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3.08 Energy Equation: BODY

Energy principles (though usually not in the context of an energy equation) are introduced in beginning physics. Newton's Second Law of Motion described "change of momentum" of a BODY in consequence of action of a sum of forces. Another perspective of such events, a perspective that introduces the construct, energy, can be developed. When a positive change of momentum of a BODY is multiplied by its positive displacement with that change, a change of "energy" of the BODY is said to have occurred as a consequence of an energy "transfer" mechanism; "work."

The theory and rationale of "work and energy" is presented in texts in a number of ways. In all cases with a BODY as the physical entity. A popular approach of HS physics is algebraic and one-dimensional. Yet forces, positions, velocities and momentum are vectors and velocity is a derivative. To look more carefully at Below a mathematical proof is presented.

Below (for a BODY as system) analytic steps are shown (Newton's Second Law is altered) to represent the event of action of a single force (acting against a BODY) to represent its displacement , is transformed into the simplest energy equation, which we will call the mechanical energy equation. The derivation produces three important ideas and their mathematical forms. These are: Kinetic Energy and Work; then Potential Energy. It will suffice for us to consider 0XZ-space. A bare minimum of forces is needed for this discussion. These are the force of gravity and the sum of all surface forces which is subscripted "app" for applied.

(1) 1

When this equation is scalar multiplied by an arbitrary differential displacement a scalar equation results in which the three terms above are transformed into differential kinetic energy, differential gravity work and differential work respectively. Below, we will complete theser mathematical operations step by step and as clearly as possible. To indicate scalar multiplication of the above equation by a differential displacement, we append " • dS " to each term.

(2) 2

We begin with term ( 1 ) which is the easiest to reduce. To examine that term we write the gravity force as its vector.

(3) Gravity varies inconsequentially up to 5 miles in
altitude. We use the surface value as a constant.

Position in our OXZ-space is represented implicitly by the vector S. Below left it is written in Cartesian coordinates and below right, the vector differential of position called a "differential displacement," dS is written.

(4) 4

Evaluation of (1) above calls for the scalar multiplication of the vector gravity force times vector differential displacement. Below on the right, K dot I equals zero and K dot K equals one.

(5) 5

Since mg_o is constant, we moved it inside the differential. We'll substitute this result into the main equation for (1) shortly. The reduction of term (2) requires a number of steps. A longer path is taken to use minimal calculus. We write velocity V, in its component form then multiply by the system mass, m.

(6) 6

Next, (mV)_body (with m constant) is differentiated with respect to time.

(7) 7

We see we have changed the form of the first, vector of (2), d(mV)/dt. Now we write the vector differential displacement of the mass, dS.

(8) 8

There are two equations immediately above. We set the scalar product of the terms "left of equality" equal to the scalar product of the terms "right of equality." We leave "left of equality" implicit and complete the product "right of equality."

(9a) 9

Below we use the definitions of velocity (in the "x" and "z" directions) to change the "look" of right-of-equalityabove. Just a change of notation.

(9b) 9.

The above operations have shown (2) to be equivalent to the intermediary result (A). To proceed we write the vector V in component form, obtain its magnitude, |V|, square that magnitude and multiply it by m/2..to have m|V|²/2.

(10) Check these steps. The result is correct but the steps are compact.

The term, m|V|²/2 , is called the kinetic energy of the BODY. When we take the derivative of the kinetic enerty we discover that it equals the terms above, (A).

(11) 11.

Whereupon we have proved:

(12) 12

We have performed the scalar multiplications of terms (1) and (2) of equation (1). So we make those changes in the equation to obtain :

(13) 12

Physical equations of thermodynamics have one of three mathematical forms; differential, difference and rate. Derivations (as above) are done using differential form. Physical equations also have meaning "with reference to equality." Terms "left" of equality describe changes of system properties, that is, activity interior to the system boundary. Terms "right" of equality identify change agents or transfer mechanisms with actions that cross the system boundary. To rearrange a physical equation is to lose track of what is what. Don't algebraically scramble the terms - nothing is gained.

Returning to our last equation, we recognize differential kinetic energy on the left. Both terms on the right are termed work - energy transfer mechanisms. The first term is called "gravity work," the second term is called extrinsic work. Mechanics applies this equation "as is."

With thermodynamics, a grand simplification is apparent. The simplicity of the gravity work term, in that it is extrinsic (outwardly observable as an elevation change), invites a new perspective of system energy. Thermodynamics takes this path and relocates differential gravity work, (term 1) to be "left of equality" and thereby become differential potential energy (energy possessed). The resulting Extrinsic Energy Equation is written below. In it we associate work distinctly with surface forces. Usually more than one force applies. The summation sign, Σ, is carried to remind us to include ALL forces. Put otherwise, were one relevant force overlooked, the answer would be meaningless.

(14) 14

In the physical sense, the change means whenever the Extrinsic Energy Equation is used, Earth is part of the system. The idea works well; in most events of humans, Earth does nothing but provide the a gravitational field. The notations KE and PE are clear and convenient. We substitute them.

(15) 15

The term right of equality is stubborn and has not been reduced to some general scalar differential form. Some texts label the term "differential work" and write it as dW with a horizontal dash through the top of the d, a strike-through. Something like: dW. The term constitutes invention of a new differential that is inexact.

An immediate uses of this equation is to associated it to a specific system/event then integrate it between appropriate limits. Terms "left of equality" integrate readily to become increments. The work term is integrated symbolically:

(16) 16

Strictly, the development was with regard to a body but it applies to an extended body and to two or more bodies as well. So the notation, c.m., for center of mass, of the system does no harm.

As you learn, you will become able to test the value of any equation without putting it to a specific task. There are no general rules or steps. Above is an equation. Lets think of physical possibilities, some ideas, the equation purports to explain. To start, lets drop the summations. Whatever works in summation must work for one.

(17) 17

The above equation encompasses a variety of events of a body.

Conditions

Zero Initial and Final Speed

Recognizing that W means ΣW, another event equation is:

The equation above the table extends to , as it is, ???

The number zero, 0, means, "does not apply or has no value." Some possible applications of zeros in the equation are:

(18) 18

Option 1 is logical.??? (typo) and mean, respectively, the initial and final speeds are equal or zero and the initial and final elevations the same. For mass or gravity to be zero is ridiculous. Option 2 asserts simply that energy of a system can change from KE to PE and the reverse. And since system mass is a factor of both terms, mass is neutral to such events. Option 3 reveals a weakness in notation of the original equation. We have a system. Hence left-of-equality being zero is legitimate and seems to say "there is no work." When in fact it says there is no net work. Had we used the summation sign, we would be discussing, 0 = ΣW_1-2 which makes matters crystal clear. This last chat (and others like it) are avoided by carry the "Σ," the effect of which is to do the summing in the equation, not elsewhere.

In "taking apart" this equation, nothing earth-shaking was discovered. However it is the simplest of equations. In larger circumstances, when "things don't make sense," one often is obliged to take apart the equation, then take apart the terms one at a time and to test the ideas of the equation location by location over the system boundary. What we have accomplished is the start of a good habit.

3.08 Energy Equation: BODY

3.07 Energy Equation: BODY