THERMO Spoken Here! ~ J. Pohl © | TOC NEXT ~ 3 |
(as with a particle as system)
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.
A summation precedes the symbol force, F in this "punch-list" equation. The summation says think 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.
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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 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 texts make no distinction, "inexact." |