Common exercises of science and physics texts describe some event as being accomplished by a person or an animal. Typically the event involves some "physical effort;" the pushing, pulling, lifting (or holding in a lifted position) of some mass. The entities involved in such events are the masses pushed, pulled, etc and the living by the living entity. Newton's Laws of Physics apply subject to choice of system. Great changes occur when a system selection includes a BEING...
Basic Thermodynamics ~ J. Pohl © | www.THERMOspokenhere.com (20-A219) |
Common exercises of science and physics texts describe some event as being accomplished by a person or an animal. Typically the event of "physical effort;" is some pushing, pulling, or lifting (or holding in a lifted position) of a mass. The entities of such events are those masses pushed, pulled, etc and the living entity that did it.
Newton's Laws of Physics apply (make sense) subject to the choice of system. Great changes of interpretation occur for system selections that include a BEING...
What do Beings Do? When a system description includes a BEING, it is usually that the BEING did (or does) something. That "something" is asked for as the "work."
BEINGS apply force:
Newton's Third Law is different in that it does not apply to a BODY. Rather it applies as a condition or rule regarding the inter-dependence of forces within systems of two or more BODIES. Thus, in essence, Newton's Laws of Motion are one Law, his Second Law of Motion (2'nd Law).
In presenting Newton's 2'nd law to students, HS physics avoids Newton's mathematics by using an algebraic equation that states force as a function of mass and its acceleration. Texts use equational notations such as:
![]() | (1) HS text forms of Newton's Second Law of Motion. |
One might wonder why physics education uses inconsistent notations. Thinking further, "Is there a superior notational form for understanding and application of Newton's 2'nd Law?" The answer is: Yes but none of the above. Below is a rationalization of Newton's 2'nd Law of Motion as Newton might have used it himself.
Among the first words of his "Laws of Motion" publication (1687) Newton made statements as a basis of the Laws. These were two axioms, two obvious truths to be accepted with no need of proof. The first axiom defined "quantity of matter" to be a measurable property of matter. Today we call "quantity of matter" mass with the symbol (mBODY). Obviously mass is relevant to its motion but it is also constant with motion while motion is not constant.
Newton's second axiom identified "quantity of motion," as being the product of the system mass times its vector characteristic, velocity. Today we call that product, momentum (mBODYVBODY). It is clear that Newton considered momentum to be the principal property of motion of a BODY. In most physics classroom texts, momentum is but slightly mentioned, being replaced by the product, mass times acceleration. The mantra taught is "force equals mass times acceleration." We now return to equations (1) to rearrange them as might Newton were he awakened from his grave.
A small thing first: most physics-text-forms of Newton's Second Law have F, f, or f-net written left-of-equality. In the paragraph just beneath the equation is written the caution that the mantra "force equals mass times acceleration," is not true because the notation characters, F, f and f-net do not represent a force. Rather, those terms represent "sum-of-forces" that act on the BODY.
Very few applications of Newton's Laws of Motion involve but one force; to use the discrete, vector-sum, notation is wise. The equation should make this perfectly clear, inately - not as a footnote. Let's use the common mathematical notation that designates an equation term as being the "discrete sum of its occurrences." That notation is to prefix the term with the Greek letter sigma, Σ. When the force idea is prefixed (in the equation) as "ΣF," the meaning is clear: "this term is the sum of all relevant forces - sum them."
From Equations (1), we obtain Equation (2) (below left):
![]() | (2) The commonly used, F, is in fact a ΣF. |
![]() |
(3)It is motion of a “BODY” we observe. Vectors (position, velocity, force) are written in the “0XYZ” vector space. |
Equation (2) is left of a better equational vantage: Equation (3). Forces and acceleration are not algebraic entities; they are vectors. The distinction, what is a scalar versus what is a vector is important. Vector entities (rank 1 tensors) are written in bold case (or with an over-bar or over-arrow). To specify a vector requires prior knowledge of the space of the system/event. A vector space, origin, coordinate axes, and a unit vector basis, must be defined. Students are familiar with Cartesian coordinates (0XYZ). Around 1850, Sir William Hamilton invented the unit vector triple, I, J and K. Since equations contain thought, equation with vector terms should identify their space. (To identify space is a necessary skill for those who program a video-games or robot events).
Finally, about Equation (3), the system of Newton's Laws was a collection of matter he modeled as a BODY (the simplest approximation of matter). Specifically the mass and acceleration of equation (3) are those of the BODY. We place that subscript behind those terms then move them left-of-equality for reasons explained below.
![]() | (4)System (with its system states) is left-of-equality. Forces might change those states: right-of-equality. |
Newton invented calculus not for fun but to define velocity and to define acceleration, the A of f = ma.
![]() | (5)
The “left” term is identically the same as the “right” term. The equation simply identifies a “shorter” way of writing things" |
Acceleration is a characteristic of the motion of something (we call it a BODY) in space (we use the Cartesian space, 0XYZ). Acceleration equals the derivative of the velocity of the something in the space (written above left).
Below Left: mass of a BODY is multiplied by the acceleration of that BODY. Acceleration is the derivative of velocity. Therefore (as shown left) mA = mdV/dt. Below Right: the mass term (a constant) is brought inside of the derivative ("d/dt") showing that mA = d(mV)/dt which is our preferred form of mA.
![]() | (6)6 |
Our next step is simply to substitute the extreme right side of (6) into the left side of (4).
![]() | (7)7 |
The above form Newton's Second Law of Motion contains momentum, his axiomatic "quantity of motion," explicitly. Forces are physical "constructs" or reasons for change of momentum. Notice that when ΣF = 0, Equation (7) becomes Newton's First Law of Motion.
Newton's perspective, the "subjects of his Laws," is made clear upon understanding the axioms he set down as basis. The first axiom established the existence of a "quantity of matter." No proof is required in an axiomatic method. Mass (a measurable scalar property of matter) is possessed by a body and is of importance in its motion. A second axiom Newton called "quantity of motion." Today we call that quantity of Newton's, momentum. Momentum is less easy to quantify that is mass. Momentum is the produce to mass times its velocity define than mass. Newton was obliged to use vectors to quantify position (relative position, he realized) as a prerequisite idea. Space needed to be quantified - a vector space. Change of position in time begat velocity. which required vector calculus. Velocity is the derivative ov the vector, position. This momentum is the principle and second idea of his laws of motion.
The form of Newton's Laws of Motion selected for this writing is arranged in accord with Newton's axiomatic approach. Calculus is made explicit; the derivative of body momentum is placed prominently, left of equality. Subscripts of the derivative to identify the system (body for now). Superscripts to the derivative to identify the vector space. Finally ΣF replaces F right of equality in s differential equation, where non-homogeneous terms belongs. Vectors... all properly notated. Phew! Also we need the initial conditions written near the differential equation.
Mathematical notation says this better than words.
![]() | (8)8 |
In use, Newton's equation is accompanied by a sketch of the physical situation. The sketch below is over- complete. In any application such completeness is rarely needed. However the sketch puts a picture to all of the ideas brought together.
There is a highly compelling reason students should use the differential equation form of Newton's Laws of Motion. In later engineering study you will find the mass equation, momentum equation and energy equation for a body have the same form. All three physical statements are first order differential equations (with their respective initial conditions). The skills and understanding gained in solving any of them are the same skills needed to solve and understand the others. All three equations take the same perspective - system.
![]() | (9)9 |
By physics, the center equation (which includes the idea "f = ma") would be written "f = ma." The left and right equations are engineering rate form expressions that account for changes of mass and energy of a BODY as system, respectively.
Further Reason: In closing, perhaps Newton did not use the formulation of Equation (7). Nonetheless we should use it. It does everything "f = ma" does and it expresses the mathematical meaning of Newton's Laws better. And since 1687 science and engineering have used the form more and more. For example, the universe is expanding. The mean distance l between conserved cosmological particles is increasing with time. The mathematical statement of the rate of increase of mean distance is written:
![]() | (10)Another first-order differential equation! |
Newton studied Cosmology and he used the rate form.
Further proof of the power of the first order differential equation, read about the mathematics of the Lotka-Volterra Predator-Prey Equations.
![]() | (10)10 |
In this writing, "BODY" (uppercase) is used rather than "body" to emphasize that Newton studied the former which is a very special, isolated, perspective of the latter. Newton used mathematical and physical analysis to study physical reality. He sought to discover the methods and perspectives of study that would reveal the secrets of nature. His approach, which we use today, might well be called Newton's Analytic Method." In any analysis, his very first step was a complete mental and mathematical identification and extraction of the body from its space to become a BODY for analysis.