THERMO Spoken Here! ~ J. Pohl © TOC     NEXT    ~   111

# 3.01 Work: BODY

WORK is a construct developed by our predecessors. Thermodynamic WORK, their idea, their plausible perspective of physical reality, is crucial to how we study physical reality now. WORK defined relevant only to a system is the consequence to that system of any force (acting at (on) the system boundary) which displaces (acts at the boundary and moves with the boundary through) a distance. Force and displacement of the surroundings are vectors. WORK is the scalar product of these vectors. Systems have energy. WORK is the measure of the energy systems have. WORK is the means by which our momentum and energy equations were made quantitative. WORK moved science from the "thought" perspective to the "measurement" perspective. System, the energy of that system, and WORK (one cause of system energy change) are extensions of Newton's Momentum Equation about the product of particle mass and velocity.

The Greek symbol, meaning "summation" precedes the symbol force, F in this "punch-list" equation. The summation sign reminds us to "sum over all forces." In Newton's time the sum was over just two categories of force. Gravity was first, of course. The next force was the body force known to act because of earth and the sun over the entirety of any system. Boundary Forces, being forces applied from nearby surroundings on and acting at locations of the system surface.

How to deal with boundary force? Some texts call boundary forces, surface forces, meaning acting on the surface. Boundary is preferred here because it implies a system has been selected while the word surface is less precise.

To simplify things, expand the body forces, and restrict our concern to the gravity force of earth (E) only - the other body forces are often absent or negligible.

Next substitute 3) into 1), ease up on notation, then scalar multiply the vector equation by a differential displacement of the particle (system) mass of the equation.

Two terms right of the equality are "differential WORK." WORK but one (related to potential energy... gravity force) have the form, with The details of this multiplication are important and have been completed precisely in many texts. That task takes a page or two when done correctly. But let us leave that for later. Vector multiplication of the term left of the equality and the first term right of the equality result in two scalar differentials. After some algebra the result is:

To discuss these terms we place them in a table.

 a) Having studied physics and calculus, we recognize this as an exact differential, "differential kinetic energy" (dKE ) of the particle. The term is scalar, not vector. This term, an energy of the mass, a particle, is on the "system side" of the equation. b) This differential term, also exact (provided gravity (g) can be considered constant, which it can in many instances) integrates easily because the constant force of gravity is directed toward earth, i.e. in the negative K (-K) direction. Hence only the "Z" component of any displacement is non-zero. Some texts identify this term as "gravity WORK." (dWgravity). Though the term is on the "boundary" side of the equation, it greatly resembles a differential energy term, like kinetic energy. This becomes differential potential energy of the particle, (dPE). It will be moved to the left of the equality as will be explained shortly. c) Unlike Term 1) and 2), this term is not an exact differential. In thermodynamics, differentials of WORK are said to be "inexact" differentials. Calculus does not have the distinction, "inexact."

To integrate 5), apply the integration idea then decide what the limits of the integration should be. Integration of an exact differential yields a simple numerical difference - the "second (final, or at the termination of the event) value" diminished by of the "first (initial, or beginning of the event) value." The result of these steps is:

About now the temptation to move gravity WORK to the left of equality and call it potential energy becomes strong. Of course we will do that - but what will it mean, how might it be justified?

A running theme of this presentation is the equations of thermo-fluid mechanics are "physical equations," and as such, whatever is written left of the equality has the characteristic, "item about the system," while terms "right" of equality are "items about boundary events of the system."

When we simply move "PE" from right to left of the equality we also change out system perspective. With potential energy as a system property, the system is changed to the become the Body

and
EARTH. How can this WORK? Earth is mostly a "silent" non-participating partner other than that it provides gravity in direct proportion with elevation. Neat, Hey? Equation 6) in increment form is:

Most texts avoid use of summations in favor of saying "net" with the idea that the summation sign (being a Greek symbol, perhaps) create confusion. Here summations are used on every term that might have more than one contributor or source. Summation signs remind us to "look around" and add all relevant effects.

The energy terms of this equation (left of equality), evolved through application of the construct WORK to Newton's Momentum of a Particle. These energy terms are said to be extrinsic meaning determinable by outside observation or energy of a system determinable with no consideration other than the particle mass and its position and velocity in space.

This WORK, the only WORK a particle can experience, is sometimes called, DISPLACEMENT WORK

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