COGNITIVE
PSYCHOLOGY
22, 342-373 (1990)
Understanding
Wheel Dynamics
DENNIS R. PROFFITT University
of Virginia
MARY K. KAISER NASA-Ames
Research Center
AND
SUSAN M. WHELAN University
of Virginia
In live experiments, assessments were made of people’s understandings about the dynamics of wheels. It was found that undergraduates make highly erroneous dynamical judgments about the motions of this commonplace event, both in explicit problem-solving contexts and when viewing ongoing events. These problems were also presented to bicycle racers and high-school physics teachers; both groups were found to exhibit misunderstandings similar to those of naive undergraduates. Findings were related to our account of dynamical event complexity. The essence of this account is that people encounter difficulties when evaluating the dynamics of any mechanical system that has more than one dynamically relevant object parameter. A rotating wheel is multidimensional in this respect: in addition to the motion of its center of mass, its mass distribution is also of dynamical relevance. People do not spontaneously form the essential multidimensional quantities required to adequately evaluate wheel dynamics. 0 1990 Academic Press, Inc.
Observing natural motions can provide information about dynamical relationships. For example, information about the weight of objects can be obtained by observing such movements as those produced by gusts of wind, or by collisions. Although people are known to be quite good at
This research was supported by Air Force Grant AFOSR-87-0238, NASA Grants NCAZ87 and NCA2-255, NICHHD Grant HD-16195, and a James McKeen Cattell sabbatical award to the first author. Carmine Louise Churchill, Heiko Hecht, and Ellen McAfee helped in running the experiments. Stephen Jacquot programmed the computer displays. David L. Gilden made valuable contributions on theoretical issues. We thank Stephen Palmer and four reviewers for their valuable comments on earlier versions of this paper. Among the reviewers was an anonymous physicist to whom we are especially indebted. Correspondence should be addressed to Dennis Proffitt, Department of Psychology, Gilmer Hall, University of Virginia, Charlottesville, VA 22903-2477. 342 0010-0285190$7.50 Copyright 0 19!30by Academic Press, Inc. All rights of reproduction in any form reserved.
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extracting dynamical information from a variety of natural events, this facility is quite limited. We have developed an account of dynamical event complexity for the natural motions encountered in classical mechanics (Proffitt & Gilden, 1989). In essence, this account defines simple object motions as those that can be adequately described in terms of the motions of objects’ centers of mass. Complex object motions require considerations of objects’ extended spatial properties such as mass distributions and orientation changes. Good contrasting examples of these two types of events are a free-falling object versus a rolling one; the dynamical understanding task for the latter presents problems not encountered in construing the former. The present studies were designed to elucidate this account of dynamical event complexity. Previous research has examined people’s dynamical understandings of only simple object motions, and as will be discussed, in these contexts competence has been found to be relatively good. In the present experiments, we assessed people’s dynamical understandings of commonplace, yet complex, object motions: the motions of wheels. The first three experiments assesseddynamical understandings of naive adults, the fourth tested bicycle racers who were especially familiar with rotational motions, and the final experiment assessed a group of high-school physics teachers. TWO CLASSES OF OBJECT MOTIONS: PARTICLE AND EXTENDED BODY MOTIONS By our account of dynamical event complexity, natural motions fall into two classes: particle and extended body motions (Proffitt & Gilden, 1989). Particle motions can be adequately represented in terms of the motions of structureless point particles. An object undergoing free-fall is a good example. Regardless of the object’s form, its dynamics can be specified by treating it as a particle with all of its mass located at its center of mass. The very act of looking at this point focuses attention on the only object motion that is of dynamical relevance. Extended body motions require dynamical representations that include information about mass distributions and the full 6 degrees of freedom of linear and rotational motions. With respect to forming dynamical understandings, extended body motions require that multiple categories of information be noticed and combined within multiplicative representations. Let us examine how these distinctions apply to wheels as they appear in particle and extended body motion contexts. Figure 1 shows two contexts for a rim-like wheel. In Panel A, the wheel is about to roll down an inclined plane, whereas in Panel B, it is hanging from a thread that will be cut. As an examination of their conservation of energy equations shows,
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FIG. 1. Two motion contexts for a rim-like wheel. In Panel A the wheel is about to roll down an inclined pane. In Panel B the wheel is attached to a thread that will be cut, resulting in freefall.
rolling and free-fall are quite distinct mechanical systems. (Here, we assume a vacuum.) Rolling: ‘/z mv* + 1/2Zw* = mg(h, - h)
(1)
Free-fall: ‘/z mv* = mg(h, - h)
(2)
where g is gravity, m is the mass of the wheel, B is its inner radius, A is its outer radius, v is its instantaneous velocity, w is its angular velocity about its center of mass, C,, and where the moment of inertia, Z, is Z = ?h m(A* + B*)
(3)
Assuming that the wheel rolls down the incline without slipping, its instantaneous velocity is v = Aw. Solving for instantaneous velocity as a function of height, h, yields: Rolling: v =
2g(ho - h) 5/2 3h + 54 - B2/A2
Free-fall: v = (2g(h, - h))“*
(4) (5)
Equations 1 and 2 state that, for the rolling wheel, kinetic energy is partitioned into both translational and rotational components, whereas for the falling wheel, kinetic energy is entirely translational. In solving for instantaneous velocity, we were able to cancel out mass for both systems, thereby revealing that mass is dynamically irrelevant in both rolling and free-fall contexts. An examination of Eq. 5 reveals the essence of point-particle systems:
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there is nothing in this equation that relates to the spatial properties of the moving object, The only object variable in the equation is the position, h, of the object’s center of mass. Thus, the act of looking at the object’s centroid draws attention to the single dimension required for dynamical analysis of the event. As is shown in Eq. 4, rolling is a context in which the wheel’s spatial properties do not cancel out. An aspect of the wheel’s shape-the ratio of its inner to outer radius, B/A, which is the inverse of what we will call compactness-influences dynamics in the rolling context. Compactness refers to the degree to which mass is distributed close to the wheel’s center of mass. The more compact the wheel, the faster it will descend the ramp. Moreover, note that the ratio, B/A, is invariant across overall changes in size. Whereas for the falling wheel, only center-of-mass position is of dynamical relevance, for the rolling wheel, two parameters of information must be combined in a dynamical representation. These parameters are the position of the wheel’s center of mass and the compactness of the wheel itself. Again, we note that this is the essence of the particle/ extended body motion distinction: particle motions reduce to centerof-mass displacements, whereas extended body motions require that these motions be combined with some spatial properties of the moving object. The distinction of object motion types depends not on whether the object is particulate or extended, but rather on the motions that it undergoes. Mechanical systems define the object parameters that are of dynamical relevance. In the simplest cases, particle motions, the only object parameter of dynamical relevance is the movement of the object’s center of mass. Extended body motions add more complexity to the dynamical understanding task. More than one category of information must be noticed, and since such dimensionally distinct quantities as shape and position cannot be added together, they must be combined through some sort of multiplicative operation resulting in a multidimensional quantity. The physical account given above is, of course, idealized. We do not propose that these equations actually represent the dynamical representations formed by observers. Rather, the exposition points out that, regardless of how people actually represent dynamical information, particle motion contexts require that they notice only where an object’s center of mass is going; extended body motions require that they notice this fact plus one or more other properties of the event, such as the object’s shape, size, mass, or orientation. Moreover, understanding extended body motions entails the additional requirement that these various dimensions be related in some fashion. With regard to human performance, we have proposed that, when mak-
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ing dynamical judgments about extended body motions, people do not spontaneously form multidimensional relationships; instead, people base their judgments on heuristics that relate to single parameters of information (Proffttt & Gilden, 1989). A clear example of this type of reasoning can be seen in how people dynamically evaluate collisions. Imagine a situation in which there are two billiard balls of unequal mass, and one of these balls, being initially stationary, is set into motion due to its being struck by the other. An analysis of the physical laws of momentum and energy conservation reveals that a kinematic relationship can be derived that specifies uniquely the mass ratio of these two balls. In particular, the postcollision projected velocities of the two balls onto an axis orthogonal to the precollision path of the striking ball defines a ratio equivalent to their relative masses. When asked to judge mass ratios while observing computer simulations of collisions, people do not spontaneously form this multidimensional relationship between velocities and angles; rather, they base their judgments on one of two heuristics that relate to these dimensions (Gilden & Proffitt, 1989). Their velocity heuristic can be stated as: following a collision, the faster moving ball is lighter. The other heuristic relates to deflection angle: after a collision, the ball that ricochets is lighter. Gilden and Proffttt found that people based their relative mass judgments on the heuristic related to the more salient dimension present in the event, either velocity or deflection angle. In many situations, such heuristical analyses led to good performance; however, when the two heuristics gave conflicting predictions, performance fell to an essentially chance level. Moreover, it was found that subjects never “averaged” the ricochet heuristic with the velocity heuristic to effect a compromise. Similar findings have been reported by Shepard (1964), who examined multidimensional judgments in a quite different context. Shepard required subjects to make similarity judgments for shapes that varied on two dimensions, size and the orientation of a feature. He found that subjects based their judgments on one or the other of these dimensions, and never related them. To summarize, we propose that dynamical judgments about extended body motions will be inaccurate because successful performance requires the formation of multidimensional quantities. Thus, people’s appreciation for natural dynamics will break down at the boundary between particle and extended body motions. It is, of course, true that people will sometimes exhibit erroneous reasoning when confronted with problems related to particle motions. However, there is a basic difference between their dynamical appreciation of these simple events and extended body motions. The dynamics of particle motions can be adequately penetrated by people if they are given the
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DYNAMICS
opportunity to view ongoing events; viewing extended body motions does not elicit more accurate intuitions. COMMON-SENSE
UNDERSTANDINGS
OF PARTICLE
MOTIONS
Recently, a literature has developed on people’s understandings of classical mechanics (Caramazza, McCloskey, & Green, 1981; Champagne, Klopher, & Anderson, 1980; Clement, 1982; Kaiser, Jonides, & Alexander, 1986; Kaiser, McCloskey, & Proffitt, 1986; Kaiser, Proffitt, & McCloskey, 1985b; McCloskey, 1983a, 1983b; McCloskey, Caramazza, & Green, 1980; McCloskey & Kohl, 1983; McCloskey, Washburn, & Felch, 1983). All of these studies presented problems dealing only with particle motion dynamics. In typical studies, people have been asked to predict the trajectories of balls rolled through curved tubes, of objects dropped from moving carriers, and of pendulum bobs whose connecting cords have been severed. For all of these events, the relevant dynamics can be assessed by considering the linear motion components of the objects’ centers of mass. Although these studies have typically been interpreted otherwise, we believe that they show that people are relatively competent with particle motion problems. We take this to be the case for four reasons. First, whereas early studies highlighted the tendency of college student populations to err on these problems, we note that typically the majority of subjects give correct responses. Second, when asked to reason about these problems in a familiar context, very few people make erroneous predictions (Kaiser et al., 1986a). Third, and of special significance, people demonstrate an excellent appreciation of particle dynamics when making judgments about ongoing events (Kaiser, Proffitt, & Anderson, 1985a; Kaiser & Proffitt, 1986). Those people who draw erroneous trajectories when shown pictorial representations of events almost always view simulations of their predictions as appearing anomalous, and judge correct trajectories as appearing natural. Finally, it should be noted that the events used in the intuitive mechanics studies are fairly complex exemplars of particle motions: external forces are applied and removed. The only experiments that have involved simple particle motions were developmental studies in which simple motions were used to ensure that young children understood the task (Kaiser et al., 1986b; Kaiser et al., 1985b). Here, it has been found that even a child of 4 years-of-age realizes that a ball exiting a straight tube rolls straight, and a dropped object falls straight down. We believe that the early studies by McCloskey and his colleagues (McCloskey, 1983a, 1983b; McCloskey et al., 1980; McCloskey et al., 1983) were so intriguing, in large part, because the errors were surprising to most of the reading audience. How, one had to ask, could a college
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sophomore with a high-school physics background actually think that a ball exiting a curved tube would continue to curve? In fact, we now know that that sophomore is unlikely to make such an error if asked to reason in a concrete context or asked to make judgments when viewing ongoing events. When viewing an ongoing particle motion event, the very act of looking at the moving object is simultaneous with noticing the single dimension that is of dynamical relevance. Although people sometimes make erroneous predictions about particle motions on paper-and-pencil tests, these same people exhibit near perfect performance when making dynamical judgments about ongoing events. As the following experiments demonstrate, the dynamics of extended body motions are not so easily penetrated. EXPERIMENT 1 Dynamical Understandings
of Wheels
Unlike particle motions, extended body motions require that more than one category of information be noticed and related in a multidimensional representation. We propose that this added complexity will be reflected in people’s ability to understand the dynamics of these events. We chose to assess people’s dynamical understandings of wheels because they are prototypical of extended body motions and are so commonplace in everyday experience. Method Subjecls. The subjects were 25 male and 25 female University of Virginia undergraduates who volunteered to participate in the study to fulfill a course requirement in an Introductory Psychology course. Thirty-five of these students had taken only a high-school physics course, five a college course, and ten had no physics training at all. Materials and procedure. Subjects were tested individually. The experimenter asked questions verbally and recorded the subjects’ verbal responses. Subjects were encouraged to ask questions if they did not understand the questions. Subjects were asked a large number of questions about natural dynamics. Fewer than half of these questions dealt with wheels. Filler questions were included to reduce the likelihood that the subjects would realize that the study was focused on their understanding of wheel dynamics. All of the filler questions were accompanied by drawings that depicted the situations being described. The structure of these questions was similar to that of the wheel questions. After each answer, subjects were asked to provide a brief explanation for their answer. The gist of these explanations was written down by the experimenter. There were four sets of questions related to wheel dynamics. They probed subjects’ understandings of the following: Factors affecting the rate that wheels roll down an inclined plane. An inclined plane was constructed, the vertical and horizontal dimensions of which were 0.23 m and 1.07 m, respectively. Four wheels were made to vary in the parameters of mass, radius, and distribution of mass. The wheels were constructed so that the set contained pairs of wheels that held two of each of these parameters constant and varied a third. Three wheels had a weight
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of 805 g, and the other weighed 2310 g; three wheels had a radius of 7 cm, and the other a radius of 3.3 cm; finally with respect to mass distribution, two wheels were solid disks (uniform mass distribution) and the other two were rims (mass distributed close to perimeter). For each pairing, the subjects were handed two wheels and told how they were the same and how they were different. For example, they might be handed two wheels of identical weight and mass distribution, but one wheel had a smaller diameter than the other. They were then asked: “If released from the top of the inclined plane at exactly the same time, will both of these wheels arrive at the bottom at the same time, or will one reach the bottom before the other, and if they descend at different rates, which wheel will reach the bottom first?” Subjects were asked to provide an explanation for their answer. After making all three judgments, subjects observed each pair rolling down the inclined plane, and were then asked to provide an explanation for the outcome that they observed, especially for those cases in which they had made an erroneous judgment. The trajectories of points located on a rolling wheel. A drawing was made that depicted a wheel rolling on a horizontal surface. Three points were drawn on the wheel: one on the circumference, one at the wheel’s center, and one midway between the center and the circumference. Subjects were given a copy of this drawing and asked to draw the motion paths followed by these three points as the wheel rolled. Bicycle stability. Subjects were asked to think back on their experiences riding bicycles and to recall that a bicycle is fairly difficult to balance when it is stationary or barely moving; however, when ridden a moderate speed, the bicycle becomes much easier to balance. They were then asked: “Why is a bicycle easier to balance when it is moving rapidly than when it is stationary or moving very slowly?” Gyroscope dynamics. Subjects were shown a toy gyroscope. A gyroscope is a wheel that is free to revolve around an axis. It is a kind of top: the axis of the gyroscope corresponds to the symmetry axis of the top; their dynamical behavior is equivalent. Subjects were shown the gyroscope and asked whether it could be made to balance on a pedestal in three different positions: axis aligned vertically, at a 45” angle, and horizontally. These questions were asked twice-once for a nonspinning gyroscope and once for one that was spinning rapidly. The gyroscope was then set into rotation and placed on the pedestal. Subjects observed the gyroscope balance on the pedestal at each of the aforementioned orientations. They were then asked to explain how the gyroscope was able to maintain its balance when it was spinning but not when it was not spinning.
Results and Discussion The results for each of the four sets of wheel questions are presented in turn. Factors affecting the rate that wheels roll down an inclined plane. As discussed in the introduction, mass and radius do not influence the time that it takes a wheel to roll down an inclined plane. Mass distribution, however, is a relevant variable. The more compact the wheel, the faster will be its rate of descent. Thus, a solid disk will descend a ramp at a faster rate than will a rim. Figure 2 presents the percentage of subjects giving each of the three possible responses for the manipulations of mass, radius, and mass distribution. On the radius problem, the distribution of subjects’ responses did not differ significantly from random guessing [x*(2) = 3.16, p > . lo].
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SAME
LIGHT
MASS
HEAVY
SAME
SMILL
RADIUS
LPIRGE
SAME
RIM
DISC
DISTRIBUTION
FIG. 2. The proportion of naive subjects giving each of three possible responses for the manipulations of mass, radius, and mass distribution.
The response distribution on the mass problem was nonrandom [x2(2) = 8.93, p < .05], due to subjects’ tendency to judge the heavier wheel as faster. Over 80% of the subjects failed to realize that distribution would affect the speed of the wheel. This systematic error lead to a highly nonrandom distribution of responses [x2(2) = 53.56, p < .OOOl].No effect on response distribution was noted for formal physics training or the gender of the subjects. As these analyses demonstrate, subjects did not show a group preference for any of the three possible responses when judging the influence of radius, and only a slight bias for the heavier wheel. The subjects performed at or near chance levels on these questions about irrelevant variables. The only strong preference that the subjects showed was a belief that mass distribution does not affect the rate that a wheel descends an inclined plane. Thus, the only shared belief held by these subjects was an erroneous one: mass distribution, the only relevant variable in this situation, was consistently judged to be irrelevant. After observing the wheels roll down the ramp, the subjects all came to the realization that mass distribution was the only influential variable. In general, their accounts of why mass distribution was influential simply described the particular wheels employed, and failed to elucidate general principles. The trajectories of points located on a rolling wheel. There was an enormous variety of responses given to the task of drawing the motion
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paths for three points located on a rolling wheel. As depicted in Fig. 3 for one cycle, the responses were categorized into six types of curves: (1) cycloid-connected arches; (2) prolate cycloid-wavy lines; (3) straight line-horizontal lines; (4) curtate cycloid-loops; (5) connected straight lines-straight lines of different orientations meeting at vertices (for two wheel rotations, these drawings looked like the letter W); (6) circle. Drawings were classified quite liberally; for example, any wavy line such as a sine curve was classified as a prolate cycloid. If there was an ambiguity as to how a drawing should be classified, then when appropriate, it was assigned to the correct category. For example, if a subject was attempting to draw the trajectory of a point on the rim of the wheel, and it was unclear whether the drawing should be classified as a cycloid or a prolate cycloid, then it was assigned to the correct cycloid category. Correct trajectories are also depicted in Fig. 8. Subjects’ responses are presented in Table 1. Summarizing from this table, only 18% of the subjects drew curves that could be classified as an appropriate cycloid for the point on the perimeter; 16% correctly drew wavy curves that could be liberally classified as prolate cycloids for the point midway between the circumference and the center; however, 94%
Cycloid
Prolate Cycloid
Straight Line
.
Curtate Cycloid
Connected Straight Lines Circle FIG. 3. Six curves used to categorize subjects’ drawings of the trajectories of three points on a rolling wheel.
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TABLE 1 Percentage of Subjects Drawing Each Category of Curve for Points Located on Perimeter, Intermediate Between the Perimeter and the Center, and at the Wheel’s Center, Respectively Curve
Cycloid Prolate cycloid Straight line Curtate cycloid Connected straight lines Circle
Perimeter
Intermediate
Center
18* 14 2 44 6 16
12 16* 2 52 4 14
94* 2 4
* Correct curve.
of the subjects correctly drew straight lines for the point at the wheel’s center. Not one subject drew curves that could be classified as correct for all three points. Bicycle stability. Bicycles are very complex machines, and a full stability analysis has not been performed (Jones, 1970; Kirshner, 1980; Lowell & McKell, 1982). The basic factors, however, are well understood. In particular, the wheels of bicycles are known to act as gyroscopes. When, for example, a rider leans to the right, this torque adds to existing angular momentum in a manner that will cause the front wheel to turn to the right, and thereby produce a torque on the wheel that acts against the lean. The essential gist of this explanation is: leaning causes the bicycle to turn; turning results in forces that act against the lean. This sequence of events is controlled by the gyroscopic nature of the wheels; the rider need not even hold onto the handle bars for the turning and righting to occur. Subjects’ responses to the question about why a moving bicycle is easier to balance than one that is stationary fell into four categories: (1) 52% of the subjects reasoned that moving forward made the bicycle harder to tip over. These references to forward momentum made no mention of how it related to turning. That is, linear momentum is influential only after a bicycle has turned in response to a lean. Thus, subjects in this category did not demonstrate an understanding of the manner in which linear momentum relates to stability. (2) 24% stated that the rider’s movements and weight distribution produced the increased stability found in moving bicycles. This is wrong. (3) 8% suggested that the wind builds up on the sides of the bicycle and rider as they move, and this wind holds them up. This, of course, is wrong. (4) The remaining 16% of the subjects gave idiosyncratic responses. All of these latter responses were incorrect but one. One subject stated that the wheels acted like gyroscopes. This subject had noticed a gyroscope that had been inadvertently left out in sight, and made the correct analogy. When asked how the
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gyroscopic properties of bicycle wheels functioned to produce stability in the bicycle, this subject had no idea. Moreover, when asked about the behavior of gyroscopes in the next question, this subject was as mystified as the other subjects. Gyroscope dynamics. The physical explanation of gyroscopic motion is difficult to describe in words. In essence, gravity causes a spinning gyroscope to precess; that is, the principle axis of a spinning gyroscope revolves sideways in a gravitational field. Gravity produces a torque on the gyroscope, pulling it down. This downward torque results in precession, a slow rotation of the gyroscope’s axis around its pedestal. This precession is the falling motion of a spinning gyroscope. Thus, the important motions to notice are not only the spinning of the gyroscope’s wheel, but also the precession of its central axis. When asked whether a nonspinning or spinning gyroscope could balance on a pedestal at three different orientations, all of the subjects correctly stated that a nonspinning gyroscope could not balance in any of these positions. For the spinning gyroscope, 100% felt that it could balance in an upright position, 60% at a 45” angle to the pedestal, and 28% at a horizontal orientation. After observing a spinning gyroscope balance at all three orientations, the subjects were asked to account for the different behavior of a spinning and a nonspinning gyroscope. Although all of the subjects admitted that they really did not know, when asked to propose an account, they gave reasons that fell into live categories: (1) 36% stated that the spinning wheel produced an upward force. (2) 14% suggested that it was due to centrifugal or centripetal force. These subjects did not know what these terms meant but did know that they referred to spinning objects. (3) 4% mentioned angular momentum, but as with the above responses, these subjects did not know what this term meant in relation to the behavior of a gyroscope. (4) 4% of the subjects suggested that the gyroscope acted like an airplane propeller. (5) The remaining 42% gave idiosyncratic responses. Clearly, none of the subjects had any intuitions about how the gyroscope is influenced by gravity. Although all of the subjects knew that the behavior of the gyroscope was affected by the spinning of its wheel, not one subject mentioned the precessional motion. This most relevant motion was completely overlooked by the subjects. EXPERIMENT 2 Understanding Mass Distribution’s Influence on Rotational Dynamics: An Assessment in an Ongoing Context The previous experiment demonstrated that people typically have no
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explicit awareness of mass distribution’s influence on the dynamics of a rolling wheel. This experiment was designed to assess whether viewing ongoing displays would evoke more accurate intuitions. Observers were shown a computer-graphics animation of a satellite spinning in space. This event is depicted in Fig. 4. The satellite changed its mass distribution by opening or closing a set of solar panels, an event that is analogous to a spinning ice skater who extends or contracts his or her arms. In a natural situation, the opening of the satellite’s solar panels would cause its spinning rate to slow, whereas closing the panels would result in an increased angular velocity. In the animated stimulus displays the opening and closing of the panels resulted in a variety of spin rates. The observer’s task for each sequence was to judge whether the resulting angular velocity was the natural outcome of the satellite’s changing shape, or whether it could only have been produced by some unseen external force. This display presented the simplest context in which mass distribution’s influence on rotational dynamics could be assessed. The satellite’s dynamics are fully specified by the law of angular momentum conservation. Angular momentum is defined as follows: L = zw
(6)
where L is angular momentum, Z is moment of inertia, and w is angular velocity. Comparing this equation with Eq. 1 reveals the simplicity inherent in the satellite’s dynamics. Unlike the rolling context, here rotational dynamics is not coupled with linear motions. Moreover, no external forces are acting on the object. Since angular momentum is conserved when the satellite changes shape, angular velocity is an inverse function of its moment of inertia: The greater the increase (decrease) in I, the greater the decrease (increase) in w. Moment of inertia weights the mass of every object particle by the
A
B
C
FIG. 4. A spinning satellite in three configurations: solar panels open, intermediate, and closed.
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square of its distance to the axis of rotation. More precisely, moment of inertia is defined as:
i=l
where mi is the mass of the ith particle, and Ri is the distance of that particle from the axis of rotation. For the satellite used in our study, rotation was about an axis passing through its center of mass and orthogonal to the plane defined by the four panels’ bases. Panels were defined as having uniform density, and the satellite consisted of only these panels; its interior was empty. Moment of inertia is defined by the shape and orientation of the panels as follows: I = (% b3h + 2 bha* + 2 abh’ sin 0, + ?A bh3 sin 2 6&z/8 bh (8) where a is the distance between the midpoint of each panel’s base and the center of rotation, b is half the width of each panel, h is the length of each panel, 8, is the angle of the panels (absolute value), at time t, relative to the vertical closed position, and m is the satellite’s total mass. The quantity, m/8bh, defines the mass density of the panels. It is a constant that does not affect the changes in angular velocity that occur when the satellite changes shape. We make two observations concerning this equation. First, the determination of the satellite’s moment of inertia entails far more complexity than was found for the wheel. The high degree of symmetry inherent in wheels allows their moment of inertia to be expressed by the simple relationship shown in Eq. 3. In general, the specification of moment of inertia requires considerably more geometry, especially if the object does not possess uniform density. Even for our relatively symmetrical satellite, we doubt that anyone would propose that moment of inertia could be accurately assessed by perception, and thus, a precise appreciation for the influence of the satellite’s changing shape on angular velocity could not be expected. Compared with particle motions, extended body motions are more complex because many object parameters must be noticed and related. The second observation is that a qualitative appreciation for the relationship between panel orientation and angular velocity can be based on the following simple heuristic: when the panels open, the satellite’s spin rate must slow; when they close, spin rate increases. The following experiment was designed to assesswhether people spontaneously notice the dynamical significance of this relationship between mass “compactness” and angular velocity when viewing ongoing events.
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Method Subjecrs. Twelve University of Virginia undergraduates (six mates and six females) volunteered to participate in the study to fulfill a course requirement in an Introductory Psychology course. Stimuli. All stimuli were viewed on a Tektronix 4129 3-D Color Graphics Workstation with a high resolution (4 lines/mm) display screen 30 cm high X 35 cm wide. The stimuli consisted of a graphic simulation of a spinning satellite. As is shown in Fig. 4, the satellite consisted entirely of four color-filled panels. These panels were described as being solar panels of uniform density. The display’s background consisted of random points that slowly drifted together, from right-to-left, across the screen. These points were described as being stars. Except for its empty interior, the satellite occluded the “stars” as it revolved in front of them. The satellite’s initial appearance on the screen always presented the solar panels in either a fully open or closed position. The satellite remained in this state until the subject pressed a key, whereupon the satellite closed or opened its panels, depending upon its initial state. The stimulus set consisted of seven pairs of events. Seven of these events began with the satellite in a panels-closed state, and presented a variety of angular velocity changes as the panels opened. The other seven stimulus events were matched reversals of the motions presented in the first set. We describe each stimulus pair relative to the motions that occurred as the panels opened for the initially panels-closed member. Stimulus 1 slowed to a stop, reversed spin direction, and achieved a final angular velocity equivalent to, but in the opposite direction from, its initial velocity. Stimulus 2 slowed to a stop. Stimulus 3 slowed too much, achieving a final angular velocity intermediate between a stopped and the canonical outcome. Stimulus 4 was the canonical event. Stimulus 5 did not slow enough, achieving a final angular velocity intermediate between the canonical event and its initial velocity. Stimulus 6 did not change its angular velocity. Stimulus 7 sped up to a final angular velocity equivalent to 3/zits initial velocity. Panels-closed stimuli had an initial angular velocity of 0.3 Hz, and panel opening was always accomplished in 2 s. Angular velocity changes were achieved via a constant acceleration during the opening interval. Procedure. Subjects were tested individually. They were told that they would be observing a simulated satellite, spinning in space. Pressing a key on the computer’s keyboard would allow them to change the state of the satellite by opening or closing its solar panels. They were told that sometimes the satellite would behave naturally; however, sometimes its movements would be possible only if it had been influenced by some unseen force, such as an invisible thruster. Their task was to judge whether the motions that occurred during the opening or closing of the solar panels were natural or would have required the application of an unseen force. Following these instructions, they were shown a random order of all 14 stimuli, in order to acquaint them with the events that they would be judging. They were then instructed that they were to make their judgments by adjusting a thumbwheel that moved a marker along a scale at the bottom of the screen. The subjects were told to use the scale relative to the set of satellites that they had just seen, placing the marker at one extreme of the scale for the most natural appearing event, and at the other extreme for the event that had most obviously been influenced by an unseen force. They were then given four practice trials with events randomly drawn from the set of 14 stimuli. Subjects were then allowed to ask questions and to proceed with the test trials. There were 28 test trials consisting of two presentations of each stimulus. The 14 stimuli were presented in two randomly ordered blocks that differed for each subject. The random orders were restricted by the qualification that no stimulus be contiguous with itself or with its opening/closing opposite. On each trial, the spinning satellite would appear in one of its open or closed states. The subject observed the satellite, and at a time of his or her own
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choosing, pressed a key causing the satellite to change states. Once the satellite had completed its transformation, the subject adjusted the scale and terminated the trial by pressing a key.
Results and Discussion
Observers demonstrated only a qualitative appreciation for the preservation of spinning direction, and no sensitivity to the influence of mass distribution on angular velocity. Figure 5 shows the mean ratings for the seven opening/closing pairs. The ratings have been placed on a lo-point scale, 10 being natural and 0 indicating the application of an unseen force. An ANOVA revealed that two of the stimulus pairs-stimuli in which changing shape resulted in a reversal in the direction of spin, or resulted in a stopping or starting of the spin-were significantly different from the other five stimulus pairs [F(1,60) = 58.4, p < .OOOl]. In addition, the ratings for the reversal stimuli were reliably lower than that for the starting/stopping stimulus pair (ANOVA was followed by a Scheffe test [p < .OS]).None of the ratings given to the other five stimuli differed from each other (Scheffe test). There was no effect for direction of transformation, opening versus closing stimuli, nor for gender. Analyses were also performed on ratings that had been normalized within subjects to obtain a common mean and variance; their outcomes were equivalent to those presented for the raw ratings. These results indicate that observers have an awareness that changing Natural
61
I
Reverses
stops
Slows too Canonical much
Slows to0 No change Speeds Up little
Event
FIG. 5. The naturalness ratings given to the satellite’s angular velocity changes resulting from configuration change. For each stimulus pair, the description provided is for the motions that occurred as the panels opened for the initially panels-closed member. For example, for the right-most category, “Speeds Up,” the opposite member of this stimulus pair would “Slow Down” as the satellite’s panels closed from an initially open position.
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the shape of a spinning object cannot, by itself, cause it to change its direction of spin, or to stop spinning (or start to spin if initially stationary); however, they have no spontaneous appreciation for the influence of mass distribution on rate of spin. For the initial panel-closed stimuli, events that slowed as the panels opened were judged to be no more natural than those that did not change in angular velocity or sped up. Similar findings were found for the initial panel-open stimuli. These results converge with those found in the previous experiment where we asked people to judge the influence of mass distribution on the rate that wheels descend an incline plane. There, as here, observers demonstrated no appreciation for the influence of mass distribution on rotational dynamics. Unlike the previous study, here people made judgments while viewing an ongoing event. For particle motion problems, animation has been found to evoke far more competent dynamical judgments (Kaiser & Proffitt, 1986; Kaiser et al., 1985a). For this extended body motion situation, however, we found that people remained unaware of the dynamical influence of mass distribution. We had intended to design an experiment to assess whether people would exhibit a greater implicit appreciation for gyroscope dynamics when viewing animated simulations of natural and anomalous versions of this event. The results of this study, however, suggested that there would be little to be gained from such a study. People would surely not perform better with the gyroscope than they did with the far simpler satellite. We did, however, create a computer simulation of a top that was shown to a number of pilot subjects. These observers found the task of distinguishing between possible and impossible gyroscopic motions to be far too difficult, performed randomly, and became extremely agitated and upset by the task. To have completed the study would have only served to demonstrate the obvious and to demoralize a number of observers. EXPERIMENT Understanding
3
the Trajectories of Points on a Rolling Assessment in an Ongoing Context
Wheel: An
In the first experiment we asked subjects to draw the trajectories of three points on a rolling wheel. One point was at the wheel’s center, one on the perimeter, and one intermediate between these two. Although the center point was typically drawn correctly, for the perimeter and intermediate points, it was found that drawings could rarely be classified as resembling the actual trajectories. In this experiment, we assessed observers’ ability to recognize these trajectories when shown computer animated point-lights, moving as if attached to an unseen rolling wheel.
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Method Subjects. The subjects were 24 University of Virginia undergraduates (12 male and 12 female) who volunteered to participate in order to fulfill a requirement in an Introductory Psychology course. Stimuli. The stimuli consisted of single point-lights moving along five trajectories on a black computer screen. These point-light motions were created on the Tektronix 4129 Workstation (4 lines/mm, display screen 30 cm high x 35 cm wide) and were viewed from an approximate distance of 45 cm. The five trajectories were the three wheel-generated motions-cycloid, prolate cycloid, and straight line-plus a curtate cycloid and a sine curve. Horizontal excursion for all trajectories was 31” of visual angle. Since for a given wheel size, amplitude of vertical motion could be used to distinguish between the wheel-generated motions, each of the four curved trajectories was presented at four amplitudes: vertical displacements of 2, 4, 6, and 8 cm (maximum visual angle, 6.5”). This manipulation created a range in the number of cycles per excursion from 1 (for the 8-cm prolate) to 12(for the 2-cm curtate). (Relating size and number of cycles could have served as a bases for distinguishing between the wheel-generated motions; however, we found no evidence that any subject attempted this strategy.). Procedure. Four straight-line trajectories were combined with the 16 curved trajectories (four sizes for each of the four curves) for a total of 20 stimuli. For each subject, these 20 stimuli were presented in four randomized blocks, thereby producing a set of 80 trials. Randomizations were different for each block and each subject. Although the subjects were not so informed, the first block served as practice, and was not included in the data analysis. Subjects were seen individually. They were informed that they would be observing the motions of single point-lights. Some of the motions would correspond to those of a point at one of three different locations on a rolling wheel, others would follow paths that could not be generated by a point on a rolling wheel. At the bottom of the terminal’s screen were depictions of three wheels, each having a dot at one of the three appropriate locations: perimeter, intermediate, and center. Also present at the bottom of the screen was a symbol designating the categorical response for a trajectory that was not wheel-generated. Whenever the subjects observed what they judged to be one of the wheel-generated motions, they were instructed to move a cursor to a position below the appropriate wheel showing the light’s location. Whenever they observed a trajectory that did not appear to have been wheel-generated, they were to move the cursor below the “other” symbol. After each trial, subjects were asked to estimate their confidence by moving a cursor on a scale that appeared below the response categories. On each trial, the point-light moved twice across the screen before the mouse controlling the cursor was enabled.
Results and Discussion Each subject’s responses were classified relative to the curve that was most frequently being observed when he or she gave each of the four responses: perimeter, intermediate, center, and other. Confidence ratings were used to break ties. There were 9 out of a possible 96 ties. Table 2 presents the percentage of subjects for each response-by-stimulus classification. Subjects performed slightly better at this task than in Experiment 1 where they drew the trajectories of points on a rolling wheel. Forty-six percent of the subjects correctly categorized a cycloid as being the perimeter point in this ongoing simulation, whereas only 18% of the subjects
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2
Percentage of Subjects Responding to Each Category of Curve by the Trajectory That They Were Actually Observing Answer Curve seen Cycloid Prolate cycloid Straight line Curtate cycloid Sine Category never used
Perimeter
Intermediate
Center
Other
46* 33 13
13 s* 67 13
96* 4 -
4 38 4 8* 38*
8
-
-
Note. Two subjects never used each of the categories, “PERIMETER” If columns do not sum to zero, it is due to rounding. * Correct curve.
8 and “OTHER.”
in Experiment 1 correctly drew cycloids for the perimeter point [x2 (1) = 6.33, p < .025]. On the other hand, performance remained poor for the intermediate point with only 8% of the subjects correctly categorizing the prolate cycloids as being intermediate in the ongoing situation as compared with 16% who correctly drew this curve in Experiment 1 [x2 (1) = 0.82, p > S]. Only 2 subjects correctly categorized all three wheelgenerated trajectories when viewing the animated displays. That overall performance remained low when subjects were viewing ongoing displays indicates that the source of wheel-generated trajectories is not obvious. This was especially the case for the intermediate point. Although performance improved for the perimeter point relative to that found in Experiment 1, it remained at a low level. EXPERIMENT 4 Dynamical Understandings
of Wheels by Bicycle Racers
Returning to the issues raised in the first experiment, there are a number of possible reasons why people have such poor understandings of wheel dynamics. Our contention is that these motions are more complex than are the particle motions for which people demonstrate far better competence. Another possibility, that we do not favor, is that people have less experience with wheels than they do with particle motions. We do not believe this to be so. Wheels are observed very frequently in our society. Jearl Walker, author of “The Amateur Scientist” column in Scientific American, has published a collection of articles on the physics of rotation in the everyday world (Walker, 1985). In the Preface to this collection, he writes: “Rotation is fascinating because in spite of its common occurrence it is
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difficult to understand . . . For example, when I spin a top I am always amazed that it stands upright in spite of the gravity tugging at it. Even when it leans over, it does not fall, instead it precesses, that is, the central axis about which it spins circles around the vertical. Such motion is magic” (p. vii). In this experiment we sought to assess dynamical understandings of wheels with individuals who were especially familiar with these motions-bicycle racers. We predicted that these cyclists would not possess appreciably better understandings of wheel dynamics than those demonstrated by subjects in the previous experiment. Method Subjects. Members of the University of Virginia Bicycle Club were contacted by phone and asked to volunteer to participate in the study. Fifteen male and 10 female cyclists were recruited in this manner. These cyclists reported that they rode their bikes an average of about 100 miles per week, the range being from 10 to 250 miles per week. Two of these subjects had no physics training, 11 had physics only in high school, and the remaining 12 cyclists had taken at least one semester of college physics, the range among this latter group being from 1 to 4 semesters. Unlike the subjects in the previous experiment who were predominantly first and second year students, the cyclists tended to be third and fourth year students with more course work in physics. Materials and procedures. The procedures were identical to those of Experiment 1 except that more detailed questions were asked about the behavior of bicycles. The cyclists were asked the same question about why a moving bicycle is easier to balance than a stationary one. Following this question they were then asked: “Suppose that you are riding your bicycle and holding your hands over your head. You lean to the right without touching your handlebars. What will happen?” If the cyclist answered that the bike would turn, he or she was then asked what caused the bike to turn. If the subject stated that the turn was caused by the front wheel turning, then he or she was asked why the front wheel turned. Finally, for those cyclists who reported that the front wheel would turn, the experimenter also asked whether the wheel would turn if the bicycle was not moving.
Results and Discussion Factors
affecting
the rate that wheels roll down an inclined
plane.
Figure 6 shows the proportion of subjects selecting each of the three responses for the three wheel pairings. An analysis of these results showed that the distribution of responses was random for both the mass and radius problems [x2 (2) = 0.55 and 3.91, respectively.]. For the distribution problem, subjects were biased toward responding that the wheels would roll at the same rate [x2 (2) = 8.00, p < .05). The cyclists’ response distributions did not differ from the subjects in Experiment 1 on the mass, radius, or mass distribution problems. (Although not significant, the cyclists were somewhat less likely than the subjects in Experiment 1 to erroneously judge mass distribution as being irrelevant [x2 (2) = 4.75, p < .lO]). As this analysis shows, the cyclists responded in a similar manner to
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SAME
LIGHT
SAME
HEAVY
SMILL
LARGE
SAME
RADIUS
MASS
RIM
DlSC
DISTRIBUTION
FIG. 6. The proportion of bicycle racers giving each of three possible responses for the manipulations of mass, radius, and mass distribution.
the subjects in Experiment 1. Both groups showed the greatest consensus in judging that mass distribution is not a factor influencing the rate of descent for a wheel rolling down an incline. The trajectories of points located on a rolling wheel. The subjects’ drawings were classified as follows: connected arches were classified as cycloids, loops as curtate cycloids, wavy lines as prolate cycloids, and straight lines as themselves. The subjects’ responses are summarized in Table 3. Thirty-six percent of the cyclists correctly drew cycloids for the outer point, 24% correctly drew prolate cycloids for the middle point, and all of the subjects drew a straight line for the center. Only one subject drew all three of these curves correctly. Bicycle stability. The cyclists’ responses fell into four categories: (1) 28% of the cyclists gave sophisticated responses that made reference to the gyroscopic properties of the bicycle wheels. (2) 20% stated that the stability was due to the fact that the wheels were revolving, but gave erroneous accounts for the nature of this effect. (3) 48% of the subjects TABLE 3 Percentage of Subjects Drawing Each Category of Curve for Points Located on Perimeter, Intermediate between the Perimeter and the Center, and at the Wheel’s Center, Respectively Curve
Perimeter
Intermediate
Center
Cycloid Prolate cycloid Straight line Curtate cycloid
36* 20 44
16 24* 60
100* -
* Correct curve.
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reasoned that moving forward made the bicycle harder to tip over. These references to forward momentum made no mention of how it related to turning, and thus, did not demonstrate an understanding of the manner in which linear momentum relates to stability. (4) The remaining subject gave an idiosyncratic response. Although the majority of cyclists, 72%, gave erroneous accounts for bicycle stability, their responses were much more adequate than were those of the subjects in Experiment 1. Fortyeight percent of the cyclists believed that bicycle stability was related to the motion of the wheels, whereas only one of the previous subjects made this attribution. In response to the question about what happens when you lean to the right while riding a bicycle with your hands off of the handlebars, 92% of the cyclists correctly stated that you would turn to the right; the other 8% judged that you would turn left. Fifty-six percent of the cyclists knew that the bicycle turned right because the wheel turned right, and 12% knew that the wheel turned right because of its angular momentum. Of these latter subjects, two of the three knew that the wheel would turn to the right only when the bicycle was moving. These two subjects had taken two and three semesters of college physics, respectively. Gyroscope dynamics. When asked whether a nonspinning or spinning gyroscope could balance on a pedestal at three different orientations, all of the subjects correctly stated that a nonspinning gyroscope could not balance in any of these positions. For the spinning gyroscope, 100% felt that it could balance in an upright position, 76% at a 45” angle to the pedestal, and 36% at a horizontal orientation. After observing a spinning gyroscope balance at all three orientations, the subjects were asked to account for the different behavior of a spinning and a nonspinning gyroscope. They gave reasons that fell into four categories. (1) 20% stated that the spinning wheel produced an upward force. (2) 16% incorrectly suggested that it was due to centrifugal or centripetal force. (3) 8% mentioned angular momentum but could not explain how this concept applied to gyroscopes. (4) The remaining 56% gave idiosyncratic responses. Of these subjects, one mentioned precession and described a demonstration given in one of her physics classes. This subject could not account for how precession resulted from gravity’s influence on the spinning gyroscope. As was found for the subjects in Experiment 1, the cyclists knew that the behavior of the gyroscope was affected by the spinning of its wheel, but only one cyclist mentioned the relevance of precessional motion. Although the cyclists had considerable experience with gyroscopic motions, their understanding of gyroscope dynamics was limited to a knowledge of a few terms memorized from previous physics courses. Across the different questions, the cyclists did not exhibit an apprecia-
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bly better understanding of wheel dynamics than did the subjects in Experiment 1. Their accounts of bicycle stability were somewhat better; however, this advantage did not generalize to their appreciation of gyroscope stability.We take these findings to imply that the poor competence found in both groups is not simply due to a lack of familiarity with wheel motions. EXPERIMENT 5 Dynamical Understandings of Wheels by High-School Physics Teachers What is the relationship between an individuals’ explicit knowledge of the physics underlying wheel dynamics, and his or her intuitions about the sorts of problems presented in Experiments 1 and 4? Do adequate intuitions about wheel dynamics follow from an explicit understanding of classical mechanics? In pilot work, we individually asked four Ph.D. physicists the problem about wheels rolling down the inclined plane. Their replies made it immediately clear that their common-sense understandings of wheel dynamics differed little from our naive subjects. Although it seemed that they were less likely to assume that mass would affect rate of descent, they guessed on the radius problem, and all four were incorrect for the mass distribution question. Clearly, they could have solved the problems if given sufficient time and writing materials, thereby allowing them to set down the necessary equations to analytically derive the answers. However, we prohibited them from doing so by requiring that they answer quickly. We were struck by the apparent finding that a deep and thorough understanding of classical mechanics did not more immediately impact on their understanding of wheel dynamics. The physicists, however, protested that they were the wrong group to test. Classical mechanics is not an abiding concern of contemporary academic physicists; they argued that they had thought very little about this domain in recent years. They suggested that, if anyone had a practiced working knowledge of the simple classical mechanics principles that our studies were investigating, it would be high-school physics teachers. Method Subjects. The subjects were 19 (10 male and 9 female) high-school physics teachers. They were attending a IO-day institute on physics instruction at the University of Virginia during the summer of 1987. This institute served the teachers’ needs to meet various State of Virginia teacher accreditation requirements. The following background information was obtained from the teachers via self report. Their average age was 40.6 years. Sixteen had obtained B.S. degrees, whereas the other
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three had obtained B.A.?,. Five had obtained an MS. degree, and four had acquired an M.Ed. They reported having the average number of semesters of preparation in the following subject areas: Physics, 6.5; Mathematics, 6.4; Chemistry, 4.9; and Biology, 2.6. They reported an average of seven years experience in teaching physics courses. Materials and procedure. The subjects were tested in a group session that lasted about an hour. They were asked a variety of questions about physical problems including all of those that were asked of the bicycle racers. We asked them to base their answers on their immediate intuitions, and not to attempt analytical solutions. The same wheels, inclined plane, and gyroscope were shown as in Experiments 1 and 4. Two additional questions were asked of the physics teachers. First, they were told that they would be shown a spinning gyroscope balancing on a pedestal. They were to watch the event, and then report: “To what would you tell students to attend when watching this event, so that they would better understand the behavior of gyroscopes when it was explained to them?” That is, the teachers were asked to report on what gyroscopic motions should be noticed in order to better understand an explanation of its dynamics. Second, they were asked: “When teaching mechanics to your students, what are the three most difficult concepts for them to understand?”
Results and Discussion Factors affecting the rate that wheels roll down an inclined plane. Compared with naive undergraduates, the physics teachers were more likely to be aware that mass distribution affected the rate of descent, but were uncertain as to what this effect should be. Figure 7 shows the proportion of subjects selecting each of the three responses for the three wheel pairs. An analysis of these results showed that the distribution of responses did not differ from random guessing for all three problems: mass, radius, and mass distribution [x2 (2) = 1.4, 0.1, and 1.4, respectively]. As poor as this performance was, it did reflect an improvement over the naive subjects in that the physics teachers were less likely than the subjects in Experiment 1 to erroneously judge mass distribution
SAME
UOHT
MASS
HEAVY
SAME
SYAU.
RADIUS
URGE
SIUE
RIM
DlSC
DISTRIBUTION
FIG. 7. The proportion of high-school physics teachers giving each of three possible responses for the manipulations of mass, radius, and mass distribution.
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PROFFITT, KAISER, AND WHELAN
as being irrelevant. The response pattern across all three response categories differed between the naive subjects and the physics teachers [x2 (2) = 10.72,p < .005]. Be that as it may, only 21.1% of the teachers correctly predicted the influence of mass distribution on rate of descent, reflecting, at best, only a marginally significant improvement in correct performance as compared with the naive subjects [x2 (1) = 3.54, p < .lO]. The trajectories of points on a rolling wheel. The subjects’ responses were classified and are summarized in Table 4. Performance remained relatively poor for this group. Forty-seven percent correctly drew cycloids for the outer point, 16% correctly drew wavy lines for the intermediate point, and all but one drew a straight line for the wheel’s hub. Only one subject correctly drew all three curves. Bicycle stability. The physics teachers did well with the initial questions about bicycle stability; however, as was the case for the bicycle racers, they exhibited little depth of understanding when questioned about specific details. Sixty-eight percent of the teachers correctly responded that a bicycle is easier to balance, when moving at a rapid as opposed to a very slow rate, because of the rotation of the wheels. Many of these teachers referred to notions of angular momentum and gyroscopic principles. When asked what happens when you lean to the right on a moving bicycle, 68% correctly responded that the bicycle will turn to the right. However, only 21% (four teachers) reported that the bicycle turned to the right because its wheel turns in response to the lean, and of these four teachers, only two provide an adequate mechanical reason for why the wheel turned. Finally, only one of these two subjects correctly stated the wheel would turn right in response to a rightward lean only when the bicycle was moving. Gyroscope dynamics. When asked about whether a nonspinning gyroscope could be balanced on a pedestal at three different orientations, 90% TABLE 4 Percentage of Subjects Drawing Each Category of Curve for Points Located on Perimeter, Intermediate between the Perimeter and the Center, and at the Wheel’s Center, Respectively Curve
Perimeter
Intermediate
Center
Cycloid Prolate cycloid Straight line Curtate cycloid Circle Other
47* 5 5 21 11 11
16 16* 42 16 11
95* 5
Note. If columns do not sum to zero, it is due to rounding. * Correct curve.
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correctly reported that it could not balance in an upright position, and 95% likewise reported correctly for the other two orientations. For the spinning gyroscope, 90% stated that it could balance in an upright position, 7% at a 45” angle, and 68% at a horizontal orientation. The teachers were asked to imagine that they were going to explain the mechanics of the gyroscope to one of their students, and that the student would first have an opportunity to watch one balancing on a pedestal. The teachers were then told to write down what they would tell this student to pay attention to while watching the event. A spinning gyroscope was shown to the teachers after the task was explained to them and before they wrote their responses. Two of the teachers wrote answers that were too vague to be useful. Of the remaining 17 subjects, all made some mention of the spinning of the gyroscope’s wheel; however, only six reported that it would be useful to notice the gyroscope’s precessional motion. Compared with the naive subjects, the physics teachers were no better at predicting the balancing of a spinning gyroscope at all three different orientations on a pedestal [x2 (1) = 1.74, p > .lO], although they were more accurate for 90” [x2 (1) = 9.57, p < .005]. What we find of greatest interest, however, is how few of the teachers, 32%, thought to direct their students’ attention to the importance of the gyroscope’s precessional motion. Most d$ficult concepts. In response to the question asking them to list the three most difficult concepts in their physics courses, the teachers mentioned angular momentum most frequently. It was mentioned by 18 of the 19 teachers and was listed first by 10 of them, as compared with four first-place listings for the next most frequently mentioned concept. GENERAL DISCUSSION
In two experiments we found that people have very poor dynamical understandings of wheel dynamics. This lack of competence is far greater than any encountered in previous studies on people’s dynamical understandings of simple particle motions. Both particle and extended body motions manifest dynamical relationships; one could imagine an idealized observer who could appreciate the dynamics inherent in both. People, however, are not such observers. Rather, they exhibit limitations related to dynamical event complexity. For people, the dynamics of particle motions are easily appreciated, whereas extended body motions are not. Tops and gyroscopes are enduring toys because their mysterious behavior is not penetrated by perceptual experience. Observing a spinning gyroscope evokes wonder and joy because it appears to defy all of our dynamical intuitions. The motions of the gyroscope are apparent, but they evoke no spontaneous intuitions as to why the device behaves as it does.
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Our previous research on perceiving wheel generated motions suggests a perceptual explanation for people’s poor understandings of some aspects of wheel dynamics. We propose that, in perceiving rotational motions, the perceptual system performs an analysis that emphasizes the extraction of form at the expense of recovering information needed for dynamical analyses. As the top panel of Fig. 8 shows, every point on a rolling wheel follows one of three possible motion paths: cycloids, prolate cycloids, or straight lines. When viewing a rolling wheel, however, these trajectories are not seen. Rather, as is depicted in the bottom panel of Fig. 8, the perceptual system analyzes these motions into two components: relative rotationsall points are seen as revolving about their contigural centroid-and a common motion that is the trajectory of the centroid (Proffitt, Cutting, & Stier, 1979). The thoroughness of this perceptual analysis can be seen in the attempts of our subjects to draw the trajectories of three points on a rolling wheel. In Experiments 1, 4, and 5, only 2 of 94 subjects drew all three curves correctly. In Experiment 3, subjects viewed moving point-lights,
FIG. 8. The top panel shows the absolute motion paths for three points on a rolling wheel. At the bottom is depicted how these points are perceived when viewed in the context of the whole wheel. All points, except the center, are seen revolving about the wheel’s center; the center manifests the wheel’s common observer-relative motion.
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and only 46% and 8% selected cycloids and prolate cycloids, respectively, as corresponding to the trajectories of appropriately located points on a wheel. This indicates that even when viewing an ongoing display, the source of these trajectories is not obvious. Moreover, as the history of mathematics shows, the cycloid was discovered very late due to its absence in perception (Struik, 1969). Galileo is typically credited with discovering the cycloid, and Rubin (1927) reports that Galileo first noticed the curve while observing a wagon rolling down a hill at night with a single torch attached to the rim of one of its wheels. A more detailed account of the percemaal system’s analysis of absolute motions into relative and common components is presented in Fig. 9 (Proftitt & Cutting, 1980; Cutting & Proffttt, 1481). This figure depicts our proposal that rotations are not only separated from common motions, but also indicates that these two motion components have quite different perceptual significances: rotations specify form, whereas common motion defines observer relative displacement. All accounts of how perception extracts form from motion rely on rotational motions. (See, for example, Ullman, 1979; Wallach & O’Connell, 195311976.)The perceived common motion is the motion path of the centroid of relative rotations. Under the assumption of uniform density, this centroid corresponds to the form’s center of mass. Thus, the perceived motion of the whole object-its common observer relative displacement-is a particle motion. Absolute
Motions
m
/\ Form
/ \ Relative Motmn
Common
Motmn
Actton
(Common ObserverRelative Displacement)
FIG. 9. The left panel depicts the perceptual analysis of wheel motions into relative and common motion components. The right panel shows our proposal that these two motion components have different perceptual significances. Rotations specify form. Common motions define particle motion dynamics.
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In essence, we propose that the perceptual system performs an analysis that extracts rotational motions for the purpose of recovering form, and common motions are used to define particle motion dynamics. The only motion that is available for dynamical analysis is common motion, and people show dynamical competence only when the relevant motions are those of the object’s center of mass. Further support for this proposal is seen in the manner in which the perceptual system defines configural centroids. The perceived center of a form is the centroid of its external boundary as opposed to the centroid of its luminance distribution (Proffttt, Thomas, & O’Brien, 1983). That is, the perceptual system effectively ignores the distribution of luminance within a form, and determines its centroid by attending to only the object’s contour. This disregard for the distributional properties of objects may be reflected in the responses given by our subjects to the question about what factors influence the rate that a wheel rolls down an inclined plane. Even though it was the only relevant factor in the situation, subjects showed the greatest agreement in their belief that mass distribution does not affect wheel dynamics. Moreover, the wheel dynamics that contribute to the stability of bicycles and gyroscopes are distributional variables. By definition, all extended body motions include mass distribution as a relevant dynamical variable. To reiterate, we propose that the perceptual system assigns quite different significances to rotational and common motions. For human perception, relative rotations specify form, and common motions carry dynamically relevant information. Thus, with respect to dynamical intuitions, the only relevant object motions are particle motions. From this perceptual perspective, mass distribution should not affect the rolling of a wheel, the absolute motion paths of points on a rolling wheel should be lost to direct intuition, and the dynamical influences of spinning wheels should be imperceptible. Although speculative, we think that this perceptual explanation accounts well for people’s poor understandings of wheel motions. CONCLUSION
The distinction that we have drawn between particle and extended body motions is a physical one. In some situations, the only object attribute of dynamical relevance is the position over time of the object’s center of mass. These are particle motion contexts. For most situations, however, aspects of the object’s shape, orientation, and rotation are dynamically relevant as well. These are extended body motion contexts. Dynamical analysis of extended body motions requires that more than one object dimension be noticed and combined into a multidimensional representation.
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We have proposed that people do not form these emergent multidimensional quantities, but rather base their dynamical judgments on heuristics that relate to only one dimension of information present in the event (Proffitt & Gilden, 1989). The use of such heuristics can result in accurate dynamical judgments in particle motion contexts, since these events are, by definition, one-dimensional. In general, extended body motion contexts cannot be adequately evaluated in this manner. The wheel is one of the simplest and most commonplace extended body systems encountered in our culture. Be that as it may, common-sense intuitions about its dynamical properties in extended body contexts are terribly muddled. When asked about the factors affecting the rate at which a wheel rolls down a ramp, university undergraduates performed at a chance level when judging the significance of irrelevant variables, but were highly consistent in their belief about the irrelevance of the only variable that is, in fact, of dynamical significance. This lack of appreciation for the influence of mass distribution on rotational dynamics was manifested in explicit judgment tasks, and when subjects viewed ongoing simulations of natural and anomalous events. Moving, from the simple wheel contexts of spinning and rolling to those observed in gyroscopic motions, produces a shift in people’s response to the motions encountered. The perplexity that accompanies the observation of a rim and a solid disk descending a ramp at different speeds is transformed into wonder when a top is observed precessing at right angles to its pedestal. In this latter event, common sense is confronted with a state of affairs that definitely runs counter to strong intuitions. Immediate intuitions about wheel dynamics are affected little by amount of experience or expertise. Bicycle racers and high-school physics teachers exhibit very similar erroneous judgments about wheel dynamics when compared with naive undergraduates. We believe that training impacts little on common-sense notions about dynamics, because these intuitions are grounded in phenomenal experience. A physicist can no more see the dynamics of a gyroscope than a geometer can see the cycloidal trajectories of points on the tire of a passing car. A physicist can provide a complete account of gyroscopic dynamics, but the multidimensional variables that are introduced in such an account are not categories of phenomenal experience. This is not to say that people cannot be trained to spontaneously deal more effectively with multidimensional systems. As experimenters, we never get the wheel problems wrong. If retested, our subjects would hopefully not make the same mistakes again after the postexperimental briefing in which they were given an explanation of the systems that they had encountered. Finally, we suspect there are people, such as certain specialized engineers, who must regularly deal with the dynamics of
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wheel systems, and who have established well-memorized sets of rules to deal with their dynamical behavior. People can be taught to reason efficiently about the dynamics of specific extended body motion systems-here, we are referring to spontaneous judgments as opposed to explicit calculation-however, we believe that the adequacy of such reasoning will always fall short of that which can be achieved in particle motion contexts. Reasoning about particle motions can always be based upon the motions of a single point: the object’s center of mass. Perceiving the motions of this point is something that people are known to do spontaneously and with accuracy (Proffrtt & Cutting, 1980). Dynamical understandings of extended body motions require that more than one object dimension be taken into account and related within a multidimensional representation. The complexity encountered in assessing multidimensional quantities, such as moment of inertia (see Eq. 7 and 8), will always pose a hindrance to common-sense reasoning about extended body motions. REFERENCES Caramazza, A., McCloskey, M., & Green, B. (1981). Naive beliefs in “sophisticated” subjects: Misconceptions about trajectories of objects. CogniZion, 9, 117-123. Champagne, A. B., Klopfer, L. E., &Anderson, J. H. (1980). Factors influencing the leaming of classical mechanisms. American Journal ofPhysics, 48, 1074-1079. Clement, J. (1982). Students’ preconceptions in introductory mechanics. American Journal of Physics, 50, 66-71. Cutting, J. E., & Proffttt, D. R. (1981). Gait perception as an example of how we may perceive events. In R. Walk & H. L. Pick, Jr. (Ed.), Intersensory perception and sensory integration (Vol. II, pp. 24!&273). New York: Plenum. Gilden, D. L., & Proflitt, D. R. (1989). Understanding collision dynamics. Journal of Experimental
Psychology:
Human Perception
and Performance,
15, 372-383.
Jones, D. E. H. (1970). The stability of the bicycle. Physics Today, 23, 34-W. Kaiser, M. K., Jonides, J., & Alexander, J. (1986a). Intuitive reasoning about abstract and familiar physics problems. Memory & Cognition, 14, 308-312. Kaiser, M. K., McCloskey, M., & Proffttt, D. R. (1986b). Development of intuitive theories of motion: Curvilinear motion in the absence of external forces. Developmental Psychology, 22, 67-7 1.
Kaiser, M. K., & Proffttt, D. R. (1986, November). Swingtime: Observers’ sensitivity to the dynamics of pendulums. Paper presented at the 27th Annual Meeting of the Psychonomics Society, New Orleans, LA. Kaiser, M. K., & Proffttt, D. R. (1987). Human sensitivity to the dynamics of collisions. Perception
& Psychophysics,
42, 275-280.
Kaiser, M. K., Proffitt, D. R., & Anderson, K. (1985a). Judgments of natural and anomalous trajectories in the presence and absence of motion. Journal of Experimental Psychology: Learning, Memory, and Cognition, 11, 795-803. Kaiser, M. K., Proffttt, D. R., & McCloskey, M. (1985b). The development of beliefs about falling objects. Perception & Psychophysics, 38, 533-539. Kirshner, D. (1980). Some nonexplanations of bicycle stability. American Journal of Physics, 48, 36-38.
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Lowell, J., & McKell, H. D. (1982). The stability of bicycles. American Journal ofPhysics, 50, 1106-1112. McCloskey, M. (1983a). Intuitive physics. Scientific American, 248(4), 122-130. McCloskey, M. (1983b). Naive theories of motion. In D. Gentner & A. L. Stevens (Eds.), Mental mode/s (pp. 299-324). Hillsdale, NJ: Erlbaum. McCloskey, M., Caramazza, A., & Green, B. (1980). Curvilinear motion in the absence of external forces: Naive beliefs about the motion of objects. Science, 210, 1139-l 141. McCloskey, M., & Kohl, D. (1983). Naive physics: The curvilinear impetus principle and its role in interactions with moving objects. Journal ofExperimental Psychology: Learning, Memory, and Cognition, 9, 146156. McCloskey, M., Washburn, A., & Felch, L. (1983). Intuitive physics: The straight-down belief and its origin. Journal of Experimental Psychology: Learning, Memory, and Cognition,
9, 636-649.
Proffttt, D. R., & Cutting, D. R. (1980). An invariant for wheel-generated motions and the logic of its determination. Perception, 9, 435-449. Proffttt, D. R., Cutting, J. E., & Stier, D. M. (1979). Perception of wheel-generated motions. Journal of Experimental Psychology: Human Perception and Performance, 5, 289-302.
Proffitt, D. R., & Gilden, D. L. (1989). Understanding natural dynamics. Journal of Experimental Psychology:
Human Perception
and Performance,
15, 384-393.
Proffttt, D. R., Thomas, M. A., & O’Brien, R. G. (1983). The roles of contour and luminance distribution in determining perceived centers within shapes. Perception & Psychophysics. 33, 53-71. Rubin, E. (1927). Visuell wahrgenommene wirkliche Bewegungen. Zeitschrift fiir Psychologie, 103, 384-392.
Shepard, R. (1964). On subjectively optimum selection among multiattribute alternatives. In M. W. Shelly & G. L. Bryan (Eds.), Human judgment & optima& (pp. 257-281). New York: Wiley. Struik, D. J. (Ed.) (1969). A source book in mathematics, Cambridge, MA: Harvard University Press. Ullman, S. (1979). The interpretation of visual motion. Cambridge, MA: MIT Press. Walker, J. (1985). Roundabout: The physics of rotation in the everyday world. New York: Freeman. Wallach, H., & O’Connell, D. N. (1953). The kinetic depth effect. Journal ofExperimental Psychology, 5, 205-217. (Also in Wallach, H. (1976). On perception. New York: Quadrangle.) (Accepted October 4, 1989)
PSYCHOLOGY
22, 342-373 (1990)
Understanding
Wheel Dynamics
DENNIS R. PROFFITT University
of Virginia
MARY K. KAISER NASA-Ames
Research Center
AND
SUSAN M. WHELAN University
of Virginia
In live experiments, assessments were made of people’s understandings about the dynamics of wheels. It was found that undergraduates make highly erroneous dynamical judgments about the motions of this commonplace event, both in explicit problem-solving contexts and when viewing ongoing events. These problems were also presented to bicycle racers and high-school physics teachers; both groups were found to exhibit misunderstandings similar to those of naive undergraduates. Findings were related to our account of dynamical event complexity. The essence of this account is that people encounter difficulties when evaluating the dynamics of any mechanical system that has more than one dynamically relevant object parameter. A rotating wheel is multidimensional in this respect: in addition to the motion of its center of mass, its mass distribution is also of dynamical relevance. People do not spontaneously form the essential multidimensional quantities required to adequately evaluate wheel dynamics. 0 1990 Academic Press, Inc.
Observing natural motions can provide information about dynamical relationships. For example, information about the weight of objects can be obtained by observing such movements as those produced by gusts of wind, or by collisions. Although people are known to be quite good at
This research was supported by Air Force Grant AFOSR-87-0238, NASA Grants NCAZ87 and NCA2-255, NICHHD Grant HD-16195, and a James McKeen Cattell sabbatical award to the first author. Carmine Louise Churchill, Heiko Hecht, and Ellen McAfee helped in running the experiments. Stephen Jacquot programmed the computer displays. David L. Gilden made valuable contributions on theoretical issues. We thank Stephen Palmer and four reviewers for their valuable comments on earlier versions of this paper. Among the reviewers was an anonymous physicist to whom we are especially indebted. Correspondence should be addressed to Dennis Proffitt, Department of Psychology, Gilmer Hall, University of Virginia, Charlottesville, VA 22903-2477. 342 0010-0285190$7.50 Copyright 0 19!30by Academic Press, Inc. All rights of reproduction in any form reserved.
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extracting dynamical information from a variety of natural events, this facility is quite limited. We have developed an account of dynamical event complexity for the natural motions encountered in classical mechanics (Proffitt & Gilden, 1989). In essence, this account defines simple object motions as those that can be adequately described in terms of the motions of objects’ centers of mass. Complex object motions require considerations of objects’ extended spatial properties such as mass distributions and orientation changes. Good contrasting examples of these two types of events are a free-falling object versus a rolling one; the dynamical understanding task for the latter presents problems not encountered in construing the former. The present studies were designed to elucidate this account of dynamical event complexity. Previous research has examined people’s dynamical understandings of only simple object motions, and as will be discussed, in these contexts competence has been found to be relatively good. In the present experiments, we assessed people’s dynamical understandings of commonplace, yet complex, object motions: the motions of wheels. The first three experiments assesseddynamical understandings of naive adults, the fourth tested bicycle racers who were especially familiar with rotational motions, and the final experiment assessed a group of high-school physics teachers. TWO CLASSES OF OBJECT MOTIONS: PARTICLE AND EXTENDED BODY MOTIONS By our account of dynamical event complexity, natural motions fall into two classes: particle and extended body motions (Proffitt & Gilden, 1989). Particle motions can be adequately represented in terms of the motions of structureless point particles. An object undergoing free-fall is a good example. Regardless of the object’s form, its dynamics can be specified by treating it as a particle with all of its mass located at its center of mass. The very act of looking at this point focuses attention on the only object motion that is of dynamical relevance. Extended body motions require dynamical representations that include information about mass distributions and the full 6 degrees of freedom of linear and rotational motions. With respect to forming dynamical understandings, extended body motions require that multiple categories of information be noticed and combined within multiplicative representations. Let us examine how these distinctions apply to wheels as they appear in particle and extended body motion contexts. Figure 1 shows two contexts for a rim-like wheel. In Panel A, the wheel is about to roll down an inclined plane, whereas in Panel B, it is hanging from a thread that will be cut. As an examination of their conservation of energy equations shows,
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PROFFITT, KAISER, AND WHELAN
FIG. 1. Two motion contexts for a rim-like wheel. In Panel A the wheel is about to roll down an inclined pane. In Panel B the wheel is attached to a thread that will be cut, resulting in freefall.
rolling and free-fall are quite distinct mechanical systems. (Here, we assume a vacuum.) Rolling: ‘/z mv* + 1/2Zw* = mg(h, - h)
(1)
Free-fall: ‘/z mv* = mg(h, - h)
(2)
where g is gravity, m is the mass of the wheel, B is its inner radius, A is its outer radius, v is its instantaneous velocity, w is its angular velocity about its center of mass, C,, and where the moment of inertia, Z, is Z = ?h m(A* + B*)
(3)
Assuming that the wheel rolls down the incline without slipping, its instantaneous velocity is v = Aw. Solving for instantaneous velocity as a function of height, h, yields: Rolling: v =
2g(ho - h) 5/2 3h + 54 - B2/A2
Free-fall: v = (2g(h, - h))“*
(4) (5)
Equations 1 and 2 state that, for the rolling wheel, kinetic energy is partitioned into both translational and rotational components, whereas for the falling wheel, kinetic energy is entirely translational. In solving for instantaneous velocity, we were able to cancel out mass for both systems, thereby revealing that mass is dynamically irrelevant in both rolling and free-fall contexts. An examination of Eq. 5 reveals the essence of point-particle systems:
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there is nothing in this equation that relates to the spatial properties of the moving object, The only object variable in the equation is the position, h, of the object’s center of mass. Thus, the act of looking at the object’s centroid draws attention to the single dimension required for dynamical analysis of the event. As is shown in Eq. 4, rolling is a context in which the wheel’s spatial properties do not cancel out. An aspect of the wheel’s shape-the ratio of its inner to outer radius, B/A, which is the inverse of what we will call compactness-influences dynamics in the rolling context. Compactness refers to the degree to which mass is distributed close to the wheel’s center of mass. The more compact the wheel, the faster it will descend the ramp. Moreover, note that the ratio, B/A, is invariant across overall changes in size. Whereas for the falling wheel, only center-of-mass position is of dynamical relevance, for the rolling wheel, two parameters of information must be combined in a dynamical representation. These parameters are the position of the wheel’s center of mass and the compactness of the wheel itself. Again, we note that this is the essence of the particle/ extended body motion distinction: particle motions reduce to centerof-mass displacements, whereas extended body motions require that these motions be combined with some spatial properties of the moving object. The distinction of object motion types depends not on whether the object is particulate or extended, but rather on the motions that it undergoes. Mechanical systems define the object parameters that are of dynamical relevance. In the simplest cases, particle motions, the only object parameter of dynamical relevance is the movement of the object’s center of mass. Extended body motions add more complexity to the dynamical understanding task. More than one category of information must be noticed, and since such dimensionally distinct quantities as shape and position cannot be added together, they must be combined through some sort of multiplicative operation resulting in a multidimensional quantity. The physical account given above is, of course, idealized. We do not propose that these equations actually represent the dynamical representations formed by observers. Rather, the exposition points out that, regardless of how people actually represent dynamical information, particle motion contexts require that they notice only where an object’s center of mass is going; extended body motions require that they notice this fact plus one or more other properties of the event, such as the object’s shape, size, mass, or orientation. Moreover, understanding extended body motions entails the additional requirement that these various dimensions be related in some fashion. With regard to human performance, we have proposed that, when mak-
346
PROFFITT,
KAISER,
AND
WHELAN
ing dynamical judgments about extended body motions, people do not spontaneously form multidimensional relationships; instead, people base their judgments on heuristics that relate to single parameters of information (Proffttt & Gilden, 1989). A clear example of this type of reasoning can be seen in how people dynamically evaluate collisions. Imagine a situation in which there are two billiard balls of unequal mass, and one of these balls, being initially stationary, is set into motion due to its being struck by the other. An analysis of the physical laws of momentum and energy conservation reveals that a kinematic relationship can be derived that specifies uniquely the mass ratio of these two balls. In particular, the postcollision projected velocities of the two balls onto an axis orthogonal to the precollision path of the striking ball defines a ratio equivalent to their relative masses. When asked to judge mass ratios while observing computer simulations of collisions, people do not spontaneously form this multidimensional relationship between velocities and angles; rather, they base their judgments on one of two heuristics that relate to these dimensions (Gilden & Proffitt, 1989). Their velocity heuristic can be stated as: following a collision, the faster moving ball is lighter. The other heuristic relates to deflection angle: after a collision, the ball that ricochets is lighter. Gilden and Proffttt found that people based their relative mass judgments on the heuristic related to the more salient dimension present in the event, either velocity or deflection angle. In many situations, such heuristical analyses led to good performance; however, when the two heuristics gave conflicting predictions, performance fell to an essentially chance level. Moreover, it was found that subjects never “averaged” the ricochet heuristic with the velocity heuristic to effect a compromise. Similar findings have been reported by Shepard (1964), who examined multidimensional judgments in a quite different context. Shepard required subjects to make similarity judgments for shapes that varied on two dimensions, size and the orientation of a feature. He found that subjects based their judgments on one or the other of these dimensions, and never related them. To summarize, we propose that dynamical judgments about extended body motions will be inaccurate because successful performance requires the formation of multidimensional quantities. Thus, people’s appreciation for natural dynamics will break down at the boundary between particle and extended body motions. It is, of course, true that people will sometimes exhibit erroneous reasoning when confronted with problems related to particle motions. However, there is a basic difference between their dynamical appreciation of these simple events and extended body motions. The dynamics of particle motions can be adequately penetrated by people if they are given the
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opportunity to view ongoing events; viewing extended body motions does not elicit more accurate intuitions. COMMON-SENSE
UNDERSTANDINGS
OF PARTICLE
MOTIONS
Recently, a literature has developed on people’s understandings of classical mechanics (Caramazza, McCloskey, & Green, 1981; Champagne, Klopher, & Anderson, 1980; Clement, 1982; Kaiser, Jonides, & Alexander, 1986; Kaiser, McCloskey, & Proffitt, 1986; Kaiser, Proffitt, & McCloskey, 1985b; McCloskey, 1983a, 1983b; McCloskey, Caramazza, & Green, 1980; McCloskey & Kohl, 1983; McCloskey, Washburn, & Felch, 1983). All of these studies presented problems dealing only with particle motion dynamics. In typical studies, people have been asked to predict the trajectories of balls rolled through curved tubes, of objects dropped from moving carriers, and of pendulum bobs whose connecting cords have been severed. For all of these events, the relevant dynamics can be assessed by considering the linear motion components of the objects’ centers of mass. Although these studies have typically been interpreted otherwise, we believe that they show that people are relatively competent with particle motion problems. We take this to be the case for four reasons. First, whereas early studies highlighted the tendency of college student populations to err on these problems, we note that typically the majority of subjects give correct responses. Second, when asked to reason about these problems in a familiar context, very few people make erroneous predictions (Kaiser et al., 1986a). Third, and of special significance, people demonstrate an excellent appreciation of particle dynamics when making judgments about ongoing events (Kaiser, Proffitt, & Anderson, 1985a; Kaiser & Proffitt, 1986). Those people who draw erroneous trajectories when shown pictorial representations of events almost always view simulations of their predictions as appearing anomalous, and judge correct trajectories as appearing natural. Finally, it should be noted that the events used in the intuitive mechanics studies are fairly complex exemplars of particle motions: external forces are applied and removed. The only experiments that have involved simple particle motions were developmental studies in which simple motions were used to ensure that young children understood the task (Kaiser et al., 1986b; Kaiser et al., 1985b). Here, it has been found that even a child of 4 years-of-age realizes that a ball exiting a straight tube rolls straight, and a dropped object falls straight down. We believe that the early studies by McCloskey and his colleagues (McCloskey, 1983a, 1983b; McCloskey et al., 1980; McCloskey et al., 1983) were so intriguing, in large part, because the errors were surprising to most of the reading audience. How, one had to ask, could a college
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sophomore with a high-school physics background actually think that a ball exiting a curved tube would continue to curve? In fact, we now know that that sophomore is unlikely to make such an error if asked to reason in a concrete context or asked to make judgments when viewing ongoing events. When viewing an ongoing particle motion event, the very act of looking at the moving object is simultaneous with noticing the single dimension that is of dynamical relevance. Although people sometimes make erroneous predictions about particle motions on paper-and-pencil tests, these same people exhibit near perfect performance when making dynamical judgments about ongoing events. As the following experiments demonstrate, the dynamics of extended body motions are not so easily penetrated. EXPERIMENT 1 Dynamical Understandings
of Wheels
Unlike particle motions, extended body motions require that more than one category of information be noticed and related in a multidimensional representation. We propose that this added complexity will be reflected in people’s ability to understand the dynamics of these events. We chose to assess people’s dynamical understandings of wheels because they are prototypical of extended body motions and are so commonplace in everyday experience. Method Subjecls. The subjects were 25 male and 25 female University of Virginia undergraduates who volunteered to participate in the study to fulfill a course requirement in an Introductory Psychology course. Thirty-five of these students had taken only a high-school physics course, five a college course, and ten had no physics training at all. Materials and procedure. Subjects were tested individually. The experimenter asked questions verbally and recorded the subjects’ verbal responses. Subjects were encouraged to ask questions if they did not understand the questions. Subjects were asked a large number of questions about natural dynamics. Fewer than half of these questions dealt with wheels. Filler questions were included to reduce the likelihood that the subjects would realize that the study was focused on their understanding of wheel dynamics. All of the filler questions were accompanied by drawings that depicted the situations being described. The structure of these questions was similar to that of the wheel questions. After each answer, subjects were asked to provide a brief explanation for their answer. The gist of these explanations was written down by the experimenter. There were four sets of questions related to wheel dynamics. They probed subjects’ understandings of the following: Factors affecting the rate that wheels roll down an inclined plane. An inclined plane was constructed, the vertical and horizontal dimensions of which were 0.23 m and 1.07 m, respectively. Four wheels were made to vary in the parameters of mass, radius, and distribution of mass. The wheels were constructed so that the set contained pairs of wheels that held two of each of these parameters constant and varied a third. Three wheels had a weight
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of 805 g, and the other weighed 2310 g; three wheels had a radius of 7 cm, and the other a radius of 3.3 cm; finally with respect to mass distribution, two wheels were solid disks (uniform mass distribution) and the other two were rims (mass distributed close to perimeter). For each pairing, the subjects were handed two wheels and told how they were the same and how they were different. For example, they might be handed two wheels of identical weight and mass distribution, but one wheel had a smaller diameter than the other. They were then asked: “If released from the top of the inclined plane at exactly the same time, will both of these wheels arrive at the bottom at the same time, or will one reach the bottom before the other, and if they descend at different rates, which wheel will reach the bottom first?” Subjects were asked to provide an explanation for their answer. After making all three judgments, subjects observed each pair rolling down the inclined plane, and were then asked to provide an explanation for the outcome that they observed, especially for those cases in which they had made an erroneous judgment. The trajectories of points located on a rolling wheel. A drawing was made that depicted a wheel rolling on a horizontal surface. Three points were drawn on the wheel: one on the circumference, one at the wheel’s center, and one midway between the center and the circumference. Subjects were given a copy of this drawing and asked to draw the motion paths followed by these three points as the wheel rolled. Bicycle stability. Subjects were asked to think back on their experiences riding bicycles and to recall that a bicycle is fairly difficult to balance when it is stationary or barely moving; however, when ridden a moderate speed, the bicycle becomes much easier to balance. They were then asked: “Why is a bicycle easier to balance when it is moving rapidly than when it is stationary or moving very slowly?” Gyroscope dynamics. Subjects were shown a toy gyroscope. A gyroscope is a wheel that is free to revolve around an axis. It is a kind of top: the axis of the gyroscope corresponds to the symmetry axis of the top; their dynamical behavior is equivalent. Subjects were shown the gyroscope and asked whether it could be made to balance on a pedestal in three different positions: axis aligned vertically, at a 45” angle, and horizontally. These questions were asked twice-once for a nonspinning gyroscope and once for one that was spinning rapidly. The gyroscope was then set into rotation and placed on the pedestal. Subjects observed the gyroscope balance on the pedestal at each of the aforementioned orientations. They were then asked to explain how the gyroscope was able to maintain its balance when it was spinning but not when it was not spinning.
Results and Discussion The results for each of the four sets of wheel questions are presented in turn. Factors affecting the rate that wheels roll down an inclined plane. As discussed in the introduction, mass and radius do not influence the time that it takes a wheel to roll down an inclined plane. Mass distribution, however, is a relevant variable. The more compact the wheel, the faster will be its rate of descent. Thus, a solid disk will descend a ramp at a faster rate than will a rim. Figure 2 presents the percentage of subjects giving each of the three possible responses for the manipulations of mass, radius, and mass distribution. On the radius problem, the distribution of subjects’ responses did not differ significantly from random guessing [x*(2) = 3.16, p > . lo].
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PROFFITT, KAISER, AND WHELAN
SAME
LIGHT
MASS
HEAVY
SAME
SMILL
RADIUS
LPIRGE
SAME
RIM
DISC
DISTRIBUTION
FIG. 2. The proportion of naive subjects giving each of three possible responses for the manipulations of mass, radius, and mass distribution.
The response distribution on the mass problem was nonrandom [x2(2) = 8.93, p < .05], due to subjects’ tendency to judge the heavier wheel as faster. Over 80% of the subjects failed to realize that distribution would affect the speed of the wheel. This systematic error lead to a highly nonrandom distribution of responses [x2(2) = 53.56, p < .OOOl].No effect on response distribution was noted for formal physics training or the gender of the subjects. As these analyses demonstrate, subjects did not show a group preference for any of the three possible responses when judging the influence of radius, and only a slight bias for the heavier wheel. The subjects performed at or near chance levels on these questions about irrelevant variables. The only strong preference that the subjects showed was a belief that mass distribution does not affect the rate that a wheel descends an inclined plane. Thus, the only shared belief held by these subjects was an erroneous one: mass distribution, the only relevant variable in this situation, was consistently judged to be irrelevant. After observing the wheels roll down the ramp, the subjects all came to the realization that mass distribution was the only influential variable. In general, their accounts of why mass distribution was influential simply described the particular wheels employed, and failed to elucidate general principles. The trajectories of points located on a rolling wheel. There was an enormous variety of responses given to the task of drawing the motion
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paths for three points located on a rolling wheel. As depicted in Fig. 3 for one cycle, the responses were categorized into six types of curves: (1) cycloid-connected arches; (2) prolate cycloid-wavy lines; (3) straight line-horizontal lines; (4) curtate cycloid-loops; (5) connected straight lines-straight lines of different orientations meeting at vertices (for two wheel rotations, these drawings looked like the letter W); (6) circle. Drawings were classified quite liberally; for example, any wavy line such as a sine curve was classified as a prolate cycloid. If there was an ambiguity as to how a drawing should be classified, then when appropriate, it was assigned to the correct category. For example, if a subject was attempting to draw the trajectory of a point on the rim of the wheel, and it was unclear whether the drawing should be classified as a cycloid or a prolate cycloid, then it was assigned to the correct cycloid category. Correct trajectories are also depicted in Fig. 8. Subjects’ responses are presented in Table 1. Summarizing from this table, only 18% of the subjects drew curves that could be classified as an appropriate cycloid for the point on the perimeter; 16% correctly drew wavy curves that could be liberally classified as prolate cycloids for the point midway between the circumference and the center; however, 94%
Cycloid
Prolate Cycloid
Straight Line
.
Curtate Cycloid
Connected Straight Lines Circle FIG. 3. Six curves used to categorize subjects’ drawings of the trajectories of three points on a rolling wheel.
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PROFFITT, KAISER, AND WHELAN
TABLE 1 Percentage of Subjects Drawing Each Category of Curve for Points Located on Perimeter, Intermediate Between the Perimeter and the Center, and at the Wheel’s Center, Respectively Curve
Cycloid Prolate cycloid Straight line Curtate cycloid Connected straight lines Circle
Perimeter
Intermediate
Center
18* 14 2 44 6 16
12 16* 2 52 4 14
94* 2 4
* Correct curve.
of the subjects correctly drew straight lines for the point at the wheel’s center. Not one subject drew curves that could be classified as correct for all three points. Bicycle stability. Bicycles are very complex machines, and a full stability analysis has not been performed (Jones, 1970; Kirshner, 1980; Lowell & McKell, 1982). The basic factors, however, are well understood. In particular, the wheels of bicycles are known to act as gyroscopes. When, for example, a rider leans to the right, this torque adds to existing angular momentum in a manner that will cause the front wheel to turn to the right, and thereby produce a torque on the wheel that acts against the lean. The essential gist of this explanation is: leaning causes the bicycle to turn; turning results in forces that act against the lean. This sequence of events is controlled by the gyroscopic nature of the wheels; the rider need not even hold onto the handle bars for the turning and righting to occur. Subjects’ responses to the question about why a moving bicycle is easier to balance than one that is stationary fell into four categories: (1) 52% of the subjects reasoned that moving forward made the bicycle harder to tip over. These references to forward momentum made no mention of how it related to turning. That is, linear momentum is influential only after a bicycle has turned in response to a lean. Thus, subjects in this category did not demonstrate an understanding of the manner in which linear momentum relates to stability. (2) 24% stated that the rider’s movements and weight distribution produced the increased stability found in moving bicycles. This is wrong. (3) 8% suggested that the wind builds up on the sides of the bicycle and rider as they move, and this wind holds them up. This, of course, is wrong. (4) The remaining 16% of the subjects gave idiosyncratic responses. All of these latter responses were incorrect but one. One subject stated that the wheels acted like gyroscopes. This subject had noticed a gyroscope that had been inadvertently left out in sight, and made the correct analogy. When asked how the
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gyroscopic properties of bicycle wheels functioned to produce stability in the bicycle, this subject had no idea. Moreover, when asked about the behavior of gyroscopes in the next question, this subject was as mystified as the other subjects. Gyroscope dynamics. The physical explanation of gyroscopic motion is difficult to describe in words. In essence, gravity causes a spinning gyroscope to precess; that is, the principle axis of a spinning gyroscope revolves sideways in a gravitational field. Gravity produces a torque on the gyroscope, pulling it down. This downward torque results in precession, a slow rotation of the gyroscope’s axis around its pedestal. This precession is the falling motion of a spinning gyroscope. Thus, the important motions to notice are not only the spinning of the gyroscope’s wheel, but also the precession of its central axis. When asked whether a nonspinning or spinning gyroscope could balance on a pedestal at three different orientations, all of the subjects correctly stated that a nonspinning gyroscope could not balance in any of these positions. For the spinning gyroscope, 100% felt that it could balance in an upright position, 60% at a 45” angle to the pedestal, and 28% at a horizontal orientation. After observing a spinning gyroscope balance at all three orientations, the subjects were asked to account for the different behavior of a spinning and a nonspinning gyroscope. Although all of the subjects admitted that they really did not know, when asked to propose an account, they gave reasons that fell into live categories: (1) 36% stated that the spinning wheel produced an upward force. (2) 14% suggested that it was due to centrifugal or centripetal force. These subjects did not know what these terms meant but did know that they referred to spinning objects. (3) 4% mentioned angular momentum, but as with the above responses, these subjects did not know what this term meant in relation to the behavior of a gyroscope. (4) 4% of the subjects suggested that the gyroscope acted like an airplane propeller. (5) The remaining 42% gave idiosyncratic responses. Clearly, none of the subjects had any intuitions about how the gyroscope is influenced by gravity. Although all of the subjects knew that the behavior of the gyroscope was affected by the spinning of its wheel, not one subject mentioned the precessional motion. This most relevant motion was completely overlooked by the subjects. EXPERIMENT 2 Understanding Mass Distribution’s Influence on Rotational Dynamics: An Assessment in an Ongoing Context The previous experiment demonstrated that people typically have no
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explicit awareness of mass distribution’s influence on the dynamics of a rolling wheel. This experiment was designed to assess whether viewing ongoing displays would evoke more accurate intuitions. Observers were shown a computer-graphics animation of a satellite spinning in space. This event is depicted in Fig. 4. The satellite changed its mass distribution by opening or closing a set of solar panels, an event that is analogous to a spinning ice skater who extends or contracts his or her arms. In a natural situation, the opening of the satellite’s solar panels would cause its spinning rate to slow, whereas closing the panels would result in an increased angular velocity. In the animated stimulus displays the opening and closing of the panels resulted in a variety of spin rates. The observer’s task for each sequence was to judge whether the resulting angular velocity was the natural outcome of the satellite’s changing shape, or whether it could only have been produced by some unseen external force. This display presented the simplest context in which mass distribution’s influence on rotational dynamics could be assessed. The satellite’s dynamics are fully specified by the law of angular momentum conservation. Angular momentum is defined as follows: L = zw
(6)
where L is angular momentum, Z is moment of inertia, and w is angular velocity. Comparing this equation with Eq. 1 reveals the simplicity inherent in the satellite’s dynamics. Unlike the rolling context, here rotational dynamics is not coupled with linear motions. Moreover, no external forces are acting on the object. Since angular momentum is conserved when the satellite changes shape, angular velocity is an inverse function of its moment of inertia: The greater the increase (decrease) in I, the greater the decrease (increase) in w. Moment of inertia weights the mass of every object particle by the
A
B
C
FIG. 4. A spinning satellite in three configurations: solar panels open, intermediate, and closed.
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square of its distance to the axis of rotation. More precisely, moment of inertia is defined as:
i=l
where mi is the mass of the ith particle, and Ri is the distance of that particle from the axis of rotation. For the satellite used in our study, rotation was about an axis passing through its center of mass and orthogonal to the plane defined by the four panels’ bases. Panels were defined as having uniform density, and the satellite consisted of only these panels; its interior was empty. Moment of inertia is defined by the shape and orientation of the panels as follows: I = (% b3h + 2 bha* + 2 abh’ sin 0, + ?A bh3 sin 2 6&z/8 bh (8) where a is the distance between the midpoint of each panel’s base and the center of rotation, b is half the width of each panel, h is the length of each panel, 8, is the angle of the panels (absolute value), at time t, relative to the vertical closed position, and m is the satellite’s total mass. The quantity, m/8bh, defines the mass density of the panels. It is a constant that does not affect the changes in angular velocity that occur when the satellite changes shape. We make two observations concerning this equation. First, the determination of the satellite’s moment of inertia entails far more complexity than was found for the wheel. The high degree of symmetry inherent in wheels allows their moment of inertia to be expressed by the simple relationship shown in Eq. 3. In general, the specification of moment of inertia requires considerably more geometry, especially if the object does not possess uniform density. Even for our relatively symmetrical satellite, we doubt that anyone would propose that moment of inertia could be accurately assessed by perception, and thus, a precise appreciation for the influence of the satellite’s changing shape on angular velocity could not be expected. Compared with particle motions, extended body motions are more complex because many object parameters must be noticed and related. The second observation is that a qualitative appreciation for the relationship between panel orientation and angular velocity can be based on the following simple heuristic: when the panels open, the satellite’s spin rate must slow; when they close, spin rate increases. The following experiment was designed to assesswhether people spontaneously notice the dynamical significance of this relationship between mass “compactness” and angular velocity when viewing ongoing events.
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Method Subjecrs. Twelve University of Virginia undergraduates (six mates and six females) volunteered to participate in the study to fulfill a course requirement in an Introductory Psychology course. Stimuli. All stimuli were viewed on a Tektronix 4129 3-D Color Graphics Workstation with a high resolution (4 lines/mm) display screen 30 cm high X 35 cm wide. The stimuli consisted of a graphic simulation of a spinning satellite. As is shown in Fig. 4, the satellite consisted entirely of four color-filled panels. These panels were described as being solar panels of uniform density. The display’s background consisted of random points that slowly drifted together, from right-to-left, across the screen. These points were described as being stars. Except for its empty interior, the satellite occluded the “stars” as it revolved in front of them. The satellite’s initial appearance on the screen always presented the solar panels in either a fully open or closed position. The satellite remained in this state until the subject pressed a key, whereupon the satellite closed or opened its panels, depending upon its initial state. The stimulus set consisted of seven pairs of events. Seven of these events began with the satellite in a panels-closed state, and presented a variety of angular velocity changes as the panels opened. The other seven stimulus events were matched reversals of the motions presented in the first set. We describe each stimulus pair relative to the motions that occurred as the panels opened for the initially panels-closed member. Stimulus 1 slowed to a stop, reversed spin direction, and achieved a final angular velocity equivalent to, but in the opposite direction from, its initial velocity. Stimulus 2 slowed to a stop. Stimulus 3 slowed too much, achieving a final angular velocity intermediate between a stopped and the canonical outcome. Stimulus 4 was the canonical event. Stimulus 5 did not slow enough, achieving a final angular velocity intermediate between the canonical event and its initial velocity. Stimulus 6 did not change its angular velocity. Stimulus 7 sped up to a final angular velocity equivalent to 3/zits initial velocity. Panels-closed stimuli had an initial angular velocity of 0.3 Hz, and panel opening was always accomplished in 2 s. Angular velocity changes were achieved via a constant acceleration during the opening interval. Procedure. Subjects were tested individually. They were told that they would be observing a simulated satellite, spinning in space. Pressing a key on the computer’s keyboard would allow them to change the state of the satellite by opening or closing its solar panels. They were told that sometimes the satellite would behave naturally; however, sometimes its movements would be possible only if it had been influenced by some unseen force, such as an invisible thruster. Their task was to judge whether the motions that occurred during the opening or closing of the solar panels were natural or would have required the application of an unseen force. Following these instructions, they were shown a random order of all 14 stimuli, in order to acquaint them with the events that they would be judging. They were then instructed that they were to make their judgments by adjusting a thumbwheel that moved a marker along a scale at the bottom of the screen. The subjects were told to use the scale relative to the set of satellites that they had just seen, placing the marker at one extreme of the scale for the most natural appearing event, and at the other extreme for the event that had most obviously been influenced by an unseen force. They were then given four practice trials with events randomly drawn from the set of 14 stimuli. Subjects were then allowed to ask questions and to proceed with the test trials. There were 28 test trials consisting of two presentations of each stimulus. The 14 stimuli were presented in two randomly ordered blocks that differed for each subject. The random orders were restricted by the qualification that no stimulus be contiguous with itself or with its opening/closing opposite. On each trial, the spinning satellite would appear in one of its open or closed states. The subject observed the satellite, and at a time of his or her own
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choosing, pressed a key causing the satellite to change states. Once the satellite had completed its transformation, the subject adjusted the scale and terminated the trial by pressing a key.
Results and Discussion
Observers demonstrated only a qualitative appreciation for the preservation of spinning direction, and no sensitivity to the influence of mass distribution on angular velocity. Figure 5 shows the mean ratings for the seven opening/closing pairs. The ratings have been placed on a lo-point scale, 10 being natural and 0 indicating the application of an unseen force. An ANOVA revealed that two of the stimulus pairs-stimuli in which changing shape resulted in a reversal in the direction of spin, or resulted in a stopping or starting of the spin-were significantly different from the other five stimulus pairs [F(1,60) = 58.4, p < .OOOl]. In addition, the ratings for the reversal stimuli were reliably lower than that for the starting/stopping stimulus pair (ANOVA was followed by a Scheffe test [p < .OS]).None of the ratings given to the other five stimuli differed from each other (Scheffe test). There was no effect for direction of transformation, opening versus closing stimuli, nor for gender. Analyses were also performed on ratings that had been normalized within subjects to obtain a common mean and variance; their outcomes were equivalent to those presented for the raw ratings. These results indicate that observers have an awareness that changing Natural
61
I
Reverses
stops
Slows too Canonical much
Slows to0 No change Speeds Up little
Event
FIG. 5. The naturalness ratings given to the satellite’s angular velocity changes resulting from configuration change. For each stimulus pair, the description provided is for the motions that occurred as the panels opened for the initially panels-closed member. For example, for the right-most category, “Speeds Up,” the opposite member of this stimulus pair would “Slow Down” as the satellite’s panels closed from an initially open position.
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the shape of a spinning object cannot, by itself, cause it to change its direction of spin, or to stop spinning (or start to spin if initially stationary); however, they have no spontaneous appreciation for the influence of mass distribution on rate of spin. For the initial panel-closed stimuli, events that slowed as the panels opened were judged to be no more natural than those that did not change in angular velocity or sped up. Similar findings were found for the initial panel-open stimuli. These results converge with those found in the previous experiment where we asked people to judge the influence of mass distribution on the rate that wheels descend an incline plane. There, as here, observers demonstrated no appreciation for the influence of mass distribution on rotational dynamics. Unlike the previous study, here people made judgments while viewing an ongoing event. For particle motion problems, animation has been found to evoke far more competent dynamical judgments (Kaiser & Proffitt, 1986; Kaiser et al., 1985a). For this extended body motion situation, however, we found that people remained unaware of the dynamical influence of mass distribution. We had intended to design an experiment to assess whether people would exhibit a greater implicit appreciation for gyroscope dynamics when viewing animated simulations of natural and anomalous versions of this event. The results of this study, however, suggested that there would be little to be gained from such a study. People would surely not perform better with the gyroscope than they did with the far simpler satellite. We did, however, create a computer simulation of a top that was shown to a number of pilot subjects. These observers found the task of distinguishing between possible and impossible gyroscopic motions to be far too difficult, performed randomly, and became extremely agitated and upset by the task. To have completed the study would have only served to demonstrate the obvious and to demoralize a number of observers. EXPERIMENT Understanding
3
the Trajectories of Points on a Rolling Assessment in an Ongoing Context
Wheel: An
In the first experiment we asked subjects to draw the trajectories of three points on a rolling wheel. One point was at the wheel’s center, one on the perimeter, and one intermediate between these two. Although the center point was typically drawn correctly, for the perimeter and intermediate points, it was found that drawings could rarely be classified as resembling the actual trajectories. In this experiment, we assessed observers’ ability to recognize these trajectories when shown computer animated point-lights, moving as if attached to an unseen rolling wheel.
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Method Subjects. The subjects were 24 University of Virginia undergraduates (12 male and 12 female) who volunteered to participate in order to fulfill a requirement in an Introductory Psychology course. Stimuli. The stimuli consisted of single point-lights moving along five trajectories on a black computer screen. These point-light motions were created on the Tektronix 4129 Workstation (4 lines/mm, display screen 30 cm high x 35 cm wide) and were viewed from an approximate distance of 45 cm. The five trajectories were the three wheel-generated motions-cycloid, prolate cycloid, and straight line-plus a curtate cycloid and a sine curve. Horizontal excursion for all trajectories was 31” of visual angle. Since for a given wheel size, amplitude of vertical motion could be used to distinguish between the wheel-generated motions, each of the four curved trajectories was presented at four amplitudes: vertical displacements of 2, 4, 6, and 8 cm (maximum visual angle, 6.5”). This manipulation created a range in the number of cycles per excursion from 1 (for the 8-cm prolate) to 12(for the 2-cm curtate). (Relating size and number of cycles could have served as a bases for distinguishing between the wheel-generated motions; however, we found no evidence that any subject attempted this strategy.). Procedure. Four straight-line trajectories were combined with the 16 curved trajectories (four sizes for each of the four curves) for a total of 20 stimuli. For each subject, these 20 stimuli were presented in four randomized blocks, thereby producing a set of 80 trials. Randomizations were different for each block and each subject. Although the subjects were not so informed, the first block served as practice, and was not included in the data analysis. Subjects were seen individually. They were informed that they would be observing the motions of single point-lights. Some of the motions would correspond to those of a point at one of three different locations on a rolling wheel, others would follow paths that could not be generated by a point on a rolling wheel. At the bottom of the terminal’s screen were depictions of three wheels, each having a dot at one of the three appropriate locations: perimeter, intermediate, and center. Also present at the bottom of the screen was a symbol designating the categorical response for a trajectory that was not wheel-generated. Whenever the subjects observed what they judged to be one of the wheel-generated motions, they were instructed to move a cursor to a position below the appropriate wheel showing the light’s location. Whenever they observed a trajectory that did not appear to have been wheel-generated, they were to move the cursor below the “other” symbol. After each trial, subjects were asked to estimate their confidence by moving a cursor on a scale that appeared below the response categories. On each trial, the point-light moved twice across the screen before the mouse controlling the cursor was enabled.
Results and Discussion Each subject’s responses were classified relative to the curve that was most frequently being observed when he or she gave each of the four responses: perimeter, intermediate, center, and other. Confidence ratings were used to break ties. There were 9 out of a possible 96 ties. Table 2 presents the percentage of subjects for each response-by-stimulus classification. Subjects performed slightly better at this task than in Experiment 1 where they drew the trajectories of points on a rolling wheel. Forty-six percent of the subjects correctly categorized a cycloid as being the perimeter point in this ongoing simulation, whereas only 18% of the subjects
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2
Percentage of Subjects Responding to Each Category of Curve by the Trajectory That They Were Actually Observing Answer Curve seen Cycloid Prolate cycloid Straight line Curtate cycloid Sine Category never used
Perimeter
Intermediate
Center
Other
46* 33 13
13 s* 67 13
96* 4 -
4 38 4 8* 38*
8
-
-
Note. Two subjects never used each of the categories, “PERIMETER” If columns do not sum to zero, it is due to rounding. * Correct curve.
8 and “OTHER.”
in Experiment 1 correctly drew cycloids for the perimeter point [x2 (1) = 6.33, p < .025]. On the other hand, performance remained poor for the intermediate point with only 8% of the subjects correctly categorizing the prolate cycloids as being intermediate in the ongoing situation as compared with 16% who correctly drew this curve in Experiment 1 [x2 (1) = 0.82, p > S]. Only 2 subjects correctly categorized all three wheelgenerated trajectories when viewing the animated displays. That overall performance remained low when subjects were viewing ongoing displays indicates that the source of wheel-generated trajectories is not obvious. This was especially the case for the intermediate point. Although performance improved for the perimeter point relative to that found in Experiment 1, it remained at a low level. EXPERIMENT 4 Dynamical Understandings
of Wheels by Bicycle Racers
Returning to the issues raised in the first experiment, there are a number of possible reasons why people have such poor understandings of wheel dynamics. Our contention is that these motions are more complex than are the particle motions for which people demonstrate far better competence. Another possibility, that we do not favor, is that people have less experience with wheels than they do with particle motions. We do not believe this to be so. Wheels are observed very frequently in our society. Jearl Walker, author of “The Amateur Scientist” column in Scientific American, has published a collection of articles on the physics of rotation in the everyday world (Walker, 1985). In the Preface to this collection, he writes: “Rotation is fascinating because in spite of its common occurrence it is
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difficult to understand . . . For example, when I spin a top I am always amazed that it stands upright in spite of the gravity tugging at it. Even when it leans over, it does not fall, instead it precesses, that is, the central axis about which it spins circles around the vertical. Such motion is magic” (p. vii). In this experiment we sought to assess dynamical understandings of wheels with individuals who were especially familiar with these motions-bicycle racers. We predicted that these cyclists would not possess appreciably better understandings of wheel dynamics than those demonstrated by subjects in the previous experiment. Method Subjects. Members of the University of Virginia Bicycle Club were contacted by phone and asked to volunteer to participate in the study. Fifteen male and 10 female cyclists were recruited in this manner. These cyclists reported that they rode their bikes an average of about 100 miles per week, the range being from 10 to 250 miles per week. Two of these subjects had no physics training, 11 had physics only in high school, and the remaining 12 cyclists had taken at least one semester of college physics, the range among this latter group being from 1 to 4 semesters. Unlike the subjects in the previous experiment who were predominantly first and second year students, the cyclists tended to be third and fourth year students with more course work in physics. Materials and procedures. The procedures were identical to those of Experiment 1 except that more detailed questions were asked about the behavior of bicycles. The cyclists were asked the same question about why a moving bicycle is easier to balance than a stationary one. Following this question they were then asked: “Suppose that you are riding your bicycle and holding your hands over your head. You lean to the right without touching your handlebars. What will happen?” If the cyclist answered that the bike would turn, he or she was then asked what caused the bike to turn. If the subject stated that the turn was caused by the front wheel turning, then he or she was asked why the front wheel turned. Finally, for those cyclists who reported that the front wheel would turn, the experimenter also asked whether the wheel would turn if the bicycle was not moving.
Results and Discussion Factors
affecting
the rate that wheels roll down an inclined
plane.
Figure 6 shows the proportion of subjects selecting each of the three responses for the three wheel pairings. An analysis of these results showed that the distribution of responses was random for both the mass and radius problems [x2 (2) = 0.55 and 3.91, respectively.]. For the distribution problem, subjects were biased toward responding that the wheels would roll at the same rate [x2 (2) = 8.00, p < .05). The cyclists’ response distributions did not differ from the subjects in Experiment 1 on the mass, radius, or mass distribution problems. (Although not significant, the cyclists were somewhat less likely than the subjects in Experiment 1 to erroneously judge mass distribution as being irrelevant [x2 (2) = 4.75, p < .lO]). As this analysis shows, the cyclists responded in a similar manner to
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SAME
LIGHT
SAME
HEAVY
SMILL
LARGE
SAME
RADIUS
MASS
RIM
DlSC
DISTRIBUTION
FIG. 6. The proportion of bicycle racers giving each of three possible responses for the manipulations of mass, radius, and mass distribution.
the subjects in Experiment 1. Both groups showed the greatest consensus in judging that mass distribution is not a factor influencing the rate of descent for a wheel rolling down an incline. The trajectories of points located on a rolling wheel. The subjects’ drawings were classified as follows: connected arches were classified as cycloids, loops as curtate cycloids, wavy lines as prolate cycloids, and straight lines as themselves. The subjects’ responses are summarized in Table 3. Thirty-six percent of the cyclists correctly drew cycloids for the outer point, 24% correctly drew prolate cycloids for the middle point, and all of the subjects drew a straight line for the center. Only one subject drew all three of these curves correctly. Bicycle stability. The cyclists’ responses fell into four categories: (1) 28% of the cyclists gave sophisticated responses that made reference to the gyroscopic properties of the bicycle wheels. (2) 20% stated that the stability was due to the fact that the wheels were revolving, but gave erroneous accounts for the nature of this effect. (3) 48% of the subjects TABLE 3 Percentage of Subjects Drawing Each Category of Curve for Points Located on Perimeter, Intermediate between the Perimeter and the Center, and at the Wheel’s Center, Respectively Curve
Perimeter
Intermediate
Center
Cycloid Prolate cycloid Straight line Curtate cycloid
36* 20 44
16 24* 60
100* -
* Correct curve.
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reasoned that moving forward made the bicycle harder to tip over. These references to forward momentum made no mention of how it related to turning, and thus, did not demonstrate an understanding of the manner in which linear momentum relates to stability. (4) The remaining subject gave an idiosyncratic response. Although the majority of cyclists, 72%, gave erroneous accounts for bicycle stability, their responses were much more adequate than were those of the subjects in Experiment 1. Fortyeight percent of the cyclists believed that bicycle stability was related to the motion of the wheels, whereas only one of the previous subjects made this attribution. In response to the question about what happens when you lean to the right while riding a bicycle with your hands off of the handlebars, 92% of the cyclists correctly stated that you would turn to the right; the other 8% judged that you would turn left. Fifty-six percent of the cyclists knew that the bicycle turned right because the wheel turned right, and 12% knew that the wheel turned right because of its angular momentum. Of these latter subjects, two of the three knew that the wheel would turn to the right only when the bicycle was moving. These two subjects had taken two and three semesters of college physics, respectively. Gyroscope dynamics. When asked whether a nonspinning or spinning gyroscope could balance on a pedestal at three different orientations, all of the subjects correctly stated that a nonspinning gyroscope could not balance in any of these positions. For the spinning gyroscope, 100% felt that it could balance in an upright position, 76% at a 45” angle to the pedestal, and 36% at a horizontal orientation. After observing a spinning gyroscope balance at all three orientations, the subjects were asked to account for the different behavior of a spinning and a nonspinning gyroscope. They gave reasons that fell into four categories. (1) 20% stated that the spinning wheel produced an upward force. (2) 16% incorrectly suggested that it was due to centrifugal or centripetal force. (3) 8% mentioned angular momentum but could not explain how this concept applied to gyroscopes. (4) The remaining 56% gave idiosyncratic responses. Of these subjects, one mentioned precession and described a demonstration given in one of her physics classes. This subject could not account for how precession resulted from gravity’s influence on the spinning gyroscope. As was found for the subjects in Experiment 1, the cyclists knew that the behavior of the gyroscope was affected by the spinning of its wheel, but only one cyclist mentioned the relevance of precessional motion. Although the cyclists had considerable experience with gyroscopic motions, their understanding of gyroscope dynamics was limited to a knowledge of a few terms memorized from previous physics courses. Across the different questions, the cyclists did not exhibit an apprecia-
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bly better understanding of wheel dynamics than did the subjects in Experiment 1. Their accounts of bicycle stability were somewhat better; however, this advantage did not generalize to their appreciation of gyroscope stability.We take these findings to imply that the poor competence found in both groups is not simply due to a lack of familiarity with wheel motions. EXPERIMENT 5 Dynamical Understandings of Wheels by High-School Physics Teachers What is the relationship between an individuals’ explicit knowledge of the physics underlying wheel dynamics, and his or her intuitions about the sorts of problems presented in Experiments 1 and 4? Do adequate intuitions about wheel dynamics follow from an explicit understanding of classical mechanics? In pilot work, we individually asked four Ph.D. physicists the problem about wheels rolling down the inclined plane. Their replies made it immediately clear that their common-sense understandings of wheel dynamics differed little from our naive subjects. Although it seemed that they were less likely to assume that mass would affect rate of descent, they guessed on the radius problem, and all four were incorrect for the mass distribution question. Clearly, they could have solved the problems if given sufficient time and writing materials, thereby allowing them to set down the necessary equations to analytically derive the answers. However, we prohibited them from doing so by requiring that they answer quickly. We were struck by the apparent finding that a deep and thorough understanding of classical mechanics did not more immediately impact on their understanding of wheel dynamics. The physicists, however, protested that they were the wrong group to test. Classical mechanics is not an abiding concern of contemporary academic physicists; they argued that they had thought very little about this domain in recent years. They suggested that, if anyone had a practiced working knowledge of the simple classical mechanics principles that our studies were investigating, it would be high-school physics teachers. Method Subjects. The subjects were 19 (10 male and 9 female) high-school physics teachers. They were attending a IO-day institute on physics instruction at the University of Virginia during the summer of 1987. This institute served the teachers’ needs to meet various State of Virginia teacher accreditation requirements. The following background information was obtained from the teachers via self report. Their average age was 40.6 years. Sixteen had obtained B.S. degrees, whereas the other
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three had obtained B.A.?,. Five had obtained an MS. degree, and four had acquired an M.Ed. They reported having the average number of semesters of preparation in the following subject areas: Physics, 6.5; Mathematics, 6.4; Chemistry, 4.9; and Biology, 2.6. They reported an average of seven years experience in teaching physics courses. Materials and procedure. The subjects were tested in a group session that lasted about an hour. They were asked a variety of questions about physical problems including all of those that were asked of the bicycle racers. We asked them to base their answers on their immediate intuitions, and not to attempt analytical solutions. The same wheels, inclined plane, and gyroscope were shown as in Experiments 1 and 4. Two additional questions were asked of the physics teachers. First, they were told that they would be shown a spinning gyroscope balancing on a pedestal. They were to watch the event, and then report: “To what would you tell students to attend when watching this event, so that they would better understand the behavior of gyroscopes when it was explained to them?” That is, the teachers were asked to report on what gyroscopic motions should be noticed in order to better understand an explanation of its dynamics. Second, they were asked: “When teaching mechanics to your students, what are the three most difficult concepts for them to understand?”
Results and Discussion Factors affecting the rate that wheels roll down an inclined plane. Compared with naive undergraduates, the physics teachers were more likely to be aware that mass distribution affected the rate of descent, but were uncertain as to what this effect should be. Figure 7 shows the proportion of subjects selecting each of the three responses for the three wheel pairs. An analysis of these results showed that the distribution of responses did not differ from random guessing for all three problems: mass, radius, and mass distribution [x2 (2) = 1.4, 0.1, and 1.4, respectively]. As poor as this performance was, it did reflect an improvement over the naive subjects in that the physics teachers were less likely than the subjects in Experiment 1 to erroneously judge mass distribution
SAME
UOHT
MASS
HEAVY
SAME
SYAU.
RADIUS
URGE
SIUE
RIM
DlSC
DISTRIBUTION
FIG. 7. The proportion of high-school physics teachers giving each of three possible responses for the manipulations of mass, radius, and mass distribution.
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as being irrelevant. The response pattern across all three response categories differed between the naive subjects and the physics teachers [x2 (2) = 10.72,p < .005]. Be that as it may, only 21.1% of the teachers correctly predicted the influence of mass distribution on rate of descent, reflecting, at best, only a marginally significant improvement in correct performance as compared with the naive subjects [x2 (1) = 3.54, p < .lO]. The trajectories of points on a rolling wheel. The subjects’ responses were classified and are summarized in Table 4. Performance remained relatively poor for this group. Forty-seven percent correctly drew cycloids for the outer point, 16% correctly drew wavy lines for the intermediate point, and all but one drew a straight line for the wheel’s hub. Only one subject correctly drew all three curves. Bicycle stability. The physics teachers did well with the initial questions about bicycle stability; however, as was the case for the bicycle racers, they exhibited little depth of understanding when questioned about specific details. Sixty-eight percent of the teachers correctly responded that a bicycle is easier to balance, when moving at a rapid as opposed to a very slow rate, because of the rotation of the wheels. Many of these teachers referred to notions of angular momentum and gyroscopic principles. When asked what happens when you lean to the right on a moving bicycle, 68% correctly responded that the bicycle will turn to the right. However, only 21% (four teachers) reported that the bicycle turned to the right because its wheel turns in response to the lean, and of these four teachers, only two provide an adequate mechanical reason for why the wheel turned. Finally, only one of these two subjects correctly stated the wheel would turn right in response to a rightward lean only when the bicycle was moving. Gyroscope dynamics. When asked about whether a nonspinning gyroscope could be balanced on a pedestal at three different orientations, 90% TABLE 4 Percentage of Subjects Drawing Each Category of Curve for Points Located on Perimeter, Intermediate between the Perimeter and the Center, and at the Wheel’s Center, Respectively Curve
Perimeter
Intermediate
Center
Cycloid Prolate cycloid Straight line Curtate cycloid Circle Other
47* 5 5 21 11 11
16 16* 42 16 11
95* 5
Note. If columns do not sum to zero, it is due to rounding. * Correct curve.
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correctly reported that it could not balance in an upright position, and 95% likewise reported correctly for the other two orientations. For the spinning gyroscope, 90% stated that it could balance in an upright position, 7% at a 45” angle, and 68% at a horizontal orientation. The teachers were asked to imagine that they were going to explain the mechanics of the gyroscope to one of their students, and that the student would first have an opportunity to watch one balancing on a pedestal. The teachers were then told to write down what they would tell this student to pay attention to while watching the event. A spinning gyroscope was shown to the teachers after the task was explained to them and before they wrote their responses. Two of the teachers wrote answers that were too vague to be useful. Of the remaining 17 subjects, all made some mention of the spinning of the gyroscope’s wheel; however, only six reported that it would be useful to notice the gyroscope’s precessional motion. Compared with the naive subjects, the physics teachers were no better at predicting the balancing of a spinning gyroscope at all three different orientations on a pedestal [x2 (1) = 1.74, p > .lO], although they were more accurate for 90” [x2 (1) = 9.57, p < .005]. What we find of greatest interest, however, is how few of the teachers, 32%, thought to direct their students’ attention to the importance of the gyroscope’s precessional motion. Most d$ficult concepts. In response to the question asking them to list the three most difficult concepts in their physics courses, the teachers mentioned angular momentum most frequently. It was mentioned by 18 of the 19 teachers and was listed first by 10 of them, as compared with four first-place listings for the next most frequently mentioned concept. GENERAL DISCUSSION
In two experiments we found that people have very poor dynamical understandings of wheel dynamics. This lack of competence is far greater than any encountered in previous studies on people’s dynamical understandings of simple particle motions. Both particle and extended body motions manifest dynamical relationships; one could imagine an idealized observer who could appreciate the dynamics inherent in both. People, however, are not such observers. Rather, they exhibit limitations related to dynamical event complexity. For people, the dynamics of particle motions are easily appreciated, whereas extended body motions are not. Tops and gyroscopes are enduring toys because their mysterious behavior is not penetrated by perceptual experience. Observing a spinning gyroscope evokes wonder and joy because it appears to defy all of our dynamical intuitions. The motions of the gyroscope are apparent, but they evoke no spontaneous intuitions as to why the device behaves as it does.
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Our previous research on perceiving wheel generated motions suggests a perceptual explanation for people’s poor understandings of some aspects of wheel dynamics. We propose that, in perceiving rotational motions, the perceptual system performs an analysis that emphasizes the extraction of form at the expense of recovering information needed for dynamical analyses. As the top panel of Fig. 8 shows, every point on a rolling wheel follows one of three possible motion paths: cycloids, prolate cycloids, or straight lines. When viewing a rolling wheel, however, these trajectories are not seen. Rather, as is depicted in the bottom panel of Fig. 8, the perceptual system analyzes these motions into two components: relative rotationsall points are seen as revolving about their contigural centroid-and a common motion that is the trajectory of the centroid (Proffitt, Cutting, & Stier, 1979). The thoroughness of this perceptual analysis can be seen in the attempts of our subjects to draw the trajectories of three points on a rolling wheel. In Experiments 1, 4, and 5, only 2 of 94 subjects drew all three curves correctly. In Experiment 3, subjects viewed moving point-lights,
FIG. 8. The top panel shows the absolute motion paths for three points on a rolling wheel. At the bottom is depicted how these points are perceived when viewed in the context of the whole wheel. All points, except the center, are seen revolving about the wheel’s center; the center manifests the wheel’s common observer-relative motion.
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and only 46% and 8% selected cycloids and prolate cycloids, respectively, as corresponding to the trajectories of appropriately located points on a wheel. This indicates that even when viewing an ongoing display, the source of these trajectories is not obvious. Moreover, as the history of mathematics shows, the cycloid was discovered very late due to its absence in perception (Struik, 1969). Galileo is typically credited with discovering the cycloid, and Rubin (1927) reports that Galileo first noticed the curve while observing a wagon rolling down a hill at night with a single torch attached to the rim of one of its wheels. A more detailed account of the percemaal system’s analysis of absolute motions into relative and common components is presented in Fig. 9 (Proftitt & Cutting, 1980; Cutting & Proffttt, 1481). This figure depicts our proposal that rotations are not only separated from common motions, but also indicates that these two motion components have quite different perceptual significances: rotations specify form, whereas common motion defines observer relative displacement. All accounts of how perception extracts form from motion rely on rotational motions. (See, for example, Ullman, 1979; Wallach & O’Connell, 195311976.)The perceived common motion is the motion path of the centroid of relative rotations. Under the assumption of uniform density, this centroid corresponds to the form’s center of mass. Thus, the perceived motion of the whole object-its common observer relative displacement-is a particle motion. Absolute
Motions
m
/\ Form
/ \ Relative Motmn
Common
Motmn
Actton
(Common ObserverRelative Displacement)
FIG. 9. The left panel depicts the perceptual analysis of wheel motions into relative and common motion components. The right panel shows our proposal that these two motion components have different perceptual significances. Rotations specify form. Common motions define particle motion dynamics.
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In essence, we propose that the perceptual system performs an analysis that extracts rotational motions for the purpose of recovering form, and common motions are used to define particle motion dynamics. The only motion that is available for dynamical analysis is common motion, and people show dynamical competence only when the relevant motions are those of the object’s center of mass. Further support for this proposal is seen in the manner in which the perceptual system defines configural centroids. The perceived center of a form is the centroid of its external boundary as opposed to the centroid of its luminance distribution (Proffttt, Thomas, & O’Brien, 1983). That is, the perceptual system effectively ignores the distribution of luminance within a form, and determines its centroid by attending to only the object’s contour. This disregard for the distributional properties of objects may be reflected in the responses given by our subjects to the question about what factors influence the rate that a wheel rolls down an inclined plane. Even though it was the only relevant factor in the situation, subjects showed the greatest agreement in their belief that mass distribution does not affect wheel dynamics. Moreover, the wheel dynamics that contribute to the stability of bicycles and gyroscopes are distributional variables. By definition, all extended body motions include mass distribution as a relevant dynamical variable. To reiterate, we propose that the perceptual system assigns quite different significances to rotational and common motions. For human perception, relative rotations specify form, and common motions carry dynamically relevant information. Thus, with respect to dynamical intuitions, the only relevant object motions are particle motions. From this perceptual perspective, mass distribution should not affect the rolling of a wheel, the absolute motion paths of points on a rolling wheel should be lost to direct intuition, and the dynamical influences of spinning wheels should be imperceptible. Although speculative, we think that this perceptual explanation accounts well for people’s poor understandings of wheel motions. CONCLUSION
The distinction that we have drawn between particle and extended body motions is a physical one. In some situations, the only object attribute of dynamical relevance is the position over time of the object’s center of mass. These are particle motion contexts. For most situations, however, aspects of the object’s shape, orientation, and rotation are dynamically relevant as well. These are extended body motion contexts. Dynamical analysis of extended body motions requires that more than one object dimension be noticed and combined into a multidimensional representation.
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We have proposed that people do not form these emergent multidimensional quantities, but rather base their dynamical judgments on heuristics that relate to only one dimension of information present in the event (Proffitt & Gilden, 1989). The use of such heuristics can result in accurate dynamical judgments in particle motion contexts, since these events are, by definition, one-dimensional. In general, extended body motion contexts cannot be adequately evaluated in this manner. The wheel is one of the simplest and most commonplace extended body systems encountered in our culture. Be that as it may, common-sense intuitions about its dynamical properties in extended body contexts are terribly muddled. When asked about the factors affecting the rate at which a wheel rolls down a ramp, university undergraduates performed at a chance level when judging the significance of irrelevant variables, but were highly consistent in their belief about the irrelevance of the only variable that is, in fact, of dynamical significance. This lack of appreciation for the influence of mass distribution on rotational dynamics was manifested in explicit judgment tasks, and when subjects viewed ongoing simulations of natural and anomalous events. Moving, from the simple wheel contexts of spinning and rolling to those observed in gyroscopic motions, produces a shift in people’s response to the motions encountered. The perplexity that accompanies the observation of a rim and a solid disk descending a ramp at different speeds is transformed into wonder when a top is observed precessing at right angles to its pedestal. In this latter event, common sense is confronted with a state of affairs that definitely runs counter to strong intuitions. Immediate intuitions about wheel dynamics are affected little by amount of experience or expertise. Bicycle racers and high-school physics teachers exhibit very similar erroneous judgments about wheel dynamics when compared with naive undergraduates. We believe that training impacts little on common-sense notions about dynamics, because these intuitions are grounded in phenomenal experience. A physicist can no more see the dynamics of a gyroscope than a geometer can see the cycloidal trajectories of points on the tire of a passing car. A physicist can provide a complete account of gyroscopic dynamics, but the multidimensional variables that are introduced in such an account are not categories of phenomenal experience. This is not to say that people cannot be trained to spontaneously deal more effectively with multidimensional systems. As experimenters, we never get the wheel problems wrong. If retested, our subjects would hopefully not make the same mistakes again after the postexperimental briefing in which they were given an explanation of the systems that they had encountered. Finally, we suspect there are people, such as certain specialized engineers, who must regularly deal with the dynamics of
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wheel systems, and who have established well-memorized sets of rules to deal with their dynamical behavior. People can be taught to reason efficiently about the dynamics of specific extended body motion systems-here, we are referring to spontaneous judgments as opposed to explicit calculation-however, we believe that the adequacy of such reasoning will always fall short of that which can be achieved in particle motion contexts. Reasoning about particle motions can always be based upon the motions of a single point: the object’s center of mass. Perceiving the motions of this point is something that people are known to do spontaneously and with accuracy (Proffrtt & Cutting, 1980). Dynamical understandings of extended body motions require that more than one object dimension be taken into account and related within a multidimensional representation. The complexity encountered in assessing multidimensional quantities, such as moment of inertia (see Eq. 7 and 8), will always pose a hindrance to common-sense reasoning about extended body motions. REFERENCES Caramazza, A., McCloskey, M., & Green, B. (1981). Naive beliefs in “sophisticated” subjects: Misconceptions about trajectories of objects. CogniZion, 9, 117-123. Champagne, A. B., Klopfer, L. E., &Anderson, J. H. (1980). Factors influencing the leaming of classical mechanisms. American Journal ofPhysics, 48, 1074-1079. Clement, J. (1982). Students’ preconceptions in introductory mechanics. American Journal of Physics, 50, 66-71. Cutting, J. E., & Proffttt, D. R. (1981). Gait perception as an example of how we may perceive events. In R. Walk & H. L. Pick, Jr. (Ed.), Intersensory perception and sensory integration (Vol. II, pp. 24!&273). New York: Plenum. Gilden, D. L., & Proflitt, D. R. (1989). Understanding collision dynamics. Journal of Experimental
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