Research on student understanding of engineering statics: The development of instructional materials to improve student understanding of center of mass

Abstract University students who have studied physics at the introductory level and beyond are often unable to apply basic concepts from mechanics to account for the motion of an extended object. Findings from an in-depth, systematic investigation were used to guide the design of curriculum to address the underlying conceptual and reasoning difficulties. Ongoing assessment was an integral part of the development cycle. The instructional sequence that evolved from this iterative process has proved effective in helping students learn to apply basic concepts from mechanics to extended objects in equilibrium. Introduction

In this paper, I describe the design and development of a tutorial sequence on the equilibrium of rigid bodies. This tutorial, Equilibrium of rigid bodies, is one of the tutorials in Tutorials in Introductory Physics by L. C. McDermott, P. S. Shaffer, and the U. Wash. P. E. G. published in 2000 that make up the set of tutorials on rotations. The tutorial described has been written to address student difficulties with the equilibrium of rigid bodies. These materials are intended for use in an introductory physics course for engineering majors prior to them taking sophomore-level engineering statics. The context is balancing. Each tutorial sequence, which includes pretests, worksheets, and homework, has been carefully designed to address specific difficulties that have found to be widespread among students in introductory physics. The final version of the tutorial sequence will be described. I summarize the findings from research, which includes post-test data for the tutorial. The tutorial has been tested in the introductory calculus-based physics course (mechanics) at the University of Washington and universities that are pilot sites for the curricula. The work described is part of the ongoing effort of the Physics Education Group at the University of Washington to improve student learning in physics through research and research-based curriculum development. Most of the students were enrolled in the introductory physics course for science and engineering majors. The study also includes two other populations: undergraduates in a sophomore-level engineering statics course and K-12 teachers.

The research described here was conducted between 1996 and 1997. However, relevant

data accumulated since 1992 have been analyzed and included. Some of the instructional materials produced are contained in the First Edition of Tutorials in Introductory Physics. Testing at pilot sites generally precedes publication of all materials by the U. Wash. P. E. G.

Background and motivation for research Several factors contributed to the motivation for the present investigation. These

included the need that we perceived to identify and address conceptual and reasoning difficulties with rigid body dynamics among science and engineering students and among teachers expected to teach the topic of balancing to precollege students.

In 1995, we began to examine student performance in solving the types of mechanics

problems assigned in the introductory calculus-based physics course. We hoped to develop tutorials that would help students learn how to solve such problems, especially those that require more than rote application of memorized algorithms. In 1996, the Physics Education Group was consulted by faculty in the College of Engineering about ways in which they could improve instruction in the sophomore engineering mechanics courses.1 We volunteered to advise them on the development of supplementary instructional materials for the statics and dynamics courses. We welcomed the opportunity to extend our investigation of problem solving to students in these courses, almost all of that had previously taken introductory physics.

As a first step in working with the engineering faculty, we observed students in sections

of the statics course as they worked together in small groups on “hands-on” activities based on the course material. It soon became apparent that the students were struggling with concepts and principles (e.g., center-of-mass and torque) that had been taught in the introductory physics course. From this experience and some preliminary studies that our group had conducted in 1994-1995, it was clear that the students were encountering additional difficulties with even more basic concepts (e.g., force) that do not appear in their study of particle dynamics. We therefore decided to postpone the development of problem-solving tutorials for the calculus-based course until we could identify and address important conceptual issues that underlie success in problem solving. This decision led to the development of a tutorial on rigid body dynamics and marked the beginning of the present investigation. I. The tutorial Equilibrium of rigid bodies

In this section, I describe the tutorial sequence: Equilibrium of rigid bodies. The tutorial sequence consists of a pretest, tutorial, and homework. A. Pretests

Several problems were administered as pretests to probe student understanding of the equilibrium of rigid bodies. Each has been designed to elicit specific reasoning difficulties with balancing extended objects.

Extended objects balanced on a beam

This problem involves two objects, a triangle and rectangle, of equal mass that are balanced on a uniform meter stick. The meter stick is supported at its center by a fulcrum. (See Fig. 1 for an excerpt of pretest problem administered.) This pretest was designed to determine the extent to which students are able to apply the concept of center of mass for a system of extended objects in a simple balancing situation. Two of the questions asked are described here.

Figure 1. Excerpt of pretest about triangle and rectangle on beam for Equilibrium of rigid bodies.

Distance prediction: triangle and rectangle on opposite sides of meterstick

Students were asked how far to move the triangle (after the rectangle, of equal mass, has been moved toward the fulcrum by ten centimeters) to restore balance. The purpose of asking this question was to (1) determine the extent to which students are able to solve a balancing problem for which there was only one variable, distance to the left and right of the fulcrum, and (2) determine a reference from which to interpret subsequent, related questions. (For example, see the following question.) Distance prediction: triangle over the fulcrum

Students were again asked how far to move the triangle (after the rectangle, of equal mass, has been moved toward the fulcrum by ten centimeters) to restore balance. (See question 2 in Fig. 1.) In this case the triangle is moved so that part is on the other side of the fulcrum, i.e., on the same side that the rectangle stands. The purpose of asking this question was to determine (1) the extent to which students are able to solve a balancing problem for which two variables, distance to the left and right of the fulcrum and amount of mass to the left and right of the fulcrum, are changed and (2) common student difficulties. Extended object supported on a fulcrum

Another problem involves a T-shaped board of uniform mass density. The board is placed on a pivot at its center of mass and held stationary by a hand. (See Fig. 2.) It does differ from the baseball problem in that students are told that the center of mass of the board is at the pivot. Prediction

The students are asked to predict the motion of the board after it is released from rest. This pretest was designed to determine the extent to which students are able to relate center of mass (relative to the pivot) in a balancing situation. Mass comparison

In this question, students are asked to compare the amount of mass to the left and right of the vertical line (shown as a dashed line in the diagram that appears in the pretest) through the pivot. This question elicits the incorrect “equal mass” reasoning for an extended object.

Figure 2. Pretest about T-shaped board on a pivot. Extended object balanced on a pivot (tilted)

A problem about a tilted bar consisting of two parts of different uniform mass densities was designed to determine the extent to which students are able to predict whether an extended object will remain at rest in any orientation when it is hung from a frictionless pivot at its CM. Two versions were administered. (See Fig. 3.) In each version, the bar was shown to be initially at rest. The bar is subsequently rotated and released from rest. On one version, version T, the bar is initially tilted. It is rotated until it is horizontal with the ground. On the other version, version H, the bar is initially horizontal and is rotated until it is at a tilt. The purpose of administering these two versions was to determine the extent to which orientation introduced additional difficulties with balancing.

Figure 3. Pretest about bimaterial bar on pivot (version H). Mass comparison

Students are asked to determine if the mass of the unshaded piece (on the left) is greater than, less than, or equal to the mass of the shaded piece. The purpose for asking this question was to elicit incorrect “equal mass” reasoning. Version T was also designed to elicit student difficulties with orientation (relative to the ground). This was accomplished by showing the longer piece (unshaded) of smaller mass closer to the ground. CM location

Another question asked of the students was to mark the approximate location of the CM of the bimaterial bar. The purpose of asking this question was to determine the extent to which students are able to relate CM to the pivot in a balancing situation for an object of non-uniform mass density.

Prediction

The last question asked students to (1) predict whether the bar would remain at rest after it is rotated and released (from rest) and (2) to draw the final orientation of the bar (if they believe it comes to rest). The purpose of asking this question was to elicit student difficulties with based on orientation relative to the ground. Asking this question on both versions also allowed us to determine if either initial orientation (horizontal or tilted) presented additional difficulties for the students. Mathematical formalism of rotational mechanics: The vector representation of torque Generic vector cross product

The purpose of this pretest is to document student difficulties with the comparison of the magnitudes of various cross products. No physical context is implied in the problem. Students are asked to compare the magnitudes of three pairs of vector cross products. Both relative magnitudes of vectors and components were readily interpretable from the grid on which the vectors where drawn. (The pretest is in Fig. 4.) This problem was administered on a single pretest.

Figure 4. Pretest about generic cross product comparisons. B. Tutorial worksheets

In the sections that follow, I describe the intial version of the tutorial worksheets for Equilibrium of rigid bodies. Excerpts are shown here.

Tutorial problem on motion prediction and free-body diagram for extended object pivoted at its center of mass

Figure 5. Tutorial questions about T-shaped board on a pivot.

In this section, I will show a sequence of questions written (1) to guide students to make the proper observations about the motion of a rigid body pivoted at its center of mass and (2) to elicit student difficulties with the concept of center of mass in a simple balancing situation. Excerpts of the problems that appear on the tutorial are in Fig. 5. Prediction and observations

Students are asked to make several predictions. They are asked to predict how the amount of mass to the left and right of the vertical line through the pivot. This question elicits the incorrect belief that there is an equal amount of mass to either side of the CM. (Students do not check their mass comparison at this point in the tutorial.) The students are also asked about net torque and to make a prediction about the motion of board after it is released as shown on the worksheet. The purpose of asking these two questions is to help guide students to begin to consider that there exists a relation between a zero net torque and rotational motion.

Students check their predictions by seeing a demonstration at the front of the room. The tutorial asks students about angular acceleration and center-of-mass acceleration. Other questions help guide students to make connections with their observations with physical relations (e.g., Newton's second law and acceleration of the center of mass). Extended free-body diagram

A diagram of the board is given for students to draw a free-body diagram of the board. The tutorial asks students to label forces as they have been asked in other tutorials. The free-body diagram provides a foundation from which students can check for consistency later on in the tutorial. Since we eventually want students to be able to make predictions about other balancing situation on the basis on the net torque due to the gravitational force, we ask students to draw an extended free-body diagram. The extended free-body diagram differs from the single particle free-body diagram by showing the location on the object that a force is exerted. To do this we ask students to represent draw forces on a diagram of the object, instead of representing it by a “dot” as is often done in particle dynamics.

In the tutorial, we asked students to consider whether the gravitational force on the board could be represented as a single force or with more than one force. An excerpt of these questions appears in Fig. 6.

Figure 6. Tutorial questions about the representation of the gravitational force on an extended free-body diagram.

The point of asking students this set of questions is to allow them to consider various

possible ways of representing a distributed force. For each way, we ask students to explain the significance of the points where the forces are exerted, i.e., at the center of mass. If they drew two forces, then the forces should be draw at the centers of mass of the two pieces represented. Although the representation of the gravitational force as two force vectors, each representing the gravitational force on one of the pieces, counters the convention used in previous tutorial instruction on forces2, we used these questions to help students think about relative masses later on in the tutorial. Tutorial problem about board placed on pivot at a point to the left of its CM

In this section, a set of questions meant to confront student difficulties using CM to make a prediction about the motion of rigid body in a balancing situation are described. Excerpts of the questions asked are in Fig. 7.

Figure 7. Tutorial problem about board placed on pivot at a point to the left of its CM. Prediction and extended free-body diagram

In order to help guide students to recognize that treating the gravitational force as a single force can be more helpful at times, students are asked to consider the situation shown in Fig. 9. In question 1, students are asked to predict what will happen to the board if it is hung from the pivot at a location to the left of its CM. Students are also asked to draw an extended free-body diagram that supports their prediction. Observations

Students are asked to obtain their own board and pivot stand and check their prediction. At this point in the tutorial, students recognize that the board will tilt to the side that the CM is located. It is not expected that students can correctly account for their observations using amount of mass to the left and right of a vertical line through the pivot. (In this case, there is an

equal amount of mass to the left and right of the vertical line.) Therefore, even though students may try to account for their observations by assuming that the side of the board on which the CM lies is heavier (not correct), the tutorial guides students to make use of the forces on their extended free-body diagrams. Tutorial question on amount of mass on the left and right of the center of mass of the T-shaped board

A set of questions meant to confront student difficulties with force and position vectors to determine torque due to a single piece of clay attached to a rigid body are shown below.

Figure 8. Tutorial questions about T-shaped board and clay on pivot.

Restoring balance with a piece of clay

The tutorial asks students to attach a piece of clay to restore balance. Students do this by attaching a small piece of clay an edge of the left side of the board. The amount of clay used is determined through trial and error. Relating CM of balanced system to pivot

Students are asked to predict if the CM of the system, consisting of board and clay, is located to left, to the right, or along a vertical line that passes through the pivot. (See question 1 in Fig. 8. The purpose of asking this question is guide students to recognize that the CM must be above, below, or at the pivot since the system is balanced. Changes in CM

Question 2 asks students to make a prediction about the motion of the system if the original piece of clay is moved closer to the pivot. The purpose of asking this question is to give students an opportunity to apply their knowledge of the location of the CM to the pivot in a case where the amount of mass on the right and left of the pivot does not change. This observation serves to provide a situation for students to refer to later in which the amount of mass on either side of the pivot remains unchanged, but the system is no longer balanced. Changes in amount of mass

The last question asks students to make a prediction about the motion of the system if more clay is added to the original piece of clay (at its original location when the system was balanced). The purpose of asking this question is to give students another opportunity to apply their knowledge of the location of the CM to the pivot. This observation serves to provide a situation for students to refer to later in which the amount of mass on either side of the pivot is changed, but the system is remains balanced. Generalizations about balancing

In this section, I describe questions meant to confront student difficulties with the incorrect belief that the amount of mass on the left and right of the center of mass is equal. Students are asked to refer to the results of previous experiments to answer the questions that appear in Fig. 9.

Figure 9. Tutorial questions meant to help students make generalizations about balancing.

The first question attempts to help students recognize that an extended object can be

balanced even though the amount of mass on one side is changed. The second question is asked to help students recognize that if the amount of mass on the left and right of the pivot is held constant, but the distribution of that mass in space is varied, the system may not remain in equilibrium. And the last question tries to confront students with the incorrect belief that the amount of mass on either side of the CM does not solely determine whether an extended object will balance. Mass comparison for T-shaped board

In this section, additional questions are described that are meant to confront student difficulties with the incorrect belief that the amount of mass on the left and right of the center of mass is equal. We chose to ask students about the mass comparison for the T-shaped board only after they had a chance to make several relevant and well-chosen observations (discussed in the previous section). We did this because this difficulty is hard to overcome and we believed they needed a framework from which to help resolve their difficulties. They did not need to resolve this difficulty to make predictions about balancing with the use of CM relative to the pivot in the exercises and experiments that appear earlier in the tutorial. Excerpts of the questions are in Fig. 10.

Figure 10. Tutorial questions to confront student incorrect belief about CM for the T-shaped

board.

The first question asks students to review their initial mass comparison prediction about the T-shaped board. Students are asked to check to see if their prediction is consistent with their generalizations about mass. This question is meant to help students recognize that the amount of mass of the two pieces of the board could differ and the board could still balance as they previously observed.

Question 2 asks to students to obtain equipment to check their predictions. The equipment consists of a T-shaped board; identical to the one the students had used, cut into two pieces as suggested by dashed line shown at the beginning of the tutorial. Students use the equipment to compare relative areas to check relative amount of mass of the two pieces. Students conclude from their observations that the amount of mass of the piece on the left is greater than that on the right.

The last question in this section (question 3), asks students to consider the extended free-body diagram for the T-shaped board for which two gravitational forces have been drawn. The purpose of this question was to introduce an alternative way in which students could account for mass comparison. Students are guided to consider the T-shaped board as an equivalent system consisting of two particles, each located at the center of mass of the left and right piece. With knowledge of the distances from the CM of the board, students are expected to recognize that since the CM of the piece of the left piece is closer to the pivot than the CM of the right piece, than it must be of greater mass. Students may use the definition of the CM or use torque equilibrium, i.e., zero net torque, to arrive at this result.

The strategy we have used here is to provide conflicting statements that students must

resolve based on their understanding of balancing. In the dialog, student 1 gives a convincing, common argument used by students who believe that an extended object balances about a point where there is an equal amount of mass to the left and right. Student 2 gives a correct explanation in which he compares the torque of the two pieces. In this section, the tutorial asks students to determine which of the students has misinterpreted the term “center of mass.” Students are asked to check their reasoning with a tutorial instructor before they continue to the next section. Tutorial questions about net torque and location of CM relative to the pivot

In the last section of the tutorial worksheet, the students are to consider the situation in which the T-shaped board is placed on the pivot at a point above its CM. The board is then rotated so that none of its edges are parallel with the ground. The students are first asked to draw an extended free-body diagram for the board in its tilted orientation shown. By drawing an extended free-body diagram for the board, students are guided to think about the location the gravitational force is exerted relative to the pivot. Additionally, being asked to draw their forces with correct relative magnitudes can challenge students.3 Prediction

After the extended free-body diagram is drawn, the tutorial asks students to predict the subsequent motion of the board and to explain how their free-body diagram is consistent with their prediction. Although most students are able to predict that the board will rotate clockwise about the pivot, many cannot account for the resulting motion using rotational dynamics. By asking students to refer back to their free-body diagram, they are able to make connections with the location that the forces are exerted and are better able to consider net torque. Net torque

Since the pivot used in the classroom is not frictionless, the board will eventually come to rest. (The tutorial assumes that students will take this into account in their prediction.) With this assumption in mind, the tutorial asks students about the net torque on the board after it stops moving. The purpose of asking this question is to help students recognize that when the board is in equilibrium, the net torque is zero even though the pivot is not at the CM. Observations

The tutorial asks students to perform the experiment and check their predictions with their observations. Students observe that the board rotates clockwise about the pivot and eventually comes to rest with its CM directly below the pivot. By checking their predictions, students must recognize that the torque produced by the gravitational force is zero since the force and position vectors are parallel. This exercise gives students an opportunity to identify the position vector for a force (relative to a pivot) and to use the vector cross product to determine torque.

II. Student performance after working through tutorial In this section, I discuss post-test data from student written examinations in the introductory calculus-based physics course (mechanics) at the University of Washington. I have administered many post-test problems on balancing and center of mass to well over 5000 students in the introductory physics course at the University of Washington and at universities that are pivot sites for Tutorials in Introductory Physics from 1998 – 2001. I will discuss results on two problems that we have found to be reliable indicator of student understanding of several key concepts and relations. Extended object balanced on a pivot Baseball bat

We have administered the baseball bat problem as a post-test since we have found that it readily elicits student difficulties with center of mass in a balancing situation. Students are asked about the CM of the bat and are asked to compare amount of mass (of the bat) located on the left and right of the pivot.

This problem has been asked in three sections of the introductory calculus-based physics course at the University of Washington (cohorts UW 121 and UWH 121). It has been asked in five standard sections in the introductory calculus-based physics course at Purdue University (cohort PRD 152). The problem was administered after students had worked through the initial version of the tutorial Equilibrium of rigid bodies. Students in all three cohorts had experienced all lecture and laboratory instruction in their course.

Cohort UW 121 consists of two standard sections of Physics 121. Cohort UWH 121 is one honors section of Physics 121. Both of these cohorts had experienced tutorials on forces and Newton’s laws and pre-lab tutorial on rotational motion. (One section in cohort UW 121 had worked through the tutorial Dynamics of rigid bodies.4) The post-test was on the final examination for both of these cohorts. Students in cohort PRD 152 had worked through the tutorial Dynamics of rigid bodies, but had not worked through the tutorial Rotational motion. The post-test was on the final examination that was administered as a multiple-choice question.

The data are in tables 1 and 2. Nearly all the students gave a correct answer for the question about the CM. Student performance on the mass comparison question was similar among students in cohort UW 121 and PRD 152. About 50% to 55% of these students gave a correct answer. Approximately the same percentage (a range of 40% to 45%) of the students in both cohorts stated that the pieces are of equal mass. Students in the honors section (cohort UWH 121) performed modestly better, with about 65% of these students who gave a correct answer. About 25% of the honors students stated that the pieces are of equal mass.

Table 1. Summary of post-test data after the tutorial about the location of the CM of the baseball bat relative to point P. Column (a) shows the percentage of students who gave the correct answer, at point P, with correct reasoning. Column (b) shows the percentage of students who stated that the center of mass is to the left of point P. Column (c) shows the percentage of students who stated that the center of mass is to the right of point P.

Question: Is the center of mass to the left of P, to

the right of P, or at P? Introductory calculus-based course - listed by cohort

(a) At point P (correct )

(b) To the left of point P

(c) To the right of point P

UW 121 (2 sections; N = 255)

90% < 5% 5%

UWH 121 (1 section; N = 50)

90% < 5% 5%

PRD 152 (5 sections; N = 1160)

95% < 5% 5%

Table 2. Summary of post-test data after the tutorial Equilibrium of rigid bodies for the

question about the amount of mass on the left and right of point P. Column (a) shows the percentage of students who gave the correct answer, A < B. Column (b) shows the percentage of students who stated that A = B. Column (c) shows the percentage of students who stated A > B.

Question: Is mA greater than, less than, or equal to

mB? Introductory calculus-based course - listed by cohort

(a) A < B (correct)

(b) A = B (c) A > B

Cohort UW 121 (2 sections; N = 255)

55% 40% 5%

Cohort UWH 121 (1 section; N = 50)

65% 25% 5%

Cohort PRD 152 (5 sections; N = 1160)

50% 45% 5%

Single particles balanced on a beam Square nut moved over the fulcrum

Students are asked to determine how to restore balance for a system of objects.5 This problem probes student understanding of changes in torque (with respect to the pivot) for objects of equal mass. Incorrect ideas about torque equilibrium and CM are elicited.

This problem has been asked in one standard section of the introductory calculus-based physics course at the University of Washington (cohort UW 121). The problem was on the final examination. The students had experienced all lecture and laboratory instruction in their course. In addition to tutorial instruction on particle kinematics and dynamics, students had worked through the tutorial Dynamics of rigid bodies and the pre-lab tutorial Rotational motion. The data are in table 3. About 65% of the students gave a correct answer. Table 3. Summary of post-test data after the initial version of the tutorial Equilibrium of

rigid bodies for distance object 2 (a square nut) should be moved to restore balance on a pegboard. Column (a) shows the percentage of students who stated that object 2 should be moved a distance equal to that moved by object 1 (another square nut) (correct). Column (b) shows the percentage of students who gave other incorrect answers or who did not answer the question.

Question: To keep the system balanced,

where should object 2 be moved to restore balance?

Cohort (a) Equal to distance moved by object 1 (correct)

(b) Incorrect or blank

UW 121 (1 section; N = 170)

65% 35%

Comparison of pretest and post-test results

The pretest and post-test results for the questions asked about the baseball bat are in this section. The standard and honors sections are compared separately. Where similar, percentages have been combined.

The data are from student responses for the mass comparison of the two pieces of the baseball bat. Table 5 shows pretest and post-test results taken from students in standard sections of the introductory calculus-based physics course at the University of Washington and at Purdue University. Table 6 shows pretest and post-test results taken from students in honors sections of the introductory calculus-based physics course at the University of Washington.

Table 5. Comparison of student performance on pretest and post-test after students in

standard sections had worked through initial version of Equilibrium of rigid bodies for mass comparison on the baseball bat problem. Column (a) shows the percentage of students who gave a correct answer on pretest. Column (b) shows the percentage of students who gave a correct answer on post-test.

Introductory calculus-based physics Percentage of students who gave a correct answer

(a) Pretest (after standard instruction) 4 sections; N = 455

(b) Post-test (after standard instruction and initial version of tutorial Equilibrium of rigid bodies) 7 sections; N = 1415

Mass comparison 15% 50% Table 6. Comparison of student performance on pretest and post-test after students in

honors sections had worked through initial version of Equilibrium of rigid bodies for mass comparison on the baseball bat problem. Column (a) shows the percentage of students who gave a correct answer on pretest. Column (b) shows the percentage of students who gave a correct answer on post-test.

Introductory calculus-based physics - UWH

121 Percentage of students who gave a correct answer

(a) Pretest (after standard instruction) 1 section; N = 20

(b) Post-test (after standard instruction and initial version of tutorial Equilibrium of rigid bodies) 1 section; N = 50

Mass comparison 25% 65%

The post-test results on the baseball bat task are encouraging. Both students in the standard sections and honors sections showed a significant improvement in their understanding of center of mass for an extended object balanced on a pivot.

The results from the triangle and rectangle post-test were also found to be encouraging. More than one-half of the students (about 65%) gave a correct answer on the post-test after working through the tutorial. Table 7. Comparison of student performance on pretest and post-test after initial version of

Equilibrium of rigid bodies for problems in which an object crosses the fulcrum. Column (a) shows the percentage of students who gave a correct answer on pretest (original version). Column (b) shows the percentage of students who gave a correct answer on post-test (modified version with pegboard and square nuts).

Introductory calculus-based physics Percentage of students who gave a correct answer

(a) Pretest (after standard instruction) 4 sections; N = 365

(b) Post-test (after standard instruction and initial version of tutorial Equilibrium of rigid bodies) 1 section; N = 170

Distance moved by object 2 to restore balance

45% 65%

The post-test data show increases in student performance on the baseball bat mass

problem. However, we thought we could do better for the students. On the second task, the comparison of the pretest and post-test data is not as straight forward as on the first. The problem with the triangle, which extends over the fulcrum, is more difficult. Because of this, we have interpreted the 65% correct as an upper limit of student success since students are more able to solve balancing problems with single particles than they are with extended objects.

Student difficulties with the mathematical representation of torque were not overcome by the tutorial Equilibrium of rigid bodies, which was used as a supplement to the traditional lecture and laboratory instruction students had experienced. These data suggest that there exists a need

to improve instruction on this topic. The development of instructional materials to address student difficulties with both graphical and analytical methods to determine vector cross products is one possible approach to this problem. More research on student understanding of vectors and vector operations is needed. Research-based problem solving worksheets, of the kind developed by Kanim6 for electric circuits and electricity and magnetism, could be used as a model from which to begin such a research and curriculum project. Conclusion and implications for instruction

I have conducted an in-depth, large population study of student understanding the equilibrium of rigid bodies. The population studied was college science and engineering majors in the calculus-based introductory physics course. The research has shown that students are often unable apply what they have learned about particle dynamics to extended objects. Serious difficulties appear in several contexts described in this dissertation. The specific conceptual difficulties that appear are summarized in the following sections.

This study examines student understanding of net torque in the context of the equilibrium

of rigid bodies. I have identified several specific student difficulties. Most students in the populations studied recognize that an object will remain in equilibrium if no net force is exerted, however, many of the same students fail to recognize that the net torque must also be zero. Furthermore, the data from the engineering statics course suggest that advanced study of engineering statics does not effectively address this difficulty.

Related to the difficulty with the conditions of equilibrium, we have identified student

difficulty with quantifying net torque as a vector sum of individual torques taken about any single point and with quantifying torque as a vector cross product of position (with respect to an origin) and force vectors for cases other than the most simple (e.g., cases for which the lines of actions of the vectors are neither co-linear nor orthogonal).

On more basic examples of mechanical equilibrium, e.g., a balancing situation, we have identified student difficulties with accounting for the torque produced by the gravitational force. Two related difficulties which we have explored in detail are the assumption that the center of mass (or center of gravity) "divides an object into two pieces of equal mass" and a belief that neutral equilibrium is determined by the object's orientation relative to the ground.

I have found that students have difficulty determining the magnitude of the torque vector. When asked to compare magnitudes of torque, many students often do not know how to interpret the three variables (position, force, and angle) nor do not recognize how to use other methods that could make the calculation expedient. For problems in which physical situations are presented, students many times do not know how to extract the relevant information from a diagram to calculate the magnitude of a torque.

The difficulties that we have uncovered during this investigation are both surprising and

serious. We have found that these difficulties are not limited to introductory students. The same difficulties have appeared in the reasoning given by classroom teachers who have taught units on some of these topics to precollege students. These difficulties have also appeared in reasoning

given by sophomore engineering majors in civil and mechanical engineering and by physics graduate students. These difficulties are prevalent and persistent and not easily overcome.

The results of the research have informed the development a set of tutorials on rotations that are included in Tutorials in Introductory Physics for use as a supplement to the traditional lecture-based course in introductory physics. Tutorials in Introductory Physics is an ongoing curriculum development project by the Physics Education Group at the University of Washington to improve student learning in introductory physics. I have shown that the tutorial Equilibrium of rigid bodies is effective in addressing some of the most serious student difficulties with equilibrium. Student results on post-test after working through the tutorials are many times near or at the same level as the graduate teaching assistants. On some tasks, the undergraduates surpass the performance of the physics graduate students.

The instructional materials have been tested at Purdue University that serves as a pilot

site for Tutorials in Introductory Physics. The pretest and post-test results from Purdue University are similar to the results obtained at the University of Washington. The instructional strategies used in the tutorial Equilibrium of rigid bodies have been adapted for use in instructional materials to prepare K-12 teachers to teach balancing. The preliminary results are encouraging. Acknowledgments This work was done under the direction of L. C. McDermott, P. R. L. Heron, P. S. Shaffer and S. Vokos. I am grateful for members of the U. Wash. P. E. G, past and present that have helped me administer pretests and develop the tutorial on equilibrium, especially P. R. L. Heron. I also thank D. Storti for allowing me to observe and to administer pretests and posttests in your engineering statics course. 1 At the time the collaboration began, the College of Engineering was undergoing a structural changes in the way in which the statics course was taught. It was for reasons related to these changes that one of the statics instructors approached the Physics Education Group before instruction commenced in Autumn 1996. The instructor in the statics course was inspired by Tutorials in Introductory Physics and the Physics Education Group's instructional efforts in introductory calculus-based physics sequence at the University of Washington and decided to learn more about the group's instructional style and try to adapt the group's methods to develop statics tutorials. 2 In the tutorial Forces, students are instructed to represent the gravitational force as a single force vector. 3 By the use of kinematical arguments and Newton’s second law, students can determine that the net force is down, toward the bottom of the page, at the instant immediately after it is released from rest, since the center-of-mass acceleration is down, toward the bottom of the page. 4 Some of the students had worked through the tutorial Dynamics of rigid bodies, while others had not. We found that this did not affect student performance on the post-test. 5 The variation involved identical objects, instead of a triangle and rectangle, hung from a pegboard. One of the objects is moved so that it is on the other side of the board. Students are asked to determine where the other identical object should be moved to restore balance.

6 S. Kanim, “Investigation of student difficulties in relating qualitative understanding of electrical phenomena to quantitative problem-solving in physics,” Ph.D. dissertation, Department of Physics, University of Washington, 1999.