Newtonian Mechanics And Causality

The universe is a mechanism run by rules

60 newtonian mechanics and causality

It is now appropriate to discuss the development of that branch of physics called mechanics. This subject is concerned with the proper description and the causes of the motion of material objects. It was the development of mechanics, particularly the work of Newton, which finally "explained" the laws of planetary motion discussed in Chapter 2. The science of mechanics not only explains the motions of celestial bodies, but also the motions of earthly objects, including falling bodies and projectiles. In fact, following the work of Newton, it appeared that the laws of mechanics could explain nearly all the basic physical phenomena of the universe. This success was widely acclaimed and contributed significantly to the development of the ' 'Age of Reason,'' a time when many scholars believed that all of the sciences, including the social sciences and economics, could be explained in terms of a few basic underlying principles.

A. Aristotelian Physics

1. Aristotle's Methods

Among the earliest contributors to the science of mechanics were the ancient Greek philosophers, particularly Aristotle. The Greeks inherited various strands of knowledge and ideas concerning natural phenomena from previous civilizations, especially from the Egyptian and Mesopotamian cultures. Although a large amount of knowledge had been accumulated (considerably more than most people realize), very little had been done to understand this information. Explanations were generally provided by the various religions, but not by a series of rational arguments starting from a few basic assumptions. Thus, for example, the Sun was said to travel across the sky because it rode in the sun god's chariot. Such an explanation clearly does not need a logical argument to support the conclusion.

It was the early Greeks, more than any of the earlier peoples, who began to try to replace the "religious" explanations with logical arguments based on simple, underlying theories. Their theories were not always correct, but their important contribution was the development of a method for dealing with physical phenomena. That method is now called the scientific method, and it is the basic mode of inquiry for science today. The essential objective of this scientific method was clearly stated by Plato in his ' 'Allegory of the Cave'' as discussed in Chapter

2, Section A2. The goal of science (or natural philosophy as it was previously called) is to "explain" the appearances, that is, to explain what is observed in terms of the basic underlying principles.

2. Prime Substances

So far as is known, Aristotle (384-322 b.c.) was the first philosopher to set down a comprehensive system on a set of simple assumptions, from which he could rationally explain all the known physical phenomena. Chapter 2, Section B3, gives a brief introduction to his system in connection with the movements of the heavenly bodies, but it needs to be discussed more fully. Aristotle wished to explain the basic properties of all known material bodies; these properties included their weights, hardnesses, and motions. His descriptions of the physical universe were matter dominated. He began from only a few basic assumptions, and then proceeded to try to explain more complicated observations in terms of these simple assumptions.

Aristotle believed that all matter was composed of varying amounts of four prime substances. These were earth, water, air, and fire. He believed that objects in the heavens were composed of a heavenly substance called ether, but Earthly objects did not contain this heavenly substance. Thus most of the heaviest objects were regarded as earth, whereas lighter objects included significant amounts of water or air. The different characteristics of all known substances were explained in terms of different combinations of the four primary substances. Furthermore, and more importantly for his development of the subject of mechanics, Aristotle believed that motions of objects could be explained as due to the basic natures of the prime substances.

3. Motion According to Aristotle

Aristotle enumerated four basic kinds of motion that could be observed in the physical universe. These were (1) alteration, (2) natural local motion, (3) horizontal or violent motion, and (4) celestial motion. The first kind of motion discussed by Aristotle, alteration (or change), is not regarded as motion at all in terms of modern definitions. For Aristotle, the basic characteristic of motion is that the physical system in question changes. Thus he regarded rusting of iron, or leaves turning color, or colors fading, as a kind of motion. Such changes are now considered to be part of the science of chemistry. We now consider motion to mean the physical displacements of an object, and rusting or fading involves such displacements only at the submicroscopic level. The understanding of these kinds of "motions" is beyond the scope and purpose of this discussion. Some aspects of molecular and atomic motions will be discussed later (but not the actual description of the particular motions present in chemical processes).

The second of Aristotle's kinds of motion, natural local motion, is at the center of his basic ideas concerning the nature of motion. For him, natural motion was either "up" or "down" motion. (The Greeks believed that the Earth is a sphere and that "down" meant toward the center of the Earth and "up" meant directly away from the center of the Earth.) Aristotle knew that most objects, if simply released with no constraints on their motion, will drop downward, but that some things (such as fire, smoke, and hot gases) will rise upward instead. Aristotle considered these downward and upward motions to be natural and due to the dominant natures of the objects, because the objects did not need to be pushed or pulled. According to Aristotle, the "natural" order was for earth to be at the bottom, next water, then air, and finally fire at the top. Thus earth (i.e., rocks, sand, etc.) was said to want to move naturally toward the center of the Earth, the natural resting place of all earthly material. Water would then naturally move to be on top of the earth.

Objects dominated by either of the prime substances fire or air, however, would naturally rise, Aristotle believed, in order to move to be on top of earth and water. Fire was "naturally" the one at the very top. Aristotle viewed all "natural" motion as being due to objects striving to be more perfect. He believed that earth and water became more perfect as they moved downward toward their "natural" resting place, and that fire and air became more perfect as they moved upward toward their "natural" resting place.

Aristotle did not stop with his explanation of why objects fall or rise when released, but he went further and tried to understand how different sorts of objects fall, relative to each other. He studied how heavy and light objects fall and how they fall in different media, such as air and water. He concluded that in denser media, objects fall more slowly, and that heavy objects fall relatively faster than light objects, especially in dense media. He realized that in less dense media (less resistive we say), such as air, heavier and lighter objects fall more nearly at the same speed. He even correctly predicted that all objects would fall with the same speed in a vacuum. However, he incorrectly predicted that this speed would be infinitely large. Infinite speeds would mean that the object could be in two places at once (because it would take no time at all to move from one place to another), he argued, which is absurd! Therefore he concluded that a complete vacuum must be impossible ("nature abhors a vacuum"). As will be discussed below, this last conclusion was later the center of a long controversy and caused some difficulties in the progress of scientific thought.

Although not all of Aristotle's conclusions were correct regarding the nature of falling objects, it is important to understand that the methods he employed constituted at least as much of a contribution as the results themselves. Aristotle relied heavily on careful observation. He then made theories or hypotheses to explain what he saw. He refined his observations and theories until he felt he understood what happened, and why.

Aristotle's third kind of motion concerned horizontal motions, which he divided into two basic types: objects that are continually pushed or pulled, and thrown or struck objects, that is, projectiles. He considered horizontal motion to be unnatural, that is, it did not arise because of the nature of the object and would not occur spontaneously when an object was released. The first type, which includes such examples as a wagon being pulled, a block being pushed, or a person walking, did not present any difficulty. The cause of the motion appears to reside in the person or animal doing the pulling, pushing, or walking.

The second type of horizontal motion, projectile motion, was harder for Aristotle to understand and represents one of the few areas of physical phenomena in which he seemed somewhat unsure of his explanations. The difficulty for him was not what caused the projectile to start its motion, that was as obvious as the source of pushed or pulled motion, but rather what made the projectile keep going after it had been thrown or struck. He asked the simple question: ' 'Why does a projectile keep moving?'' Although this seems to be a very obvious and important question, we will see below that there is a better and more fruitful question to ask about projectile motion.

Aristotle finally arrived at an explanation for projectile motion, although one senses that he was not completely convinced by his solution. He suggested that the air pushed aside by the front of a projectile comes around and rushes in to fill the temporary void (empty space) created as the projectile moves forward, and that this inrushing air pushed the projectile along. The act of throwing or striking the object starts this process, which continues by itself once begun. This entire process came to be known as antiperistasis. This particular part of Aristotle's physics was the first to be seriously challenged. Aristotle himself wondered if this were really the correct analysis, and therefore also suggested that perhaps the projectile motion continued because a column of air was set into motion in front of and alongside the object during the throwing process. This moving column of air, he suggested, might then "drag" the projectile along with it. This bit of equivocation by Aristotle is very unusual and emphasizes his own doubts about his understanding of projectile motion.

Aristotle's final kind of motion was celestial or planetary motion. He believed that objects in the heavens were completely different from earthly objects. He believed heavenly objects to be massless, perfect objects made up of the celestial ether. He accepted the idea of the Pythagoreans that the only truly perfect figure is a circle (or a sphere, in three dimensions), and thus argued that these perfect objects were all spheres moving in perfect circular orbits. This celestial system has been described in more detail in Chapter 2, Section B. It is important to note that Aristotle considered heavenly objects and motions to obey different laws than earthly, imperfect objects. He regarded the Earth as characterized by imperfection in contrast to the heavens. In fact, he believed that the Earth was the center of imperfection because things were not completely in their natural resting places. The earthly substances earth and water had to move downward to be in their natural resting place, and air and fire had to move upward to be in their natural resting place. The Earth was imperfect because some of these substances were mixed together and not in their natural places.

4. Significance of Aristotelian Physics

Because Aristotelian physics has been largely discarded, there is a tendency to underestimate the importance of Aristotle's contributions to science. His work represents the first "successful" description of the physical universe in terms of logical arguments based on a few simple and plausible assumptions. That his system was reasonable, and nicely self-consistent, is testified to by the fact that for nearly 2000 years it was generally regarded as the correct description of the universe by the Western civilized world. His physics was accepted by the Catholic Church as dogma. (Aristotle's physics was still being taught in certain Catholic schools and even in some Islamic universities in the twentieth century.)

But much more important than the actual system he devised was the method he introduced for describing the physical universe. We still accept (or hope) that the correct description can start from a few simple assumptions and then proceed through logical arguments to describe even rather complicated situations. Important also were the questions he asked: "What are objects made of?" "Why do objects fall?" "Why do the Sun, Moon, and stars move?" "Why do projectiles keep moving?'' It was Aristotle, more than any other individual, who first phrased the basic questions of physics.

5. Criticisms and Commentaries on Aristotle's Physics

With the decline of Greek political power and the rise of Rome, scientific progress in the ancient world slowed to a crawl. Although the Roman Empire gave much to Western civilization in certain areas, it reverted largely to the earlier practice of explaining physical phenomena in religious terms. What scientific progress there was (and there was little compared with the large contributions of the Greek philosophers) came from the eastern Mediterranean. Alexandria, in Egypt, was the site of a famous library and a center for scholars for several hundred years. It was in Alexandria that Claudius Ptolemy (about 500 years after Aristotle) compiled his observations of the wandering stars and devised his complex geocentric system, as discussed in Chapter 2.

Another 350 years after Ptolemy, in the year 500, also in Alexandria, John Philoponus ("John the Grammarian") expressed some of the first recorded serious criticism of Aristotle's physics. John Philoponus criticized Aristotle's explanation of projectile motion, namely, the notion of antiperistasis. Philoponus argued that a projectile acquires some kind of motive force as it is set into motion. This is much closer to our modern understanding, as we shall see, that the projectile is given momentum, which maintains it in motion. Philoponus also questioned Aristotle's conclusion that objects of different weights would fall with the same speed only in a vacuum (which Aristotle rejected as being impossible). Philoponus demonstrated that objects of very different weights fell with essentially the same speeds in air. It is remarkable, however, that these first serious criticisms came more than 800 years after Aristotle's death.

With the fall of the Roman Empire and the beginning of the "dark ages" in Europe, and in particular with the destruction of the library and scholarly center in Alexandria, essentially all scientific progress ended. Some work, especially in astronomy and in optics, was carried on by Arab scholars. Finally, another 700 years later, the Renaissance began and European scholars rediscovered the early Greek contributions to philosophy and art. Much of the early work had to be obtained from Arabic texts. In the thirteenth century, St. Thomas Aquinas (12251274) adroitly reconciled the philosophy of Aristotle to the Catholic Church. By the beginning of the fourteenth century, Aristotle's philosophical system was church dogma and thereby became a legitimate subject for study. Fundamental questions regarding the nature of motion were finally raised again, particularly by various scholars at the University of Paris and at Merton College in Oxford, England. William of Ockham (1285-1349), and Jean Buridan (1295-1358) were noted scholars of motion and became serious critics of some of Aristotle's ideas. Buridan resumed the attack started 800 years earlier by Philoponus on Aristotle's explanations of projectile motion. Buridan provided examples of objects for which, he emphasized, the explanation of antiperistasis clearly would not work; for example, a spear sharpened at both ends (how could the air push on the back end?), and a millwheel (with no back end at all). Buridan also rejected the alternate explanation that a column of moving air created in front of and alongside a projectile kept the object moving. He noted that if this were true, one could start an object in motion by creating the moving air column first—and all attempts to do this fail. Buridan, like Philoponus, concluded that a moving object must be given something which keeps it moving. He referred to this something as impulse, and felt that as an object kept moving it continually used up its impulse. When the impulse originally delivered to the object was exhausted, it would then stop moving. He likened this process to the heat gained by an iron poker placed in a fire. When the poker is removed from the fire, it clearly retains something that keeps it hot. Slowly, whatever is acquired from the fire is exhausted, and the poker cools. As is discussed in Section B3 below, Buridan's impulse is close to the correct description of the underlying "reality," eventually provided by Galileo and Newton.

The level of criticism of Aristotle's physics increased. In 1277, a religious council meeting in Paris condemned a number of Aristotelian theses, including the view that a vacuum is impossible. The council concluded that God could create a vacuum if He so desired. Besides the specific criticisms by Buridan, scholars of this era produced, for the first time, precise descriptions and definitions of different kinds of motion and introduced graphical representations to aid their studies. These contributions, chiefly by the Scholastics in Paris and the Merton-ians in England, provided the better abstractions and idealizations that made possible more precise formulations of the problems of motion; for example, Nicole Oresme (1323-1382) invented the velocity-time graph (see below) and showed a complete understanding of the concept of relative velocity. This ability to idealize and better define the problem was essential for further progress, as discussed below.

B. Galilean Mechanics

1. Graphical Representations of Motion

A familiar example of motion is shown in Fig. 3.1, as a graph of speed versus time. The example is a hypothetical trip in an automobile. The vertical axis (the ordinate) indicates the auto's speed at any time in miles per hour (mi/h). The

Time (hours)

Figure 3.1. Hypothetical automobile trip.

Time (hours)

Figure 3.1. Hypothetical automobile trip.

horizontal axis (the abscissa) indicates the elapsed time in hours. The car starts the trip by accelerating (speeding up) to about 30 mi/h and then proceeding at this speed until reaching the edge of town, about one-half hour after starting. At the edge of town, the driver accelerates up to about 60 mi/h, on the open road. After about 1 hour from the beginning of the trip, the driver notices a State Police car and slows down (decelerates) to the speed limit of 55 mi/h. About one-half hour later, the driver pulls into a rest stop, decelerating to zero speed. After a short rest of about 15 minutes, the driver starts again, immediately accelerating to 55 mi/h, and continues until reaching the town of his destination. At the town limits the driver slows down to 40 mi/h until reaching the exact spot of his final destination, where he comes to a complete stop.

The example is simple and easy to follow. The important point, however, is that the entire trip can be recorded on the speed versus time graph, and the graph can be used to determine the motion of the car at any time during the trip. The basic kinds of motion are demonstrated in the hypothetical trip. Motion at a constant (nonchanging) speed is called uniform motion. The times spent at constant speeds of 30, 60, and 55 mi/h are all examples of uniform motion. The rest stop is also an example of uniform motion (with zero speed). Simple uniform motion is shown in the graph of Fig. 3.2. The graph is a horizontal straight line. The example shown is for a uniform speed of 30 mi/h. If the automobile were traveling at 60 mi/h, the graph would also be a horizontal straight line, but twice as high. If the automobile were not moving, then the graph would also be a horizontal line, but at zero mph; that is, rest is also uniform motion. (As will be discussed in Section D2 below, uniform motion must also be constant in direction.)

The times spent speeding up and slowing down in Fig. 3.1 are all examples of accelerated motion. Although slowing down is often called deceleration, it is more useful to consider it as negative acceleration. From the graph, one can see that some of the accelerations involve a rapid change of speed, whereas others correspond to a slower change of speed. Thus there are different kinds of accel-

Time (minutes)

Figure 3.2. Uniform motion.

Time (minutes)

Figure 3.2. Uniform motion.

Time (seconds)

Figure 3.3. Accelerated motion.

Time (seconds)

Figure 3.3. Accelerated motion.

erations. A specific example of accelerated motion is shown in Fig. 3.3. It corresponds to a car (for example) which is accelerating from rest to more than 50 mi/h in several seconds. The speed is not constant, but continually increases during the time shown on the graph. The "steepness" of slope of the graph indicates the magnitude of the acceleration. Thus at 1 second the slope of the graph is greater than at 5 seconds, and therefore the acceleration is greater at 1 second than at 5 seconds, in this particular case.

If the speed changes at a steady or constant rate (increase or decrease), then the motion is uniformly accelerated motion. Examples of uniformly accelerated motion are shown in Fig. 3.4. The essential characteristic of uniformly accelerated motion is that the graph is a sloping straight line. The actual speed does not matter. What is important is that the speed is changing by a constant amount during each time interval. Thus, for example, the line in Fig. 3.4 that represents a car starting from rest (0 mi/h) indicates that its speed increases by 10 mi/h in each second: from 0 to 1 seconds the speed increases from 0 to 10 mi/h, from 1

Time (seconds)

Figure 3.4. Uniformly accelerated motion.

Time (seconds)

Figure 3.4. Uniformly accelerated motion.

to 2 seconds the speed increases from 10 to 20 mi/h, from 2 to 3 seconds the speed increases from 20 to 30 mi/h, and so on. By contrast in Fig. 3.3, the increase of speed is less in each successive one-second interval; the motion is accelerated but not uniformly accelerated.

The graph of speed versus time for the motion of an object tells us a great deal about the motion at any one instant, but it does not directly indicate the result of the motion—namely, the distance traveled. Frequently, one is as interested in knowing the distance traveled as in knowing the speed at any one instant. Furthermore, in studies of motion, it is often much easier to measure distances traveled than to measure speeds at all times. Thus we are interested also in obtaining a distance versus time graph for the motion of an object and relating the information on the distance versus time graph to the information on the speed versus time graph.

It is actually quite easy to determine the distance traveled from a speed versus time graph. The distance traveled by a car with a constant speed of 30 mph in one hour of driving is 30 miles. In two hours, the same car would travel a total of 60 miles. We obtain these simple results, by using (in our head) the standard formula for distance traveled by an object with speed v (for velocity) over a time t; namely, distance = speed X time, or symbolically d = v X t

Now if we plot on a speed versus time graph the motion of an object with uniform motion (speed) v, we obtain a simple horizontal straight line, with zero slope, as previously discussed for Fig. 3.2. In Fig. 3.5 we show that the formula d = v X t corresponds simply to the area under the horizontal line, from time = 0, up to time = t. The light area in this case represents the distance traveled in two hours. This correspondence is true in general, that is, whether the speed is uniform or not. Consider the graph of Fig. 3.6.

The actual speed of the object is shown by the curved line on the graph. The distance traveled in six hours is represented by the light area. Shown also on this

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