Science 122 Program 15 Newton and the Laws of Motion

# Newton and The Laws of Motion

## Program 15Lesson 2.7

#### 6. Summary

 Newton at 25 Newton at 65

# Coming Up

In this lesson we will focus on the life and influences of Sir Isaac Newton. We will look at the seventeenth century as a period of blossoming creativity, but examining the contributions of other major scientists of this rich period in history. We will take a quick trip through the life of Isaac Newton through a resume, stopping to look more closely at events which led to the publication of his life's work.

After a short focus on vectors, we will see Newton's laws of motion from a slightly different perspective which is intended to compliment the material in the textbooks and on the video program. Finally we will present in outline form, background information on the premises and assumptions of the Newtonian paradigm and its wide ranging influences.

# 1. Introduction

This is a rather long lesson. This is because the life of a man whose influence was so widespread in both space and time is impossible to chronicle in detail in any time frame. We have presented information in such a way that you should be able to place it in context without being overwhelmed by it. If you begin to feel overwhelmed, go back and look at the lesson objective to keep it in perspective.

The place of Sir Isaac Newton in history is well justified, both for his scientific contributions and also for the effect that his viewpoints have had on subsequent thought.

No one since Aristotle has had such influence on a global world view. Unlike Aristotle, who work was speculative although brilliant, Newton developed precise mathematical tool, specific definitions of mass, space, and time, and employed a variety of inductive and deductive techniques to answer the question of the ages regarding the motion of the planets. In doing so he removed the last trace of Aristotle's authority on matters of motion and cosmology, set a new way of thinking about the universe, established the Copernican system with planets moving according to Kepler's laws, described using Galileo's kinematics.

Newton's synthesis of gravity ranks as one of the greatest, if not the greatest achievements of the human mind in its conceptual simplicity, its logical and mathematical elegance, its potential to describe complexity in simple terms, and the paradigm shift which it culminated.

### 1.1. One of History's Greatest Thinkers

Newton's writings on gravity and motion show an extraordinary intelligence and talent for logical discourse and for rationalizing intuition about the physical world. Newton's laws, simply and concisely stated in three sentences, are the basis for all of physics. These, along with an equally concise law of gravitation, form the most complete formal system for describing forces and motion ever stated. Everything from the stresses on a bridge to the orbital characteristics of satellites stem from Newton's laws. We are able to calculate the masses of the sun, planets, and even distant stars using Newton's law of gravitation.

The mind of such a thinker is hard to envision. So great were Newton's powers of concentration and problem solving that he was revered almost as a Demigod in his native England and considered somewhat superhuman worldwide. His name brought images of intellect the way Einstein's name does to us today.

The similarity of the two great physicist's discoveries and their impact on their times is interesting. We encourage you to look in the library for biographies of the two men. Even if you don't read the books, look at the preface and introduction to see what their biographers have to say about them. Either one of these men would make a great topic for the final paper for the course. Consult with your instructor for assistance in choosing or limiting your topic.

### 1.2. Created science of mechanics

Mechanics is the science which deals with the actions of forces on bodies and with motion. Subcategories of mechanics are kinetics, kinematics, and statics.

You may wish to spend ten minutes looking up these words in the dictionary. It will help you to understand what mechanics is.

Newton's description of the relationship between forces, motion, and equilibrium, as stated in the three laws, was found to apply not only to motion in the earthly realm such as that described by Galileo for projectiles. The planets also moved according to those principles and relationships.

### 1.3. Explained the motion of planets

The precision and elegance of Newton's explanation of the principles of planetary motion ended forever the problem of prediction of the location of a particular planet. The motions could be calculated as accurately as desired for anytime in the past, present or future after only three observations.

### 1.4. Set stage for future analysis

Newton's methods of analysis became the model from which new methods were derived. He formalized the physics of Galileo by providing a set of rules and a system of operation which could be applied to all physical phenomena, not only mechanics, but also to the study of electricity, magnetism, and light.

The chemical behavior of matter also came under closer scrutiny once it was realized that all matter, large or small, near or far is subject to the same laws.

### 1.5. crystallized scientific revolution

To compare the revolution to the freezing of water may seem far fetched, but by this time you should be accustomed to these abstractions. The analogy this brings to mind is this. Imagine a liquid such as water which is put into a freezer. Eventually the water will freeze, but what if it is continually stirred. Have you ever seen an Icee machine. Inside is a slush composed of crystals and liquid which is kept well below freezing temperature but prevented from freezing by continual stirring. Now imagine that the stirring is stopped. This is also the method by which we make ice cream, gelato, or sorbet.

What will happen? The mixture will freeze, or crystallize, almost immediately.

Now do you see the metaphor? If not, we hope you will before you finish this lesson.

From Copernicus to Newton is a little over one century, a short time for such a radical shift in world view of an entire culture. Newton's work provided such a well documented and convincing proof that opposition to heliocentrism, which was admittedly weakened from the heyday of the Scholastics, amongst intellectuals and laypersons alike virtually vanished overnight with the publication of Newton's book, The Mathematical Principles of Natural Philosophy, published in 1686.

# 2. The Seventeenth Century

In an Lesson 2.2 (program 10) we described the sixteenth century and the changes which were taking place in thought patterns and new discoveries. You may wish to review this to put yourself in perspective for the seventeenth century, one of the most creative and prolific periods in all of history, and the beginning of the modern era of the dominance of science and technology.

### 2.1. Creative and Prolific Period

Galileo and Newton were not the only natural philosophers of the sixteen hundreds. Changes of the magnitude of the paradigm shift which accompanied the scientific revolution do not take place because of the ideas of an individual, no matter how influential.

An individual is born of his or her culture and society. The beliefs and practices of a culture and its associated society represent in some way the collective influences of the individuals.

In the seventeenth century the authority of the Church was eroding, largely through the growth of Protestantism all over Europe. The Protestant movement was weakest in Italy, traditionally the seat of power of the Church. In northern Europe and the British Isles the movement was strong, and in England was in its second century.

Even in Italy Galileo was a criminal. He was tried and convicted by the Church more for his politics than for his science. It was his refusal to acknowledge the authority of the Church that got him in trouble more than his ideas on motion, or even his defense of heliocentrism.

Galileo's Starry Messenger , Two Chief World Systems, and Two New Sciences were read all over Europe in universities and other such subversive settings. These books detailed Galileo's experiments and conclusions regarding astronomy kinematics, and mechanics.

#### 2.1.1. Galileo's writings on astronomy, kinematics, and mechanics

Galileo had attacked the geocentric system of the Scholastics indirectly through his studies of motion. He used the results of his motion studies to discredit Aristotle, then used that to discredit Aristotle's cosmology. This was a clever tactic, and well executed, but Galileo presented no replacement model, no explanation for Kepler's laws, of which he was aware but ignored in his writings, and no connection between his descriptions of motion and the celestial motions, other than if Aristotle was wrong about sublunar motion then he was probably wrong about celestial motion.

Galileo's work also considered the concepts of time, air pressure and temperature and led to the development of the pendulum clock, the barometer, and the thermometer, of which he designed and constructed the first working model.

#### 2.1.2. Toricelli, Pascal, Boyle on pneumatics

Galileo also wrote about air pressure which stimulated the development of the barometer by Toricelli and later work by Pascal on the nature of changes in air pressure with altitude.

In England was Robert Boyle, who instigated a radical paradigm shift in chemistry, and whom we will meet again further downstream. Boyle's law describes the relationship between the pressure of a gas and it's volume and will become important when we study the kinetic theory of gases in section four of the course.

#### 2.1.3. Descartes on analytic geometry, optics, philosophy

We have already noted Descartes' contributions to deductive logic, analytic geometry and dualism in lessons 2.4 and 2.5 (programs twelve and thirteen.) He also contributed to the growth of the field of optics, which was of growing interest in the middle to late sixteen hundreds.

#### 2.1.4. Huygens on astronomy, light, centripetal force, pendulum clock

Christian Huygens (1629 - 95) was another of the geniuses of the century. Son of a multilingual Dutch poet, he published a wave theory of light which was at odds with Newton's corpuscular (particle) theory. The wave theory of Huygens was shown to the a correct one in the early nineteenth century by Thomas Young while Newton's particle theory was shown to also be correct in the early nineteenth century by Albert Einstein. The resolution of this Certs/Miller Lite controversy caused a disruption of physics for a third of a century before quantum theory emerged. We will explore the waters of that river near the end of our journey.

Among other ideas contributed by Huygens were the concept of centrifugal force, although Newton wasn't aware of if and came to the same conclusions independently, he suggested that the freezing and boiling point of water be used as standards for construction of a thermometer, an idea that was used by Fahrenheit and Celsius to make accurate thermometers, and the pendulum clock, the accuracy of which motivated the voyages of Captain Cook a century later to test Newton's astronomy in the South Pacific. He also made improvements in the telescope lens, and built a telescope which which he discovered the rings of Saturn and one of its moons.

#### 2.1.5. Hooke on mechanics and elasticity

In England, Robert Hooke (1635 - 1703), charter member of the Royal Society of London, theorized that the motion of the planets was a mechanical problem. he also invented the spiral watch spring, constructed the first machine to do arithmetic and described cells in plant tissues. He is most famous for his ridicule of Newton's theories of color which opposed his own (Newton's were correct) and for Hooke's law which states the proportionality between the force and stretch of a spring.

Both of these play direct roles in Newton's theories and we shall encounter the name of Hooke again in this lesson and in the future.

#### 2.1.6. Roemer measured the speed of light

Olaus Roemer (1644 - 1710) was a Danish astronomer who measured the speed of light for the first time, thereby establishing that it did not travel instantaneously from place to place. This idea was suggested by Galileo, but because of the great speed of light he was unable to prove it.

### 2.2. Why sudden blossoming?

In this study guide we can only highlight the principal reasons. We will no doubt leave out something that someone somewhere thinks should be included. In the interests of brevity we offer our defense.

Hopefully the reader will be able to see the complexity and connections which were emerging as the printed word spread ideas far beyond their source as improvements in the printing press and bookbinding made them accessible to a higher percentage of the population than ever before, and in more languages. It was not uncommon in Europe for the well educated citizen to be fluent in four, five, or six languages.

#### 2.2.1. requires understanding of whole social, political, economic, religious changes

It is not really surprising that creativity, curiosity and scientific inquiry blossomed during this period when the whole of the social, political, economic, and religious climate is taken into account. Any attempt to put this in proper perspective is beyond the scope of this course. We encourage you to study the history of the period.

#### 2.2.2. pyramiding effect of centuries of work

This burst is the pyramiding effect of the accumulation of knowledge, like a river downstream of where several large rivers join. This effect continues today at accelerating pace, leading to terms such as future shock. The seventeenth century is the point where things began to fall into place, like in a football game when the team finally begins to play as a team after struggling in the early season.

#### 2.2.3. craftsmen as well as men of wealth were turning to science

Another important reason was that the increasing commerce between nations and the distribution of the wealth of the new world attracted men of ability, interest, and means. Craftsmen were interested in the improvement of methods and products as the negative value placed on manual labor by the ancient Greeks began to give way to a pride of accomplishment for all.

Men of wealth were attracted to science as a trendy and somewhat exciting hobby of diversion from the aristocratic excesses and bureaucratic minutiae. Not only that, but there was money to be made in inventions and new products.

#### 2.2.4. good mathematical and experimental tools

In England especially, but to a lesser degree everywhere in Europe, the skills of the artisans and craftsmen combined with the money of the wealthy was producing precision instruments of all kinds which could be used to make increasingly accurate measurements. Most notable were improvements in telescopes and clocks, but also in barometers and thermometers. The vacuum pump was invented in the later part of the century and provided new opportunities to study air pressure and motion in the vacuum.

#### 2.2.5. Galileo

Galileo's methods of well formulated problems and a new way of viewing facts and experiments had far reaching influence. His work directed attention to mathematics as a fruitful and convenient language of physical science as his bullheaded arrogance fostered scorn of sterile introspection and blind subservience to dogma.

#### 2.2.6. Better Communications

Earlier we hinted at the role of communications. From the beginning of the course we have alluded to physical science as shared reality. To share reality requires communications, as we noted in lesson five (program 5) when we discussed the role of language in the beginnings of science. The growth of knowledge depends upon the exchange and criticism of ideas and methods.

##### 2.2.6.1. Printing presses now widespread, but still relatively young art

Although the printing press was invented in China, it was the use of movable type which allowed quick typesetting which spread of the printed word. The first Gutenberg bible was published near the middle of the fifteenth century. The printing press was widespread, but printing was a relatively young art which was growing rapidly.

##### 2.2.6.2. Scientific societies

A major innovation in the spread of communications between scientists was the formation of scientific societies. The first of these was the Royal Society of London for Improving Natural Knowledge which was begun by Hooke and Boyle along with architect Christopher Wren and astronomer Edmund Halley. Today it is known as the Royal Society, a government subsidized organization which funds and stimulates research and advises the British government on science and technology.

The strength of this and others like it which arose late in the century was in its regular meetings during which anyone could present the results of their experiments or speculations. These meeting defined the paradigm for methods and ideas. They were, and still are, full of cooperation, presentations, debates, and quarrels. They promoted the healthy skepticism and self correction that we spoke of in lesson 2.

Another strength of the societies is that of sheer numbers. Finding money to support scientific inquiry is easier the more specific the idea, the better the problem is presented, and the more intellectual support it has. The same is true for attacks by antagonists, the common front being harder to penetrate than an individual idea.

The societies also initiated the idea of publishing the results of their proceedings in printed form. These publications contained the text of presentations and the debates which followed. Combined with the rapidity of publishing afforded by the movable type of printing presses, these journal spread the knowledge all over Europe almost as soon as they were heard on the floor of the society's chambers.

It is interesting to note that the internet was formed for the same purpose in the nineteen eighties for the same reason, except that in our case, the publication of printed text is too slow to keep up with the pace of change. By the time a scientific journal is published and distributed these days, a good percentage of it will be out of date. Some folks think there is just too much stuff to read that it is impossible to read it all and equally impossible to know which of it to read, just like they said about the printing press.

### 2.3. New Questions

By the sixth decade of the century, Aristotle had fallen out of favor as the authority on motion, both earthly and celestial, and his theories on matter were soon to suffer the same fate. At the same time, the circular perfection of Plato as regards the heavens had been shattered by Kepler's laws. So what remained of the old theories that was worth keeping?

Not much.

Plato's question no longer held much interest. Instead the questions of the times were directed toward the forces which act on the planets to account for their observed paths and how to explain the observed effects of gravity here on earth.

What was missing was the connection between the two. On one hand was earthly motion, uniformly accelerated as Galileo had described it. On the other was the elliptical motions of the planets, as described by Kepler, and driven by some mysterious force, perhaps emanating from the sun acting like one of Gilbert's magnets.

This is where Newton comes in, to make the connection in the greatest synthesis of all times.

# 3. Curriculum Vitae

As we did with Galileo, we present the chronology of Newton's life in annotated outline form. We will give some attention to the significant events The video program will elaborate some of the details. As before, it is not our intention that you should "know" these things, or that you should know any particular detail.

As you watch the video, and as you look over the resume, try to build a picture of Newton the man as well as Newton the scientist. He is one of the most interesting, intelligent, and influential men in all of history, scientific or otherwise. It is interesting, but also important in understanding the nature of science to see how the man who has had such a great impact on our lives was influenced and shaped by his environment.

### 3.1. Personality Traits

Newton was a loner, whose self esteem, even in adulthood was very low. He preferred to keep his work secret, or within a circle of a few not particularly close friends. His work was very personal, and someone said of him that he seemed depressed as indulging in a private vice, the way one might if they were obsessed with the dark arts, or some other socially unacceptable behavior.

His powers of concentration were legendary, even in his own lifetime. He was known on several occasions to disappear into his room where he would remain for days at a time, refusing food and allowing no entry while he worked on a problem. He is said to have had the ability to focus on one problem and think of virtually nothing else until the problem was solved. Most of us are not so fortunate as to isolate ourselves from distraction in that manner.

### 3.7. 1672 Rejection by Royal Society

After five years, Newton had organized his thoughts, formed his conclusions, and built up the courage to present his theory of light and colors to the Royal Society. He did so. Unfortunately, the theory was contrary to the ideas which were popular at the time. Leading the opposition in the debate against Newton's theory was Hooke, whose ideas were largely responsible for the popular view, and whose prestige carried much weight in influencing the Society.

Newton was so shaken by the criticism and rejection of his ideas that he vowed to do no more science and to make no more presentations for public debate. He just didn't have the personality to take the sometimes harsh criticism of his skeptical and sometimes belligerent peers.

Although he continued to attend the Society's meetings, he quietly continued his studies of alchemy, theology, prophecy, mathematics, celestial mechanics, while he taught the latter from his position at Cambridge.

Newton thought Hooke's assault on him had been a little too personal and it deeply affected him. The two of them never had much of a relationship after that, although it was this event which would later lead, albeit indirectly, to the publication of the Principia, as his major work is usually called.

### 3.8. 1675: Hooke's "inverse square assertion"

Three years later, Hooke asserts that he knows that an inverse square force of attraction would yield motion which was consistent with Kepler's laws, but he couldn't prove it.

Newton goes home, locks himself in the room and wrestles with it, as much to spite Hooke as for any other reason, and proves it using geometry. We will take a closer look at his methods in the next lesson.

Considering his previous encounter with Hooke, Newton tells no one what he has done.

### 3.9. 1684: Halley's dispute with Wren, Hooke

Nine years later Edmund Halley, after whom the comet is named for reasons we will see in the next lesson, found himself in a debate with Christopher Wren and Hooke. The dispute centered around this inverse square assertion of Hooke nearly a decade before, concerning the nature of the force necessary to make elliptical orbits. Recall that it was Hooks who began the discussion on this topic in the first place twenty years earlier. Since then it had become one of the most debated and intriguing questions of the time.

Halley, who was one of a handful of Newton's friends, sought Newton's advice, for ideas which he could use to counter Hooke's arguments in their debates. Newton, remembering that he had done it nearly a decade before, told Halley so. Unfortunately, Newton said, he could not remember the details, but yes inverse square force emanating from the sun would indeed produce Kepler's orbits.

At Halley's insistence, Newton went home, worked it all out and prepared a brief summary in a report to Halley. Halley recognized the significance of Newton's proof. After much persuasion from Halley, the promise of financial support for writing and publication of the results, Newton returned to Woolsthorpe once again. There he worked feverishly in a burst of creativity seldom if ever seen in any age. In retrospect this is not surprising when one considers that all of that creative energy had not been channeled for a dozen years since the episode at the Royal Society where Newton's self esteem had been lowered even further by Hooke and his supporters.

Eighteen months later, he returned with hundreds of pages of writings. These included logical discourse, formal definitions, geometric proofs. mathematical derivations unlike anything anyone had ever seen before. The concise definitions and statements of axioms and subsequent theorems, corollaries and discussions represent what most scholars consider to be the most extensive and self consistent science book ever written, surpassing Aristotle, Ptolemy, Aquinas, and Galileo.

The Book was Principia Mathematica Philolophis Naturalis, "The Mathematical Principles of Natural Philosophy."

### 3.10. 1687: Principia Mathematica Philolophis Naturalis

The frontispiece of the first edition of Principia Mathematica Philolophis Naturalis.

The book is a masterpiece of organization and clear thought. It was written in the scholarly Latin of the times. Its style was difficult at best, even for the well educated scientist who was fluent in Latin. It is said that Newton wrote it that way deliberately because he did not want criticism from those who were not able to grasp it in it's entirety. There were a few who did grasp it. The book was a sellout upon publication. Second and third editions appeared soon thereafter.

This is in sharp contrast to Galileo, who fifty years earlier had published his book in the vernacular Italian instead of scholarly Latin because he wanted it to be accessible to the masses, who were more likely to be literate in their native Italian than in the Latin.

The Principia contains three books. Newton describes them in the preface:

" . . . I offer this work as the mathematical principles of philosophy, for the whole burden of philosophy seems to consist in this --from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena; and to this end the general propositions in the first and second Books are directed. In the third book I give an example of this in the explication of the System of the World (and) I derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, I deduce the motions of the planets, the comets, the moon, and the sea."

It is difficult for us to sum it up any better than that.

### 3.11. 1690: Nervous Breakdown

The pressure of success may have been too much for Isaac's frail personality. The term nervous breakdown is very general and doesn't really tell us what happened. He sort of withdrew and didn't function well as he became belligerent,and unfriendly, lost his ability to concentrate, that sort of thing. He underwent a long recovery, but never fully recovered and did little scientific work, none of any note. He turned his attention to theology and mysticism.

The rest of his life, although revered as a legend, and even as a Demigod by some, he did nothing of note, at least nothing which could compare with the grand syntheses represented by Principia.

### 3.13. Member of Parliament (1689 - 1701)

The only statement of record in the annals of Parliament was a request to close a window to prevent a draft. What a moment that must have been as the great and esteemed Isaac Newton rose to speak and all heads turned, ears straining to hear the words of wisdom!

"Please shut the window, there is a draft"

# 4. Newton's Laws

Being the basis of all of physics, Newton's laws are well documented in many sources, including the texts for this course. We will not attempt to duplicate that coverage here. In the library you will find numerous books which describe the laws and their use at levels from elementary school through the level of multi-years of calculus. We recommend that you consult outside sources if you need more explanation than is provided in the video program and the texts. The video program will elaborate on the laws, using demonstrations and graphics in an attempt to look at them, their meanings, their roots, and their implications. Here we will present the outline of the material only. It is designed to focus on those aspects of the laws which are particularly relevant to our purposes. It will help you to study if you try to relate the outline to the textbooks. This is also a good time to communicate with your classmates, electronically or otherwise, to discuss the laws and their utility.

### 4.1. Laws of Nature

The laws, as we often refer to them, as if they were capitalized, are not only the basis of all of physics, and for that matter, all of physical science. They are also perhaps the best example of what we like to call "rationalized intuition." One of the reasons for the success of Newton's physics is that the laws are practically self evident once you have seen them. Once you allow your mind to understand them, they become immediately obvious and not contrary to common sense at all. Part of Newton's genius was his ability to state these simple but abstract principles in such a way that someone three hundred years later can read and say "Yeah, I knew that."

It may not be quite so easy for all of us because of our personal paradigms. Most of us have not spent much time thinking about the nature of forces and motion. If you have not, you will have to work a little harder, or at least start paying attention to the forces you must exert on objects such as doors and grocery carts. If you start to look at your interactions with the world through Newtonian eyes, you will soon begin to see how obvious the laws really are.

That our species has the ability to rationalize our intuitions and to understand such abstract things as force and motion is awesome. That it took so long for one such as Newton to arrive with the right talents and motivations into a time period when the major philosophical questions about the physical universe were being rephrased is not surprising.

It seems that the best ideas are the simple and elegant ones that we look back on and say, "Yeah, that seems obvious, why didn't I think of that." We will see many more of these . It is the nature of our brain and its ability to learn when directed by a point or view that we have called a paradigm.

As it turns out the laws, although simple and intuitive, are among the most difficult to "prove". Like Galileo, any attempt to test the laws must deal with friction and its effects on motion. However, the understanding of frictional forces which came as a result of applying Newton's mechanics to them, allows us to deal with the frictional forces quite effectively, as wee will see in Section Three.

### 4.2. Focus on Mathematics: Forces and Vectors

One of the most useful tools that Newton invented in the concept of vectors. Once the concept was conceived, it became useful in applications far beyond its original intention. Is is an essential feature of all physical analysis. The concept, although formalized by Newton, is actually little more than Galileo's separation of projectile motion into a vertical freefall component and an horizontal inertial component. You may want to review Galileo's synthesis and analysis of the motion of projectiles in Lesson 2.6 (Program 14).

In the Principia Newtons states as as the first corollary to the laws, for which he offers both an elegant Euclidian argument and also sets the stage for the type of geometric analysis which is to follow.

 A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately.

His argument, although Euclidian, is not comprehensible to the typical freshman physical science student. If you wish to look at it, there are many copies of translations of the Principia in libraries around the world.

We wish to offer a less Euclidian example, but one which is probably easier to understand. We call it the bug on the board.

Imagine an ant crawling on a flat board. The ant is walking directly across the board, perpendicular to its edges as shown.

The time it takes for the ant to walk across the board depends only on the speed of the ant and the width of the board, because velocity equals distance divided by time, v = d/t. Now suppose the board is being carried, flat side up so that the board and the ant together are moving to the left at a constant speed, like this:

Notice how the arrows are used to indicate the direction of the motion, but also the speed. The longer the arrow, the greater the speed.

First of all, can't you see that moving the board to the left in no way influences the length of time required for the ant to cross the board. Likewise, the ant will travel a certain distance to the left, along with the board, whether or not it is crawling across the board.

The combined motions of the ant can be viewed as the diagonal of a parallogram, in this case a rectangle, much the way Galileo did for projectile motion. The difference is that the motion is at a constant speed in both directions, and so the actual path of the ant is a straight line rather than a curved trajectory.

Now read Newton's first corollary again.

 A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately.

Restate it thus: An ant, moving in two directions simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those motions separately.

Can you think of other examples, other than the projectile, where this corollary might apply?

See how Newton has used a very simple method to produce a complex motion out of two individual motions. The usefulness of this will become apparent soon if it is not already.

What is important to note, is that it does not matter whether the two motions are combined or undertaken separately, the result is the same, meaning that the bug winds up at the same place at the same time regardless.

This means that either situation is equivalent, so we can choose whichever one is most useful for our purposes. In the same way that we can measure in inches or centimeters, or use foreign currency to make purchases, the equivalence is what is important.

We can write this as a form of addition but we must be careful to note that the symbols mean that we are combining operations which take into account both the length and direction of the arrows. If we label the rectangle like this

then we can write the combined operation like this:

Ant + Board = Result, or we could abbreviate and write A + B = R. Remember this is just a way to describe the operation that really represents the motion of the ant combined with the motion of the board. The plus sign is used a little differently than in ordinary arithmetic. Here it is the combined effect of two operations.

Of course we could calculate the precise length of the result (and even it's angle, if we want) since we know the Pythagorean theorem. If |A|, |B|, and |R| represent the lengths of the arrows A, B, and R, then the length of the three arrows is related by

### It's a Pythagorean triangle!

OK, don't panic, this is just to show you how the idea of vectors, new with Newton, forms a link between the physical movement of the ant and board and the geometry of the ancients.

This is what we mean by a new mathematical technique. We will see how this applies as time goes by.

For a more detailed account of the use of vectors, look in the texts.

#### 4.2.1. Amount and direction of force are both important

There are two reasons why this is a useful technique. First, it allows us to find the result of many forces or motions and express it as a single force or motion, thus simplifying the situation. We did this to understand the motion of the bug.

Second, it allows to take an existing force or motion and break it into components in two or more directions, thus simplifying the situation. Galileo broke the motion of the projectiles into two components, freefall and inertial, and studied the motion independently.

#### 4.2.2. Geometry of Forces

When there is more than one force it may be expedient to sum them all up and represent all the forces as one net force, or the combined result. There are techniques to do this that need not concern us here, except to note that Newton invented them.

When the forces all act on one line, we say they are colinear whether they are in the same direction, or in opposite directions.

Colinear forces pulling on a ring.

In this case it is easy to find the result. How would you do that?

If the forces are not along the same line, then they are called planar forces if they are all directed along the same flat surface.

Planar forces pulling on a ring.

Here it is not so easy, but not very difficult either to find the result. You do not have to do this. You just need to recognize that it can be done. In any of these cases involving forces we call the result, or resultant vector the net force.

### 4.3. Three Laws of Motion (Mechanics)

#### 4.3.1. First Law: Inertia

This is basically a restatement of Galileo's principle of inertia. It is really a special case of the second law, stated redundantly by Newton to emphasize that it is contrary to Aristotle's view that a force is needed to sustain motion. Newton said this:

### "Every body continues in its state or rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it."

Three different ways to say it are shown below

##### 4.3.1.1. a = 0 <=> (The symbol (Greek letter "sigma") means "summation". Here it means the net force (the vector sum of forces).

This can be read in several ways. On the video program we will discuss the meaning of the double arrow. One way to read this is to say, "if an objects is not accelerating then there is no net force acting on it, and vice versa." Another is "acceleration is zero if and only if the net force on it is zero.

#### 4.3.2. Second Law: Force/Motion

The second law defines in a quantitative way what happens when the net force is not zero. In that case, acceleration occurs in direct proportion to the force. The affect on a larger mass would be inversely proportional to the mass. What do you think we mean by inversely proportional?

One aspect of the second law is that since velocity is a vector quantity (both speed and direction are important) then a change of direction constitutes a change in velocity, even if the speed remains constant. So this relationship also applies to circular motion at a constant speed, which is a condition where the speed remains constant while the direction of motion is constantly changing. This law adds a cause of motion to Galileo's kinematics, but it also defines mass as the measure of an objects inertia.

We can now see inertia as the resistance to a change in motion. The larger the mass, the more inertia an object has. Force is that which has the ability to change motion. Newton stated the law this way.

### "The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed."

He used the phrase 'change of motion' in place of 'acceleration and mass', but that distinction shouldn't bother us now. We will return to it in lesson 3.3.

##### 4.3.2.1. .
###### 4.3.2.1.1. acceleration depends on motive force and resistance
• mass doubled with constant force: acceleration reduced by one half
• force doubled with constant mass: acceleration doubled
###### 4.3.2.1.2. defines both force and mass in terms of acceleration
• mass is a measure of resistance to change in motion
• measures inertia
• a force of one Newton will accelerate a mass of one kilogram at a rate of one meter per second per second
• 1 Newton = 1 kg x 1 m/s2
• 1 N = 1kg*m/s2
• net force is a cause of change in motion
###### 4.3.2.1.3. includes circular motion
• force is required to maintain circular motion even at constant speed
• a change of direction is also a change in velocity

#### 4.3.3. Third Law: Action/Reaction

The third law required the most insight, is the most difficult to grasp and has the most implications. Some thought can convince us that if the other two laws are true, then this one must also be. Newton stated it this way:

### "To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts."

In modern language we commonly hear this stated thus: "For every action there is an equal but opposite reaction." This is only partially correct, and misses that point that every action involves at least two objects, and they exert equal and opposite forces on each other.

Here's an example. Imagine an empty Newtonian universe, like ours, where the laws apply. Suppose that you live outside this universe and can manipulate things inside it, kind of like cyberspace. Into this universe you throw a rubber ball. What will happen to it?

Nothing, right. If it is the only object in the universe then it must continue to travel in a straight line at a constant speed. Of course it has to. What would stop it if it was the only object in the universe. According to the first law a net force is required to change its motion, but there is nothing to exert that force because the ball is the only thing in the universe.

If you want to change its motion you will have to put something else in the universe for it to interact with. Let's do that.

Let's put another ball, equal in mass and moving at exactly the same speed, but in the opposite direction, and on a collision course with the first ball. The second ball, like the first will continue in a "right" (straight) line until something alters it, and the only other thing is the other ball.

Here's the picture.

What will happen when they collide? It may be difficult to predict with certainty, but suppose we say that if the balls are exactly the same mass, if their speeds are exactly equal, and if they strike each other head on, then each ball will bounce off the other and wind up headed in exactly the opposite directions at the same speeds (assuming the balls are made out of a perfectly elastic material, but its our universe and we can make it so). We can actually demonstrate this, but it makes sense that the balls would behave this way.

After they collide the situation is this:

Now we rely on the second law to help us to understand what has happened. Each ball has changed it's motion which required that a force was exerted on it. This could only come from the other ball since the two balls are the only things in the universe. So the two balls exert forces on each other. The size of the force has to be the same, doesn't it. If I clap my two hands, which one gets hit the hardest? For that matter what is the sound made by clapping only one hand.

So the idea here is that whenever a force is exerted, there must be something exerting the force and something on which the force is exerted. The word 'exert' is somewhat misleading since it implies that forces must be actively exerted. In reality, inanimate object, even if stationary, can exert forces. For example, if one of the balls was replaced with an immovable wall, it would not alter the analysis of the forces. The difference would be that the force exerted by the ball on the wall would not be great enough to cause the wall to move because of it's material strength.

##### 4.3.3.1. forces occur only in pairs

WHAT WILL HAPPEN IF EQUAL AND OPPOSITE FORCES ARE NOT EXERTED?

##### 4.3.3.3. centrifugal vs. centripetal force

There is much unnecessary confusion surrounding the concepts of centrifugal and centripetal forces. The confusion arises when we are following a curved path and our senses become confused about the direction of the forces. A similar confusion results when we feel as if we are being pushed backwards into the car seat when it is pushing us forward as the car accelerates.

Centrifugal and centripetal forces are easy to understand when we view them from outside, from our stationary reference frame. The two forces are an action-reaction pair. They act with equal strength, in opposite directions, and on two different objects.

Centrifugal means directed away from the center, centripetal means directed towards the center. In the pictures below the two objects are the bucket and the string, or the string and the hand. In fact there are many different action-reaction pairs involved in the action of swinging the bucket.

We can sum this up simply by noting that the centripetal force is the force exerted on the bucket by the string. It is the force that keeps the bucket from flying off and continuing in its inertial straight line motion as it becomes a projectile.

The centrifugal force is the force exerted on the string by the bucket. It is the pair of forces that keeps the string tight.

What keeps the water from spilling out of the bucket? What causes the water to be flung out of the bucket in the spin cycle of the washing machine?

# 5. Newtonian Paradigm

The system of the world described by Newton in Principia was so well thought out, and so well presented that it established a new paradigm virtually overnight. Truly enough, the times were prepared for it and this is what we meant before when we said the Newton "crystallized" the scientific revolution; the ideas in the Principia were not his alone because he shared the heritage and relied heavily on methods and ideas from the past and from his own times. One thing that it did was to put these ideas together in one place and find the common ground among them. Kepler's laws here, Galileo's laws of freefall and Inertia there, Euclidian geometry, Aristotelian and Platonic logic, Descartes' analytic geometry, algebraic symbolism from the middle East and other things. Almost everything we do in science today is there in the book. The experimental method, the experiment as the final arbiter of reality, inductive and deductive logic, precise definitions of motion and forces, properties of matter as regards force.

We have studied the laws of motion in this lesson. The law of gravity and its implications will be the next two lessons, the first two of section three of the course which we call "The Newtonian Paradigm."

We will briefly introduce the topic here in the hopes that you will begin to appreciate how different things have become very, very quickly, both in the course of history, and in the history of this course.

There are many facets to this paradigm, and once again you do not have to be familiar with the details of them all. A general statement that sums it up might be something like: Natural laws rule a mechanistic universe.

You should be aware of the aspects of the Newtonian, so even if you do not take the time to read the details or to follow the outlines, you should at least look at the eight major topics. You may want to refer to this section throughout the course when you come across the phrase "Newtonian Paradigm".

### 5.1. Rules of Nature

Although Newton stated these rules, parsimony and experimental truth had been stated by others. The concept of universiality, expressed in 1 and 2 are unique to Newton.

### 5.7. Influences and parallels

Newton's work culminated Western thought which began with the Pythagoreans. In his synthesis, he distilled the essence of those ideas which form the basis of our modern science. So influential was Newton's work that he (along with Descartes) launched a new paradigm which had implications and influences far beyond the physical science. That he created the science of physics which has remained largely unaltered to this day is statement enough of the genius. The influcence on other areas is similar in many ways to the influence of great thinkers before his such as Plato and Aristotle.

For that reason we sometimes refer to this world view that arose out of Newton's work as "The Newtonian Paradigm". Here is an outline of some of the parallels and influences. You may recognize some of them. Others you will no doubt encounter later in your academic adventures.

#### 5.7.4. John Locke (1632-1704)

##### 5.7.4.3. ideas expanded by Berkeley and Hume5.7.4.4. known as prophet of reason in Enlightenment5.7.4.5. important influence on U.S. Constitution
• set up system and laws, it runs itself
• triangular balance of power

# 6. Summary

In this lesson we have covered a lot of ground. You will not be able to remember it all, nor are you expected to, any more than you are expected to memorize the script of a movie when you see it. When you see a movie you are generally more interested in understanding the plot and the interactions between the characters, than in memorizing the script. It is the same here.

We have seen how scientific inquiry and creativity in general blossomed in the seventeenth century as the scientific revolution came into full swing. In 1600 Bruno was executed for heresy. In 1632, Galileo was placed under house arrest for challenging the authority of the Church. In 1662 the Royal Society of London was formed to encourage open discourse and debate about scientific issues. By 1700 scientific inquiry was the fad, with aristocrat and artisan involved.

We also looked at Newton, the insecure genius with low self esteem who had to be persuaded into changing the world. And change it he did, perhaps more than any other man, except for Jesus, who, if you believe the stories, was more than just a man.

We took a look at Newton's laws, all the while encouraging you to study the way the texts present this material along with the video program. Hopefully these different perspectives will allow you to comprehend both the meaning and the significance of the laws.

We left an outline of the relevant features of the Newtonian paradigm and its influences at the end. If you skim this outline, it should stimulate some ideas that you may want to follow up on.