Re: The Principle of Equivalence Explained

-jr writes>
Hello NoEinstein. Plane geometry was a turning point in my mind as a
kid. My eyes opened. Trig was another explosion for me. Certainly as
an architect you must have had training there. You should also have
training in vector analysis of some type. Which set me on a mission.
As an architect you have a primary interest in “feet on the ground
units of measure”, like grams and kilograms or pounds and ounces,
etc. Quantities like torque, stress and leverage are dealt with by
you regularly. You have a strong arsenal of mathematical tools already
at your disposal. You may not require an understanding in
calculus to put together ideas, but if you are without it, you do
yourself a great disservice. Calculus, literally blew me away.

It is true that Kinetic Energy and linear momentum are defined in
units of mass and velocity. Weight is defined in units of mg. More
generally Force is defined in units of ma. Any magnitude for mg or ma
converts to an instantaneous magnitude of momentum. So when you
measure acceleration at a precise moment in time, you have an
instantaneous velocity. Prior to Newton (and Liebnitz who provided
the superior notation in use today) we could only figure average or
median rates of change. In the case you cite with your dented clay
example the magnitude of Force mg is taken at its instantaneous final
velocity, or mv.

So Pounds can be used as a common “feet on the ground” measure for any
quantity that is defined in terms of mass and motion. Here with KE and
LM the precise difference is in the exponents and the coefficients and
that energy is a scalar and momentum a vector. KE = .5mv^2. Linear
Momentum = mv. Both are conserved. The conserved aspect is
especially functional, but the above difference can be used
advantageously through the valid mathematical operations of
differentiation and integration, as well, where you currently use only
addition and subtraction and their improved forms of multiplication
and division.

Pi is a quantity that comes up in the physical math in many cases.
What does Pi have to do with physical math quantities like KE and LM?
No amount of addition and subtraction will provide you a clue. Where
differentiation and integration will. (Actually cones and conic
sections will also supply a clue… but a more primitive one).

Consider a Euclidean circle with area,

(1) [pir^2], and circumference,
(2) [2pir]

Consider PE or KE

(3) [1/2][mv^2], and linear momentum,
(4) [mv]

Note that by letting the exponent ^2 in (1) operate as the coefficient
2 in (1) by multiplying one case of Pi by the exponent ^2 as the
coefficient 2, and reducing the exponent magnitude in (1) by 1 we have
(2). This is called differentiation in calculus. In words here: we
differentiate the area of a circle and we get its boundary length. In
reverse we take the integral of the circumference and acquire the
area. A circle circumference is an efficient enclosure of area.

So what is the differentiated and integrated pair relationship between
energy and linear momentum, if any? In (3) we multiply the coefficient
by the exponent to obtain 1. We reduce the exponent by 1 and we have
(4).

So the relationship between potential and kinetic energy, and linear
momentum, is analogous, in terms of the mathematical operations of
differentiation and integration, to the relationship between the
circle’s area and its boundary, again, with respect to differentiation
and integration. Is this worth knowing to you? The fact that Pi
mysteriously appears everywhere in the physical math, insured my
interest. Differentiation and integration are legitimate mathematical
operations, even in complex cases. And where else could you acquire a
relationship between the circle and its boundary and energy and
momentum? And a scalar and a vector.

Is the relationship that is revealed by differentiation here
significant? Is least action significant? Are vectors significant?

Do the same thing to (differentiate) [16t^2].

I am not ignoring your ideas wrt to EMR. This is much simpler to
initially entertain and complex enough to warrant initial exclusion of
anything else. I will address that topic as well but I would like to
establish a meaningful dialogue first. I am not trying to teach you
the calculus. Only to establish its value. Have a good time.
johnreed

.

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