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 calculusbased 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.
B.
Pretest about generic cross product comparisons.
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 freebody 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 Tshaped 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 freebody 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.
Introductory calculusbased course - listed by
cohort
UW 121 (2 sections; N =
255)
UWH 121 (1 section; N =
50)
PRD 152 (5 sections; N =
1160)
Table 2.
Question: Is the center of mass to the left of P, to
the right of P, or at P?
(a) At point P (b) To the left
(c) To the right
(correct )
of point P
of point P
90%
< 5%
5%
90%
< 5%
5%
95%
< 5%
5%
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.
Introductory calculusbased course - listed by
cohort
Cohort UW 121
(2 sections; N = 255)
Cohort UWH 121
(1 section; N = 50)
Cohort PRD 152
(5 sections; N = 1160)
Question: Is mA greater than, less than, or equal to
mB?
(a) A < B
(b) A = B
(c) A > B
(correct)
55%
40%
5%
65%
25%
5%
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.
Cohort
UW 121 (1 section; N =
170)
Question: To keep the system balanced,
where should object 2 be moved to
restore balance?
(a) Equal to distance
(b) Incorrect or
moved by object 1
blank
(correct)
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 (a)
Pretest (b)
Post-test
students who gave a (after
standard (after
standard
correct answer
instruction)
instruction
and
initial version of
tutorial Equilibrium
of rigid bodies)
4 sections; N = 455
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 (a)
Pretest (b)
Post-test
students who gave a (after
standard (after
standard
correct answer
instruction)
instruction
and
initial version of
tutorial Equilibrium
of rigid bodies)
1 section; N = 20
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).
Percentage of students
who gave a correct
answer
Distance moved by
object 2 to restore
balance
Introductory calculus-based physics
(a) Pretest
(b) Post-test
(after standard
(after standard instruction
instruction)
and initial version of
tutorial Equilibrium of
rigid bodies)
1 section; N = 170
4 sections; N = 365
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.