PHYSICS
CHAPTER TWO: CONCEPTS OF FORCE IN HISTORY
2.1 Aristotelian Force
It was Aristotle who first developed a systematic set of ideas about the physical world, which is often referred to as Aristotelian physics. Related to his concept of force is his classification of motion as natural, voluntary, and forced (Jammer, 1957).
In the natural motion category, Aristotle believed that objects intrinsically either have a natural tendency to fall down to the earth, which he called gravity, or a natural tendency to rise into the sky, which he called levity. He thought that heavy bodies fall faster because the falling speed is in proportion to the physis (nature) or weight of the objects. The earth and the sky are natural places objects would move to according to their internal natural tendencies.
Voluntary motion refers to motion of living organisms such as animals and humans, who are agents able to exert force to make other inanimate things move. Nonliving objects are obstacles that stop or guide motion, but they do not exert forces.
In the forced motion category, an object moves because of the moving force applied to it by an agent. The object continues to move after the agent is no longer in contact with it because force is still transmitted to the object through a medium such as air (which is called the antiperistasis theory). Motion in a vacuum is thus not possible. A force does not move an object unless it overcomes the object's inertia, an intrinsic resistance of the object. A constant force applied to an object produces a constant speed which is also inversely proportional to the inertia of the object. In the absence of force, an object would stop immediately. So, forced motion is made consistent to the other two types of motion by Aristotle through his notion of antiperistasis and his theory of entelechy or agency (Leclerc, 1972).
In sum, for Aristotle, no motion is possible without force acting on the moving object, or in other words, motion and force are inseparable and a moving object is always an effect of some kind of entelechy, either visible or invisible.
2.2 Impetus Theory
In his mechanics, Aristotle confined himself only to the concept of force as the agency involved in motion and ignored the Platonic concept of dynamis or force as inherent in matter (Jammer, 1957). It was upon questioning this idea that Philoponus of Alexandria in the Middle Ages developed a new concept of force. Philoponous argued that when an object is thrown in the air, a force is imparted to the object. This force is responsible for the motion after the hand is no longer in contact with the object. Further, this force is prone to dissipation. Philoponus's idea was then developed by Jean Buridan and Nicole Oresme into the so-called Impetus Theory. In its complete form, the Impetus Theory states: 1) impetus is a kind of (invisible) power, existing in the physical world, which is able to move objects in the same direction as the impetus itself; 2) it can be imparted to an object from another object; 3) depending on the amount of impetus imparted to an object, the object moves correspondingly to that amount; 4) impetus can be dissipated or weakened by the resistance in the medium; 5) because impetus can be either rectilinear or curvilinear, the motions produced by the impetus can also be straight or curved (as the motion in celestial spheres).
2.3 Newtonian Force
It was Kepler, Galileo, Leibniz, Descartes, Newton, and many others who, collectively, completed the mechanical theories which we now know as Classical or Newtonian mechanics. The concept of force in Newtonian mechanics, which was developed over three hundred years ago, has remained a powerful scientific concept. This concept, being highly precise in its description of mechanical events, to many, represents the very quintessence of the physical world that we are living in.
The basic concepts of Newtonian mechanics are of critical importance for the discussions in this book and no reader would be expected to fully understand the theories proposed next without understanding the basic Newtonian concepts. For this reason, a brief summary is given here.
The Newtonian concept of force is given in three simple mechanical laws:
1) A body remains in its kinetic state, either at rest or in motion, unless there is an external force acting on it; which is known as the Law of Inertia.
2) If a (net) force F is applied to an object with mass m, then the acceleration a of the object caused by F follows the relation F=ma.
3) If object A exerts a force F on object B, then object B exerts a force -F on object A, equal in magnitude and opposite in direction to F.
The first law emphasizes the essential physical fact that all kinetic changes of an object are the result of dynamic force; in other words, dynamic properties of the world are primary, whereas kinematic properties are only secondary.
The second law is often regarded as a Newtonian definition of force, in which we can see that F is completely defined through mass m and acceleration a. Both m and a have precise definitions: m is the amount of matter in the object as measured in grams or kilograms (i.e., units of gravitational force), a is the amount of velocity change in one second, as measured in 'meter/per second/per second', or m/s². But the Newtonian force F is not only defined through m and a, it is also measured in the units of m and a, i.e., it has the unit kgm/s², which is called Newton in physics. This is because F is not measurable in itself and can only be measured through its effect (i.e., acceleration a) on an object (with mass m). Therefore, in Newtonian mechanics, we see a great paradox: on one hand, the Newtonian force so defined has finally given the things we need, the mechanical laws that are highly precise and objective, but on the other hand, it has in fact become much more abstract than any of the earlier concepts of force because it is now defined without reference to any mechanical agent, as if we had gained the power of precision and objectivity by pushing the concept itself to a higher level of abstraction (Jammer, 1957). In Newtonian mechanics, Gravity is a Newtonian force.
Note, the first law and the second law of Newton complement each other in the description of the mechanical state of an object with respect to the absence or the presence of a Newtonian force. The first law stipulates how an object would not change without force and the second law describes how an object would change with force.
The third law of Newton gives the interactive property of Newtonian force and emphasizes its dualistic nature in existence, magnitude, and directionality. Hence, for each Newtonian force F, there is another Newtonian force -F, inseparable in existence with, equal in magnitude and opposite in direction to F.
Besides the concept of Newtonian force, the concept of momentum, as defined by the multiplication of mass m and velocity v, i.e., mv, is also one of the key concepts in Newtonian mechanics. Historically, the concept of momentum can be regarded as an outgrowth of the impetus concept of the Middle Ages. But momentum is in fact quite different from impetus which has no quantitative definition. In comparison, momentum is a quantitatively precise and well-defined mechanical concept. Both the concept of impetus and the concept of momentum, however, were developed to capture the essence of the motion of an object in the absence of any force influence and to describe the persistence of motion as a dynamic phenomenon.
Energy is another Newtonian concept, which is divided into kinetic energy and potential energy. Kinetic energy is defined as ½mv² and potential energy is defined as ½mh², where h is the height of the object with mass m above the ground. Both forms of energy can however be seen as quiescent Newtonian force because, for an object, to have either forms of energy is to be able to potentially have Newtonian force and to influence another object. Therefore, in a broad but veracious sense, Newtonian force is a dynamic process in which both kinetic energy and potential energy (in fact, we could also include the thermal, the acoustic, the electrical, etc.) undertake their transmission, distribution, and transformation.
Note, since both the Newtonian force, F=ma, and the kinetic energy, E= ½mv², are defined in terms of mass m and velocity(note, a=dv/dt), a relationship between Newtonian force and kinetic energy, F=dE/dt, exists (Bueche, 1988). With this relationship, the total amount of Newtonian force in a time period can be conveniently and equivalently expressed by the amount of energy consumed (or more precisely, transformed) in this time period by the integral
E=Fdt.......................(1)
Newtonian force in less idealized but more realistic situations is generally a function of time and space, i.e., as in Vector Calculus or Field Theory, it can be expressed by F(s,t), the density force function in time and space, which can be further described by F(s,t), the magnitude function, and tan-1(F(s,t)y /F(s,t)x) , the directional function. Then, Equation (1) can be generalized to
E=F(s,t)dsdt................(2)
This equation will be quite useful in the final chapter when we discuss the formation of dynamic concept through dynamic perception.
Therefore, Newtonian mechanics is characterized by its objectivity and quantitative precision, with which none of the earlier theories are comparable. The key Newtonian concepts of force, momentum, and energy, all of which are under the quantitative definitions in space, time, and matter, display in measurable space, time, and matter an intrinsically connected pattern under the exact description of mathematics. The substantial quality of an object, as measured by mass, is found not only to play an indispensable role in the definition of every dynamic concepts, but also to be the link between dynamic concepts and kinematic concepts in Newtonian physics. The Newtonian concept of force can be regarded as a dynamic process which can be accurately measured by the change of the mechanical state of an object in motion.
In order to distinguish the Newtonian concept of force in physics from the commonsense representations of force in the human cognitive system, I will refer to the latter as P instead of F.
2.4 Force-related Research in Psychology
Research on commonsense knowledge about the dynamic or mechanical properties of the world has a very short history. Only in the past twenty years or so have researchers started paying attention to human cognition with respect to the dynamic properties in the world.
In particular, educators in physics education noticed the incompatibility between students' commonsense knowledge and the Newtonian mechanics they tried to teach and went further to find out exactly what the differences are between students' natural knowledge and Newtonian mechanics in order to improve their teaching.
Pioneering work by Michotte (Michotte,1963) on causality had led to a discovery of causal reasoning in infants by Leslie (Leslie,1982) and his Theory of Body mechanism or ToBy in the human cognitive system. From Barllargeon's (Barllargeon,1986) work on infant perception of solid objects to Spelke's theory (Spelke, 1988) on the origins of physical knowledge, from Talmy's penetrating analysis of force dynamics in language (Talmy, 1988) to Freyd's representational momentum in adult cognition (Freyd, et. al. 1988), a quest to explore the mind's conception of dynamic information is very well on the agenda of cognitive psychology.
Dynamic Misconceptions:
One important study (Clement, 1982) of people's commonsense conceptions of physical events was conducted by Clement, who reported that most physics students at the time of entering a physics class already have intuitive preconceptions concerning mechanical events. In addition, these preconceptions, though mostly incorrect according to Newtonian mechanics, are highly stable and extremely widespread among students. The particular case he gave is what he called the 'motion implies force' conception, in which a continuous motion of an object implies the presence of a force in the same direction as the motion. Clement presented several examples to show the pervasiveness and the diversity of situations in which students use such a conception. Figure 1 shows the typical response of students when they were asked to indicate in a pendulum all the forces on the swinging bob (b) and the correct answer of Newtonian physics (a). The 'motion implies force' conception was found to be extremely persistent: even after one semester of Newtonian mechanics, about 50% of science and engineering students still got the pendulum problem wrong.
Several other researchers also have reported various types of commonsense conceptions which are inconsistent to the concepts of Newtonian mechanics. McCloskey, Caramazza, and Green (McCloskey, Caramazza, and Green,1980) reported that out of the 48 students tested, 36% of them believed that a ball dropped out of a flying airplane would travel straight down to the ground. In another study, about half of the 2000 students tested believed that identical objects which were exactly at the same height but traveling in different directions would have different forces acting on them (Viennot,1979); For at least half of the 26 students tested, a book sitting on a table has no force acting on it besides gravity (Minstrell, 1982); A ball traveling in a curved tube with high speed is believed by 50% of 48 students to continue its curved path after it comes out the tube (McCloskey, 1983); High school students were found to solve dynamic problems utilizing diverse components of their own ideas, rarely consistent to the laws of Newtonian mechanics (White, 1983).
Halloun and Hestenes, for a period of three years, administered various tests on physics students commonsense beliefs (Halloun and Hestenes, 1984a, 1984b). They found that when it comes to the kinematic concepts, commonsense beliefs are often vague and lack sufficient differentiation; when it comes to dynamic concepts, they found that the students hold views such as motion is always associated with some cause, constant force produces constant velocity, and acceleration is due to increasing force (notice the similarity to the Aristotelian physics!); force can be classified into external force exerted by an agent and internal force (impetus) in the object; Gravity is an intrinsic property of objects to fall to the ground; a resistive force is one to a moving object.
Mechanical 'misconceptions' have been of great concern to researchers in Education because their goal is to teach the students Newtonian concepts. However, educators rarely raise the kind of questions that cognitive psychologists do, but, their findings are of great value to cognitive psychology.
Extraction of Dynamics from Kinematics:
Following Gibson's ecological approach to perception, many researchers in psychology have directed their attention to the physical events in the environment, ranging from mechanical events such as collision, natural events such as water boiling, to biological events such as a tree growing or a person walking, even intentional expressions used as deception. Obviously, both spatiotemporal information and dynamic information are present in events like these. Thus, event perception involves not only the taking-in and processing of spatiotemporal information such as shape and motion by the visual system, but also the extraction of dynamic information from spatiotemporal information. Various experimental results seem to support a specific view about event perception: dynamic, biological, and even intentional properties of events can be predictably determined by the display patterns themselves (Michotte, 1963; Natsoulas, 1960, 1961; Johansson, 1980; Todd and Warren, 1982; Runeson and Frykholm, 1983; Kaiser and Proffitt, 1984, 1987; Gilden and Proffitt, 1989; Pittenger, 1990; Runeson and Vedeler, 1993).
In (Todd and Warren,1982), for example, subjects were found to be able to tell which one of the two objects in a collision event is more massive from the patterns of their velocities. Michotte, though having not explicitly discussed his experiments in term of the dynamic information, had in fact demonstrated the close link between kinematic information and dynamic information in perceptual processes. Runeson and Frykholm (1983) showed films of actors lifting different boxes to subjects and found they could accurately estimate the weights of these boxes. Pittenger (1990) discovered that subjects could easily identify a pendulum constructed incorrectly in its period-length relationship and report the anomaly not in kinematic terms but in dynamic ones. Kaiser and Proffitt (1984, 1987) reported people's ability to discriminate two-dimensional dynamically canonical and anomalous collisions; All subjects from age 5 to adults performed above chance level in singling out collision events in which the dynamic laws are violated, based on what they saw in the visual displays.
An important question for event perception researchers is how exactly the dynamic information is perceived from kinematic information. Two very different views have recently emerged. The first view is that dynamic event perception, constrained by the event context and the internal processing limit of the perceptual system, is achieved by making use of simultaneous available but segmented, elementary, and heuristic-like protocols (Gilden, 1992). The second view holds that kinematic information can sufficiently specify the dynamic information. Runeson suggested an approach to event perception which has been referred to as the Kinematic Specification of Dynamics or KSD (Runeson, 1977). A classical example (Runeson, 1977) is the one-dimensional collision event which can be described mathematically as,
m1v1+m2 v2=m1v1'+m2v2' ................(3)
where m1 and m2 are mass of the two participating objects, v1 and v2 are velocities of the two objects before collision, and v1' and v2' are velocities after collision. With simple manipulations, this equation can be expressed differently as
m1/m2=(v2'-v2)/(v1-v1') ......................(4)
In Equation (2), the left-hand side belongs to dynamics and the right-hand side can be seen as the specification of the dynamics by a velocity difference ratio (kinematics). Thus, at least in this example, physics and mathematics are the vehicle which has made the specification clear. Here, the general idea is quite in tune with Gibson's picture of an ecological approach to perception, the pick-up of useful information in the environmental optic array (for this example, the velocities of the two objects) and the detection of the relevant informational invariant (the connection of the velocity difference ratio to the relative mass of the two objects). Thus, in the above example, to perceive dynamics is to detect the invariant of the velocity difference ratio with reference to the relative mass, which essentially is the result of the law of momentum. This view has recently been generalized into a principle, KSD Principle (Runeson, 1983), in an attempt to treat all events this way.
In sum, mechanical events, by their very nature, are multidimensional phenomena, including spatial, temporal, and dynamic information. Finding out the connection between kinematic information and dynamic information can be considered as an environment-centered approach to the problem of event perception. On the other hand, we can also take into consideration of the human cognitive system and try to find out how the knowledge or representations of dynamics are structured in that system.
Dynamic Representations:
Traditionally, there has been little work done in cognitive psychology which directly concerns people's dynamic conceptions of the world. It is only recently that discussion about the mental representation and processing of mechanical information has emerged (Barllargeon, Spelke and Wasserman, 1985; Leslie, 1982; Leslie and Keeble, 1987; Leslie, 1993; Freyd and Finke, 1984; Freyd, 1987).
It has been found that infants of 5 month old understand that a solid object cannot move through the space occupied by another solid object (Baillargeon, Spelke, and Wasserman, 1985). In other words, at 5 month old, infants already see an object as solid being able to resist the forceful intrusion of another solid object. The reverse has also recently been shown (Leslie and Keeble, 1987) that 6-month-old infants perceive a direct launching event as causal, i.e., different from merely a spatiotemporal structure. Thus, a moving object is being conceived of as a carrier of force capable of moving another object upon impact. It has also been found (Spelke, 1988) that infants perceive the unity of two rigidly moving surfaces differing in texture, substance, and form just as they perceive the unity of two rigidly moving surfaces of the same texture, substance, and form, whether through vision or touch. This suggests that cognitive representations for the picking up of dynamic information exists for both vision and touch, for it is only through attending to dynamic information that the infant was able to infer the mechanical unity and rigidity between the two movable surfaces. Recently, a series of experiments by Freyd (1987) and Freyd, Pantzer, and Cheng (1988) showed that people access their dynamic representations even when they are viewing a picture or a photo (containing frozen scenes of unstable events). These authors further claimed that even the representation of static scenes can be seen as the representation of forces in equilibrium.
This Book:
In spite of the large amount of research effort on human dynamic perception and conception in recent years, little, I have found, has been directed toward finding out the exact nature, organization, or structure of dynamic representations in the cognitive system. The research results in this area are often quite general, vague, and, in most cases, only superficial when it comes to what a dynamic representation is like, what lies at the root of dynamic representations, and how these representations are organized. For example, the well-known Impetus Theory of motion, developed by McCloskey and commonly accepted by most researchers in the field, tells us little about the so-called impetus and how it is conceived by the mind; In fact, the idea of impetus is little more a verbalization since its essentially nonphysical quality (as the result of its origin in the Middle Ages) prevents us from giving any content to it or even carrying out any further scientific investigation into it.
I believe that a more fruitful approach to the study of dynamic representations lies in how the substantial quality or mass of a mechanical object is represented in the cognitive system and how the cognitive system's conceptions of mechanical motion and mechanical interactions are related to the conception of mass. Such an approach would enable the full use of Newtonian concepts to accurately describe and the clear understanding of the exact nature of the experimental stimuli or situations, to manipulate the adequate kinematic and dynamic experimental variables, and to observe and analyze more objectively the experimental responses of the subjects.
Believing that the conception of mass plays a role in all dynamic concepts, I first carried out two experiments investigating how sensitive people are to the mass of an object in mechanical events of single object motion. My hypothesis was that people are very sensitive to the information of object mass and their conceptions of an object and its mechanical behavior would be directly related to their conception of this information. Along with the mass of an object, the speed and the cohesiveness of an object were also independent variables of these two experiments. With the variable of mass and the variable of speed, we can further investigate whether the subjects would conceive the mechanical state of an object in motion. My second main concern here is how people conceive the mechanical states of an object when the object is suddenly acted upon by an external force. A mechanical state of an object is characterized by both the object's kinematic properties, i.e., its speed and direction of motion, etc., and the object's dynamic property, i.e., its mass. Under the sudden influence of an external force, the mechanical state (often, the kinematic properties) of an object often changes.