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# 3.00 Work, Energy Discussion: BODY

An outline of how we will proceed.

By previous studies we know Newton's Laws of Motion provide a valuable means of understanding motion. In the 60-90 years after his death, scholars extended the utility of the 2nd Law. Though this extension can be told in a casual way, physics or engineering applications require a quantitative development.

Mathematical extension of the 2nd Law statement begins regarding the rate of change of momentum of a BODY under action of forces. Newton's 2nd Law is "extended" simply. Consider that the BODY is displaced. Multiply Newton's 2nd Law by that displacement.

This writing needs revision!

Changes of matter, physical reality and physical events occur in the physical are complex. Our scientific studies of real events do not adress reality; it is too complex. Our studies have a The studies select matter assumed principle to the event)adress begin with To study reality begins Every study nature and its Instruction and study of reality, at a beginning level, happens in special, limited ways. Masses as we see and feel them are treated differently from what they are. Mass is treated in accord with a model of that mass. The simplest model used for real masses is that of a BODY. To view reall matter as a BODY is to disregard the consequence of its spatial variation; to imagine it to exist as "mass at a point." System is our perspective and BODY is our model of the reality we see. Characteristics of a BODY are its position in space and its velocity; these quantities are vectors.

To express a vector quantitativly requires statement of a "number" as the vector magnitude (being zero or greater) and a statement of the vector's orientation in space, that is, its "direction." We use the idea "direction" all the time in daily speech. To state direction mathematically requires preparation.

"Directions," are stated relative to a pre-determined (agreed upon) origin and a set of unit vectors attached to that origin. For example, "where I am standing and the directions: east, north and up." That information comprises specification of a coordinate system: an origin and a unit-vector basis.

The vector, velocity, is related to changes (instantaneous changes) of the time-wise vector of position of the BODY. The relation of velocity to positions of a BODY is defined by calculus. With these aspects as preparation, we applied Newton's Laws of Motion which include force, the cause of change of motion of a BODY (as a construct). (1) Newton's 2'nd Law applies to the vector property of a BODY; its momentum. (2) The 2nd Law multiplied by velocity accomplishes a new and powerful physical perspective. The derivative of momentum becomes energy-rate and force times velocity becomes work-rate.

Newton's Second Law is mathematical, being a first-order vector differential equation with momentum of the BODY as the dependent variable, time as its independent variable, and the sum of forces (often called the "forcing function") as a non-homogeneous term. The equation is general but made specific in application once a BODY is selected, a time domain is chosen, coordinates and a vector basis are ascribed, and the initial condition and the forcing function are prescribed.

While Newton's 2'nd Law is expressed with the elegance of mathematics, it is more than mathematics; it is physical. Its solutions (adjusted by way of experimentation) provide valuable approximations of actual events of physical reality. Equation solutions of the 2'nd Law actually predict the outcomes of motion.

Extension of Newton's Second Law:   A fact of mathematics is that when an equation is multiplied by a constant, or a variable or some operator acts, that equation is unchanged. Put otherwise, the information of the equation after the operation is no more that it was previously. However, when Newton's 2'nd Law is multiplied by an arbitrary vector differential displacement the resulting scalar equation is altogether different from the original. How can that be?

The reason is Newton's 2nd Law is not merely mathematical. The mathematical operation, multiplication of the 2nd Law by a differential displacement represents physical change. The original physical entities, the terms of the 2nd Law, become new independent physical entities: differential kinetic energy (differential potential energy ~ about this later) and differential work.

Mathematics is the tool for equation development. When Newton's 2nd Law ( differential in form) is scalar vector multiplied by an arbitrary displacement three ideas (each potentially related to the other two) emerge. Two ideas are energy forms which are kinetic energy, potential energy and work. A general development, being a bit complex, is done later. First we consider some special cases.

Work is the Measure of Energy   This ultimate idea is seen the easiest. The ultimate idea from the transformation of Newton's 2nd Law is that matter has energy and that work is the measure of energy. This is to say that work can be made quantitative and by that fact so also can energy of a BODY.

Work with Vertical Motion:  To move a BODY vertically and uniformly, In its general occurance, work of a BODY results in changes of both kinetic energy and potential energy. For the special occurance wherein work occurs to a BODY that moves in a plane horizontal to Earth, no change of potential energy occurs. Some call this work, acceleration work.

• intro the Momentum Equation
• Extend the Momentum equation BODY to include the energy transfer mechanism, work. This will need three parts.
• We will do Work horizontally to identify KE and "acceleration" work.
• Do work vertically to identify gravity work PE.
• Put these together
• Many examples ~ BODY
• Examples will ruin the idea BODY
• One Example will point need of Internal Energy.

We have studied events (including permanence or no event) for selections of physical reality (systems) with that matter approximate as a BODY. Two characteristics of a BODY are its position in space and velocity. Both characteristics are vectors; velocity requires calculus for its definition. Newton's Laws of Motion included the construct, force, as the cause of change of motion of a BODY. (1) Newton's 2'nd Law applies to the vector property of a BODY, its momentum. (2) The 2nd Law multiplied by velocity accomplishes a new and powerful physical perspective. The derivative of momentum becomes energy-rate and force times velocity becomes work-rate.
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