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This development addresses the system model, BODY. For all events of the BODY, Newton's Second Law applies. Newton categorized forces as acting either "at a distance" or "at the surface" of the system. Forces are categorized as "BODY forces" which act at a distance and "surface forces" which act at the system surface. Our form of "f = mA" is written:
|(1)Newton's Momentum Equation: BODY as system.|
BODY forces are electro-magnetic, electro-static and gravitational in nature. Our concern is limited to forces of gravity.
|(2)The instantaneous rate of change of momentum of a
BODY in time equals|
the gravity force and sum of surfaces forces acting.
Sometime in our history someone (capable at basic calculus) decided to scalar multiply the above vector-differential equation (Newton's Momentum Equation) by an arbitrary vector-differential displacement of the BODY. That differential displacement is written as: dS. The first step of that multiplication is:
|(3)This notation indicates a multiplication of the entirety|
of Eqn (2) by “·dS”.
To continue the multiplication, three scalar vector products must be effected.
|(4)Scalar vector multiplication is a linear operation. Equation (3) becomes these three terms.|
Our task is to simplify Equation (4). The terms are written in general or (mathematically speaking) implicit form. Scalar multiplication of the time derivative of the vector momentum by the differential displacement (the term left-of-equality) transforms that term into the scalar differential:
The first term right of equality is the vector product of the ever-present gravity force times the arbitrary displacement. This scalar multiplication yields:
To proceed from equation (6), let's recognize that -mgodz is also equal to - d[mgo(z - zo)]. This is because zo is a constant reference elevation and m and go are also constants. We'll see why we do this shortly below. With these changes our equation is now:
An important option occurs with regard to the differential term "-d(mgZ)." As above, being right of equality, it is properly called a "gravity work" term. It is work associated with displacement of the body force, gravity, for any movement of the system. But we see the scalar multiplication has eliminated all but vertical motion. So with mass and gravity constant, this "work" term is quantitative in terms of elevation which is a spatial, visually observable (extrinsic) property.
One option is to use the above equation as is. Some areas of mechanics use this approach. A second, tempting option, the choice of thermodynamics, is to move the "gravity work" west, across the equality boundary where it belongs - as an energy of the mass.
It is very important to understand and remember that this stratagem about gravity force amounts, in effect, to our change of system as being some body to system being that body and another big one - EARTH! Next, kinetic and potential energy are defined and the equation becomes:
This development has followed the "plan of attack" of engineering: "do the simpler things first." Above left are the "exact differential" forms for kinetic and potential energy. Integration of an exact differential is perfunctory. Our problems are right-of-equality. The integration of the summation of surface forces through their differential displacements is more difficult task.
To apply (7) to a specific case, the equation must be associated with a physical situation, then integrated. Integrated, "left terms" readily become "deltas" or increments. Integration of the work terms requires special attention.
For the purpose of further study and discussion, the above integrations are often notated for an event in brief as: W1-2. This yields the nicer equation form:
By our manipulations:
i) System momentum, a vector, was transformed to a new system (scalar) property, kinetic energy.
ii) The vector effects of gravity force on systems were recast, sent left, to become the scalar property, potential energy. By this new perspective, Earth was added to our system.
iii) The remainder of the right side, or "force side," of Newton's modified Second Law of Motion, in the new equation, is the new construct, surface work.
Practically every student has done calculations of kinetic energy change, potential energy change and work. In this writing we adapt these math and physics thoughts to more complicated events. Our path will be to progressively present "problem child" physical circumstances that coax our models. Our destiny, which we don't reach, is to adapt Newton's ideas to the realm, to include comprehensively, systems with inner time dependent distributions of mass and energy, special "inside the matter" energy types (intrinsic energy), work, friction and heat.