = v(average).t substitute (1) into this equation s = (v + u)t/2 ... (2)

Now, rearranging a = (v - u)/t, we get t = (v - u)/a, which we substitute into (2)

s = (v + u)(v - u)/2a 2as = v^2 - u^2 v^2 = u^2 + 2as

We can also substitute (3) into equation (2) instead:

s = vt/2 + ut/2 s = (u + at)t/2 + ut/2 s = ut/2 + at^2/2 + ut/2 s = ut + at^2/2

Thus, we have the kinematic equations, which describe acceleration, velocity and displacement independently of force:

v = u + at v^2 = u^2 + 2as s = ut + at^2/2 s = (u + v)t/2

Linking Energy and Work with other concepts

If we push against a wall all day, and the wall is never displaced, we cannot say that any work has been done.

From this we say that force is the rate of change in work - not with respect to time, but with respect to how much the mass to which the force has been applied has been displaced:

Work = ∫F.ds Work = Fd + c if F=0, Work = 0. Thus, c=0 Work = Fd

This is measured in Schnitzels (Sz) or kgm^2s^-2, and is interchangeable with energy.

From this, we can work out how much work is required to lift an object, and thus how much latent energy results from the lifting of this object:

F = mg E = Work = Fd d = height above the ground = h Therefore,

E=mgh

This is the equation for what we call Gravitational Potential Energy, and it is the amount of energy inherent in the mass which will be restored as kinetic energy if allowed to freefall height h to the ground.

(Note: g is acceleration due to gravity, which, if you recall, is a constant, and is measured at 9.81m/s)

If we now consider the kinetic energy (the energy in the movement of the object) just before it hits the ground:

E=mgh

If we now substitute in an appropriate kinematic equation, describing the fall of mass m at acceleration g through height h:

v^2 = u^2 + 2as u=0 a=g s=h v^2 = 2gh gh = v^2/2

substitute into E = mgh

E = mv^2/2

This is the equation describing the energy of motion: kinetic energy.

Continued in Volume II

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