When physical intuition fails

Chandralekha Singh

Citation: Am. J. Phys. 70, 1103 (2002); doi: 10.1119/1.1512659
View online: http://dx.doi.org/10.1119/1.1512659
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When physical intuition fails
Chandralekha Singh
a)
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Received 14 March 2002; accepted 16 August 2002
We analyze the problem-solving strategies of physics professors in a case where their physical
intuition fails. A nonintuitive introductory-level problem was identied and posed to twenty physics
professors. The problem placed the professors in a situation often encountered by students, and their
response highlights the importance of intuition and experience in problem solving. Although
professors had difculty in solving the problem under the time constraint, they initially employed a
systematic approach, for example, visualizing the problem, considering various conservation laws,
and examining limiting cases. After nding that familiar techniques were not fruitful, they made
incorrect predictions based on one of two equally important factors. In contrast, other more familiar
problems that require the consideration of two important principles for example, conservation of
both energy and momentum for a ballistic pendulum were quickly solved by the same professors.
The responses of students who were given the same problem reected no overarching strategies or
systematic approaches, and a much wider variety of incorrect responses were given. This
investigation highlights the importance of teaching effective problem-solving heuristics, and
suggests that instructors assess the difculty of a problem from the perspective of beginning
students. 2002 American Association of Physics Teachers.
DOI: 10.1119/1.1512659
I. INTRODUCTION
Physical intuition is elusiveit is difcult to dene, cher-
ished by those who possess it, and difcult to convey to
others. Physical intuition is at the same time an essential-
component of expertise in physics. Cognitive theory suggests
that those with good intuition can efciently search the in-
formation stored in memory to pattern-match or map a given
problem onto situations with which they have experience.
Over the course of their training, professional physicists de-
velop a high degree of physical intuition that enables them to
analyze and solve problems quickly and efciently. Standard
introductory physics problems are easy for professors be-
cause they know how to distill those physical situations into
familiar canonical forms. Introductory students often
struggle over the same problems because they lack this dis-
tillation ability, and because the canonical forms are not
familiar.
Problem solving can be dened as any purposeful activity
where one is presented with a novel situation and devises
and performs a sequence of steps to achieve a set goal.
1
The
problem solver must make judicious decisions to reach the
goal in a reasonable amount of time. There is evidence to
suggest that a crucial difference between the problem-
solving capabilities of physics professors experts and intro-
ductory physics students novices lies in both the level and
complexity with which knowledge is represented and rules
are applied.
24
Physics professors view physical situations at
a much more abstract level than beginning students, who
often focus on the surface features and get distracted by ir-
relevant details. For example, students tend to group together
all mechanics problems involving inclined planes, regardless
of what type of physical principles are required for solving
them.
4
Many studies have focused on investigating the differ-
ences between the problem-solving strategies used by expert
physicists and introductory physics students.
24
The prob-
lems chosen in these studies are typically those which phys-
ics professors nd easy to solve using their intuition. Here
we analyze the problem-solving strategies of physics profes-
sors in a case where their physical intuition fails. An intro-
ductory level problem was identied for which the physical
intuition of most experts is lacking. We compare the
problem-solving strategies of professors and introductory
physics students in this context. According to cognitive
theory, expertise in a particular domain consists of having a
large stock of compiled knowledge to deal with a wide vari-
ety of contingencies.
5
No matter how expert people are at
coping with familiar problems, their performance will begin
to approximate that of novices once their stock of compiled
rules in memory has been exhausted by the demands of a
novel situation.
5
In these situations, experts cannot easily
invoke compiled knowledge from memory because the ap-
plicability of a particular principle is not entirely obvious.
They must process information on the spot in a manner simi-
lar to novices.
We posed an introductory physics problem related to rota-
tional and rolling motion to twenty physics professors and
several introductory physics students. The question posed
was inspired by a numerical problem found in the textbook
by Halliday, Resnick, and Walker.
6
It is interesting because
despite being at an introductory physics level, it is unlike the
type of problems most professors have thought out before.
Of the twenty professors interviewed, not one had useful
intuition that could guide them to the correct solution, nor
could they easily identify how to solve the problem.
II. THE PROBLEM ON ROTATIONALAND
ROLLING MOTION
Ignore the retarding effects of air resistance. A rigid wheel
is spinning with an angular speed
0
about a frictionless
axis. The wheel drops on a horizontal oor, slips for some
time, and then rolls without slipping. After the wheel starts
1103 1103 Am. J. Phys. 70 11, November 2002 http://ojps.aip.org/ajp/ 2002 American Association of Physics Teachers
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rolling without slipping, the center of mass speed is v
f
. How
does v
f
depend upon the kinetic coefcient of friction
between the oor and the wheel?
We suggest that the reader attempt to solve the problem
before referring to the solution in Appendices A and B.
III. DETAILS OF THE STUDY
The above problem was posed to twenty college profes-
sors. Each discussion lasted between 15 and 30 minutes, de-
pending upon the faculty members interest in pursuing it.
Although the discussions were not taped, extensive notes
were written down after each discussion to ensure that each
faculty members thought processes and problem-solving
strategies were captured accurately. Another part of the study
involves administering this problem
7
in the form of a recita-
tion quiz to 67 calculus-based introductory physics students
after they had taken an exam on rotational and rolling mo-
tion. In addition to asking students to explain their reasoning,
we discussed with several students their intuition and ap-
proach individually to better understand how they had inter-
preted and answered the problem.
A. Response of professors
After posing the problem, we asked the professors for their
intuition along with their reasoning. If they were quiet for a
long time, we encouraged them to articulate what they were
thinking. Most admitted that they did not have much intu-
ition about how the nal speed v
f
should depend on the
coefcient of friction, . We then asked them how they
would approach the problem. Seven faculty went to the
chalkboard and drew a picture of the situation. Only three
made an attempt to solve the problem quantitatively rather
than reasoning qualitatively. They may have been hesitant to
attempt a quantitative solution while under pressure because
they were originally asked about their intuition, and also the
direction/principle was not obvious. Although some were
quick to point out their gut feelings and the corresponding
reasoning, others were more cautious. Many noted that they
did not have extensive experience dealing with problems in
which the slipping part rather than the rolling part is im-
portant. Three admitted having seen this type of problem
before despite acknowledging a lack of intuition. A few also
mentioned that they were not good at thinking when put on
the spot. Some expressed frustration at the fact that a simple
conservation principle did not seem obvious for this prob-
lem.
What is fascinating about most professors responses is the
manner in which they approached the problem. They almost
always visualized the problem globally and pondered over
the applicable physics principles. More than half mentioned
the idea of using some conservation principle, for example,
angular momentum conservation, however, during the dis-
cussion, none could gure out how to apply it to the prob-
lem. Many thought about the very high and low friction lim-
iting cases and several drew analogies with familiar
situations which may employ similar underlying principles.
Many invoked energy dissipation arguments. However, sev-
enteen out of the twenty professors concentrated almost ex-
clusively on one of the two essential features of the problem,
either the frictional force or the time to start rolling. The
response of professors can be classied into ve broad cat-
egories: 1 Five professors focused on friction and noted
that a larger friction coefcient would imply higher energy
dissipation and, therefore, smaller v
f
. 2 Five professors
focused on the time to start rolling. They noted that a smaller
friction coefcient would imply a larger slipping time before
the wheel locks, resulting in a larger energy dissipation and a
smaller v
f
. 3 Three professors focused on the fact that
without friction, the wheel would keep slipping and never
roll. Based upon this fact, they concluded that a larger fric-
tion coefcient implies a larger v
f
. Although the conclu-
sions in categories 2 and 3 are the same, we have sepa-
rated them because professors in category 3 did not
explicitly invoke slipping time or energy dissipation argu-
ments. 4 Three professors correctly observed that v
f
de-
pends on two opposing factors: the time to start rolling and
the magnitude of the frictional force. One of them believed
that friction will dominate and a higher friction coefcient
will imply smaller v
f
he also noted that some conservation
principle might be applicable, for example, angular momen-
tum conservation. Another said that he was not sure which
one of these opposing effects will dominate. A third profes-
sor said that because a larger friction coefcient implies a
larger acceleration but a smaller time before rolling, the dis-
tance traveled during slipping would be the same regardless
of . He suggested that a higher friction would probably
imply a smaller v
f
. 5 Four did not express any clear opin-
ion about whether v
f
should be larger or smaller if the fric-
tional force is larger. Three of them wondered whether the
angular momentum conservation is applicable. However,
they could not convince themselves about how and for which
system this principle may be applicable. Three of them con-
sidered the limiting cases no friction implies that the wheel
never rolls and very high friction implies it rolls immedi-
ately. Two briey entertained all possible dependencies of
v
f
on , but no clear reasoning was provided.
Professors often used reasoning that involved real-world
analogies. One professor noted that the problem reminds him
of airplane wheels during landing. He said that he is won-
dering which principle is most appropriate in this case for
several minutes he made various hand gestures simulating
the landing of a plane while trying to think about the appli-
cable principle. He noted that the rst thing that comes to
his mind is the angular momentum conservation, but then
concluded that because the ground exerts a torque, the angu-
lar momentum cannot be conserved. Then, he said that per-
haps he should think about the energy dissipation, but noted
that it was not clear to him if the energy lost is higher when
is higher or when the slipping time is longer. Another
professor who believed that a higher friction coefcient im-
plies larger v
f
drew an analogy with walking. He said that
while walking, the harder you push the ground, the faster
you can walk due to the reaction force of the ground. Simi-
larly, the larger the frictional force that the ground exerts on
the wheel, the faster the v
f
should be. Immediately after
being posed the problem, another professor drew an analogy
with a pool ball which initially slips before rolling. He ad-
mitted that he did not have any intuition, but drew a picture
showing the directions of v, and the frictional force and
then wrote down the correct kinematic equations. He did not
bother solving the equations but said that because the accel-
eration is larger for the higher friction case while the time to
start rolling is smaller, the distance traveled before rolling
should be the same regardless of an incorrect inference.
One professor recalled seeing this type of a problem in a
textbook and noted that most likely angular momentum con-
1104 1104 Am. J. Phys., Vol. 70, No. 11, November 2002 Chandralekha Singh
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servation should be used to identify the dependence of v
f
on
. However, during the discussions, he was unable to deter-
mine how to use this principle and for which system the
angular momentum conserved. Another professor who jok-
ingly noted that he even remembers the page number on
which this problem is in a book said that he does not remem-
ber how v
f
should depend on . He said that he was not sure
whether the angular momentum is conserved for this system
and therefore he might attempt a kinematics route. He pre-
ferred not to go to the chalkboard during the discussion and
said that he works best when not under pressure. Another
professor who preferred to go to the chalkboard immediately
drew the correct picture. He noted that no friction implies the
wheel never rolls while high friction implies that it rolls
immediately. He also noted that the frictional force f
k
in-
creases v, because it is the only force on the wheel and it
decreases because it causes a torque in a direction opposite
to
0
. Then, he wrote down f
k
mgma and mgr
I. At this point, he tried to relate the linear and angular
accelerations using ar which is not correct because the
wheel is not rolling at this time. When it led to Imr
2
,
which does not have to be true, he asked for more time to
think about it. Another professor initially said that he is won-
dering whether there is a conservation principle, for ex-
ample, angular momentum conservation, that can be em-
ployed. After pondering for sometime, he admitted that
angular momentum conservation often is tricky to discern.
Because he was not sure how to use it, he decided to use
Newtons law/equation of kinematics but then he got con-
fused about how to calculate the linear acceleration of the
wheel. He thought that friction should slow the wheel so
there must be an additional force on the wheel that should
increase its speed. He decided not to go to the chalkboard.
Later, when we discussed the problem solution, he admitted
that drawing the picture would have helped. Pointing to the
acceleration he jokingly said: this is where my intuition
fails.
B. Student response
The student response can be classied in six broad catego-
ries: 1 Twenty-ve students 37% believed that friction
will act in a direction opposite to the velocity and slow the
wheel down. Therefore, larger implies smaller v
f
. 2
Eighteen students 27% provided responses that were simi-
lar to the expert response category 2 and noted that because
the frictional force is responsible for making the wheel roll,
higher should imply higher v
f
. 3 Six students 9% pro-
vided responses that were similar to the expert response cat-
egory 3 and noted that because lower friction implies
longer slipping time, v
f
will be lower in this case. 4 Four
students 6% provided reasoning different from that in cat-
egory 1 to claim that higher would imply smaller v
f
. 5
Seven students 10.5% believed that v
f
will be independent
of which is the correct response, but only one student
provided qualitatively correct reasoning. 6 Seven students
10.5% provided responses that did not appropriately ad-
dress the question that was asked. For example, one noted
that v
f
will be larger while the wheel is slipping and
smaller when it grips.
Individual discussion shows that students seldom em-
ployed a systematic approach to problem-solving, and cer-
tain types of oversights common in student responses were
rare in the response of professors. Unlike professors, students
rarely examined the limiting cases, contemplated the appli-
cability of a conservation law, or used analogical reasoning.
Many students did not take the time to visualize and analyze
the situation qualitatively and they immediately jumped into
the implementation of the solution based upon supercial
clues. Many thought that the problem was relatively easy
because there was friction on the oor and they were asked
for the nal speed of the wheel once it starts rolling. For
example, 37% of the students thought that friction will re-
duce the linear velocity because the two must oppose each
other. Individual discussions show that several students in
this category did not differentiate between linear and angular
speed. When they were explicitly asked about whether there
was a horizontal speed at the time the wheel hits the oor,
some started to worry that they were confusing the linear and
angular speeds. Some assumed that the wheel will develop a
linear speed as soon as it hits the ground. When asked ex-
plicitly about what will cause it to develop the linear speed,
some noted that the impact will produce a linear speed as
soon as the wheel hits the ground, others said that there has
to be a force in the direction of motion without actually
identifying it, and a few admitted that they could not at the
moment think of a good reason for it. Inadequate time spent
in visualizing the problem caused some students to confuse
the vertical speed of the falling wheel with its horizontal
speed.
Written responses and individual discussions show that
many students in all categories often focused only on the
linear speed and largely ignored what changes the rotational
speed to accomplish the rolling condition v
f
r
f
. Such
responses were rare from the professors. Professors almost
always had a more holistic view of the problem, they always
tried to visualize the problem, and considered the changes in
both the linear and angular speed to establish rolling.
IV. DISCUSSION
This investigation shows that even professors, who have a
vast amount of physics knowledge, when forced to think on
their feet due to the novelty of the problem, have difculties
similar to those encountered by students in some ways. In
solving problems about which they lack intuition, they have
difculty with the initial planning decision making of the
problem solution. We emphasize that the problem posed was
an introductory physics problem for which the planning of
the solution only requires determining the appropriate intro-
ductory physics concepts applicable in the situation. It does
not involve invoking any techniques learned in upper-level
or graduate courses.
The problem posed had two important variables that were
inversely related to v
f
: the force of friction and the time to
start rolling. Professors had great difculty thinking about
the effect of both parameters in the problem. In particular,
they often focused only on one feature of the problem fric-
tion or the time to roll and did not consider the other one
properly. Those who focused on the time to roll often noted
that a high friction would lead to quicker rolling so less
energy will be dissipated in that case and v
f
will be larger.
Those who focused on friction and did not account for the
time to roll, typically concluded that a high friction would
lead to more energy dissipation and hence a smaller v
f
. Only
three professors mentioned that both of the above factors will
1105 1105 Am. J. Phys., Vol. 70, No. 11, November 2002 Chandralekha Singh
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inuence v
f
. Only one of them concluded that it was not
obvious how will affect v
f
. The other two ended up with
incorrect inferences.
On the other hand, unlike students, professors in general
had little difculty considering the effect of friction on both
the linear and rotational aspects of rolling motion simulta-
neously. Their training and experience made it quite natural
to sense that both types of motion will be affected by friction
and consideration of both is important for establishing roll-
ing. The fact that in an unfamiliar situation, even professors
struggled to focus on more than one important aspect of the
problem while in a familiar situation both aspects came natu-
rally to them points to the importance of familiarity and ex-
perience in problem solving.
The rotational problem posed is analogous to one for
which professors have no trouble intuiting the solution: the
case of a completely inelastic collision between two objects.
In this case, the nal speed is determined solely by linear
momentum conservation, and is independent of the collision
time. To check the intuition of professors for the more famil-
iar domain of linear motion, ve of the twenty faculty mem-
bers were asked about the completely inelastic collision of a
bullet with a block resting on a horizontal surface. They were
asked about how the nal speed of the bullet and the block
moving together should depend upon the time it takes the
bullet to come to rest with respect to the block due to the
changes in the block material keeping its mass unchanged if
the material of the block is softer it will take longer for the
bullet to come to rest with respect to the block. All of them
responded correctly, noting that the linear momentum con-
servation guarantees that the time the bullet takes to come to
rest with respect to the block is not relevant for determining
the nal speed of the blockbullet system moving together.
The spontaneity of expert response to this problem, along
with their difculty in grasping how it may be applicable to
the rst problem posed to them, suggests that experience and
familiarity with a particular type of problem are still very
important in the problem-solving skills of professors.
Although professors behaved as students in some aspects,
the problem-solving strategies employed by them were gen-
erally far superior. In particular, they often started by visual-
izing and analyzing the problem qualitatively and searching
for useful conservation principles before resorting to other
routes. They were much more likely to draw analogies and
map the unfamiliar problem onto a familiar one. They often
examined limiting cases; a strategy that was rarely employed
by students. It is true that this problem excluded the zero
friction limit because for that case the time for the wheel to
start rolling is innite. Thus, the nal rolling condition is
never met in this limit and the problem does not have a
solution. Therefore, relying on this limit does not yield use-
ful clues and can lead to incorrect inferences as noted in
several professors responses. Nevertheless, examining the
limiting cases and applicability of general principles is an
excellent problem-solving heuristic which can often make
further analysis of the problem easier. Some professors also
mentioned or attempted to use kinematic methods. Despite
their inability to solve the problem under time pressure, their
holistic view and systematic problem-solving approach and
knowledge-base helped them narrow down the problem
space and prevented a wide range of oversights that were
common in the student response. It was clear that although
their initial intuition was wrong, given enough time, their
systematic approaches would invariably lead to the correct
solution. On the other hand, a majority of students did not
employ a systematic approach to problem solving. Individual
discussions show that many students jumped into the imple-
mentation of the solution without even taking the time to
visualize the problem. Several students thought that the prob-
lem was relatively straightforward because they only focused
on the fact that the effect of friction on a nal speed was
required. Many only focused on the linear motion and they
ignored what was responsible for changing the rotational
motion to establish the rolling condition. Professors adopted
a much more global approach to the problem, and considered
both the linear and rotational aspects of the problem.
The surprised reaction of several professors after nding
out that v
f
is independent of hints at why the idealized
situations, for example, motion on a frictionless surface, are
very difcult for students to internalize. For example, one
professor noted that he found the answer counterintuitive
because it implies that the nal velocity of the wheel will be
the same for ice and for a high friction surface. Of course, in
a realistic situation, factors such as air-resistance and rolling
friction would make v
f
dependent on . Only after one has
carefully considered the limitations of the idealizations in the
light of our everyday experience can one feel comfortable
making the corresponding inferences.
V. SUMMARY
Expertise in physics is founded upon the pillars of intu-
ition, knowledge, and experience. Physicists continually
transform their experiences into knowledge. Intuition plays
the role of a catalyst, greatly speeding up the process by
allowing for shortcuts to be taken during problem solving.
We identied an introductory-level physics problem for
which a group of twenty physics professors displayed a
nearly universal lack of intuition. Although professors would
have performed better without the time constraint, our goal
here was to elicit the thought-processes and problem-solving
strategies of experts as they venture into solving a nonintui-
tive problem. In quizzes and examinations, students often
work under a similar time constraint.
The inherent difculty of the problem posed in this study
is comparable to problems professors can solve without
much difculty. This study suggests that the perceived com-
plexity of a problem not only depends on its inherent com-
plexity but also on the experience, familiarity, and intuition
we have built about a certain class of problems. It has often
been said that problems are either impossible or trivial, de-
pending on ones success at solving them. Introductory stu-
dents lack the vast experience, knowledge-base, and intuition
that the professors have about a majority of introductory
physics problems. As instructors, we should not be surprised
that beginning students have great difculty solving the bal-
listic pendulum problem, which requires invoking both the
momentum and energy conservation principles. For profes-
sors, who have built an intuition about this class of problems,
it appears easy. For students, who lack intuition about
these problems, it is difcult to focus on several aspects of
the problem simultaneously. There are likely to be less sur-
prises if we put ourselves in students shoes and analyze the
difculty of a problem from their perspective.
There are indeed few introductory-level problems for
which expert intuition is so universally lacking. The collec-
tive response of twenty professors to the problem suggests
that none have frequently encountered or carefully thought
1106 1106 Am. J. Phys., Vol. 70, No. 11, November 2002 Chandralekha Singh
Downloaded 10 Sep 2013 to 14.139.196.4. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
about a problem like it. A survey of most of the contempo-
rary introductory textbooks supports this hypothesis. The re-
sponse of professors in this study can shed some light on the
kinds of difculties that able students face as they solve
problems and strive to develop physical intuition of their
own. Finally, although professors and students both had dif-
culties in solving the problem, expert problem-solving
strategies were generally far superior. Although professors
did not immediately know how to solve the problem, they
demonstrated that they know how to solve problems, and
given enough time, their systematic approaches would have
inevitably led to the correct solution. It may be useful to
design instructional strategies that explicitly teach problem-
solving heuristics and help students build and employ intu-
ition in physical problems as we help our students learn vari-
ous physics concepts.
ACKNOWLEDGMENTS
We are very grateful to all of the faculty with whom the
problem was discussed for their time and to F. Reif, R. Gla-
ser, H. Simon late, A. Lesgold, R. Tate, P. Shepard, and J.
Levy for useful discussions. We thank F. Reif, R. Glaser, J.
Levy, and R. Johnsen for a critical reading of the manuscript.
This work was supported in part by the Spencer Foundation
and the National Science Foundation.
APPENDIX A: SOLUTION METHODS FOR A HOOP
Imr
2

The answer is that v

f
is independent of . This answer
suggests the applicability of a conservation principle. The
problem can be viewed as a rotational inelastic collision with
the oor, analogous to a linear inelastic collision. We can
invoke the conservation of angular momentum principle
about a xed axis through the point where the wheel initially
touches the ground see Figs. 1 and 2. The angular momen-
tum of the wheel is constant about this axis during the time
the wheel slips, there is a kinetic frictional force, but because
the line of action of this force passes through the axis, it does
not produce a torque about the axis. Let m, r, and I be the
mass, radius, and moment of inertia of the wheel about its
center of mass, respectively. For simplicity, we will assume
that the wheel can be approximated as a hoop so that I
mr
2
. Let
0
be the initial angular speed of the wheel about
its center of mass, and v
f
and
f
be the linear and angular
speed about its center of mass, respectively, when it starts to
roll see Fig. 1. The initial angular momentum before the
wheel touches the ground is just due to the spin and
L
0
I
0
mr
2

0
. When the wheel is rolling, the angular
momentum about the chosen axis has two contributions: one
due to the spin and the other due to the linear motion
r

cm
(mv

f
), where r

cm
is the displacement of the center of
mass of the wheel from the xed chosen axis see Fig. 2.
The magnitude of the latter contribution is rmv
f
see Fig. 2
so that
L
f
I
f
rmv
f
Imr
2

f
2mr
2

f
, A1
where the rolling condition v
f
r
f
has been used. If we use
the fact that L
0
L
f
, we nd that
f

0
/2 independent of
see Fig. 3.
Another approach to this problem is to use the equations
of linear and rotational kinematics and the condition for roll-
ing. Let t0 be the time when the wheel drops on the oor
and t be the time during which it slips before starting to roll.
If the wheel is spinning in the clockwise direction when it
drops on the oor, the frictional force will act to the right and
will increase its linear velocity to the right with a constant
acceleration aF
k
/mg where g is the magnitude of the
acceleration due to gravity from its initial value of zero
only spinning. The initial angular velocity
0
will decrease
with a constant angular acceleration rF
k
/Ig/r, be-
cause the frictional force at the rim of the wheel causes a
counterclockwise torque rF
k
. From the equations of kine-
matics, we nd
v
f
atgt, A2

0
t
0
gt/r, A3
because the wheel starts to roll without slipping at time t,
v
f
r
f
. If we substitute the values of v
f
and
f
from Eqs.
A2 and A3, we obtain t
0
r/(2g). Then by substitut-
ing t in Eqs. A2 and A3, we nd v
f

0
r/2 and
f

0
/2, independent of .
The above result can also be veried by noting that the
energy dissipated by friction during slipping is independent
of . By using the work-kinetic energy theorem, we obtain
Fig. 1. Schematic diagram of the wheel at four different times: a spinning
on a frictionless shaft, b hitting the oor, c slipping on the oor, and d
rolling on the oor.
Fig. 2. Schematic diagram of the wheel showing r
cm
, v

f
, r, and the xed
axis about which the initial and nal angular momenta are calculated in
Appendix A.
1107 1107 Am. J. Phys., Vol. 70, No. 11, November 2002 Chandralekha Singh
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W
k
K
f
K
i
, where W
k
W
lin
W
rot
, K
f
K
f , lin
K
f , rot
,
and K
i
K
i, lin
K
i, rot
are the total work done by friction, and
the total nal and initial kinetic energies, respectively. W
lin
,
W
rot
, K
lin
, and K
rot
are the work done by friction for the
linear and rotational motion, and the linear and rotational
kinetic energies, respectively,
W
lin
F
k
xmg gt
2
/2m
0
2
r
2
/8, A4
W
rot
F
k
rmgr
0
tgt
2
/ 2r
3m
0
2
r
2
/8, A5
where x and are the linear and angular displacements of the
wheel, respectively, during the time it slips, and we have
used the equations of linear and rotational kinematics to re-
late x and to t. Thus, the total energy dissipated by friction
during slipping, W
k
m
0
2
r
2
/4, is the same regardless of
although the power dissipated energy dissipated per unit
time depends upon it. For large , t is small but the power
dissipated is high, which ensures that the total energy dissi-
pated is independent of . Therefore, v
f
is the same regard-
less of .
APPENDIX B: GENERAL SOLUTION FOR
ARBITRARY MOMENT OF INERTIA I
To show that v
f
is independent of regardless of the
moment of inertia I of the wheel, we note that
mgr/I, so that
f

0
mgrt/I. Using the condition
for rolling without slipping, v
f
r
f
, we obtain
t

0
r
g 1mr
2
/I
. B1
If we substitute the value of t in v
f
and
f
, we nd that they
are independent of :
v
f

0
r
1mr
2
/I
, B2

0
1mr
2
/I
. B3
We can calculate the total work done by friction and the
work done for the linear and rotational components of mo-
tion with xat
2
/2 and
0
tt
2
/2 and nd that they are
independent of :
W
lin
F
k
x
I
2
mr
2

0
2
2 mr
2
I
2
, B4
W
rot

I mr
2

0
2
2 mr
2
I
2

I
2
mr
2

0
2
mr
2
I
2
, B5
W
k
W
lin
W
rot

I
0
2
mr
2
2 mr
2
I
. B6
We can also calculate the change in the total kinetic energy
of the system and show that it is equal to the total work done
independent of :
K
i
I
0
2
/2, B7
K
f
mv
f
2
I
f
2
/2I
0
2
I
2 mr
2
I
, B8
W
k
K
f
K
i
. B9
The I dependence actually the dependence on the shape of
the object because it is the ratio I/(mr
2
) that is important of

f
/
0
1/(1mr
2
/I) is particularly interesting. In the limit
I/(mr
2
)0 the mass of the object is localized close to the
axis,
f
0, so that maximal energy is dissipated by fric-
tion. The largest value I can take is Imr
2
, which corre-
sponds to the case in Appendix A. Qualitatively, the depen-
dence of
f
on I/(mr
2
) can be understood by noting that less
energy is dissipated if the angular speed has not decreased
signicantly when the rolling begins (v
f
r
f
). If the shape
of the object is changed so that I/(mr
2
) decreases while all
other parameters are kept xed, the angular speed will de-
crease more before the rolling condition is established.
The calculations can be repeated for the case where the
initial linear speed is nonzero when the object touches the
ground, that is, v
0
0 and
0
0 as in the case of a non-
spinning bowling ball thrown on the oor at an angle or a
struck pool ball that initially only has a linear speed. The
independence of v
f
on still holds in fact, it holds even for
cases where the object may initially have both nonzero linear
and angular speeds. Interestingly, in this case, the I/(mr
2
)
dependence of v
f
and
f
is opposite that of the case noted
above for
0
0 and v
0
0. Here, v
f
/v
0
1/(1I/(mr
2
)).
Therefore, in the limit as I/(mr
2
)0 the mass of the object
is localized close to the axis, v
f
v
0
, so that negligible
energy is dissipated by friction before the wheel starts roll-
ing. Qualitatively, the dependence of v
f
on I/(mr
2
) can be
understood by noting that less energy is dissipated if the
angular speed increases quickly to catch up with the linear
speed so that v
f
r
f
without the linear speed having de-
creased signicantly. Obviously, the angular speed will in-
crease quickly if I/(mr
2
) is small.
a
Electronic mail: singh@bondi.phyast.pitt.edu
1
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Fig. 3. Graph of linear speed v and scaled angular speed r vs time. Larger
values of lead to shorter locking times, but the nal speed v
f
is indepen-
dent of .
1108 1108 Am. J. Phys., Vol. 70, No. 11, November 2002 Chandralekha Singh
Downloaded 10 Sep 2013 to 14.139.196.4. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
in solving problems, Am. J. Phys. 64, 14951503 1996; J. P. Mestre, R.
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6
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New York, 1997, 5th ed., p. 291, Prob. 16P.
7
The problem posed to the students had the following additional sentences
at the end: For example, will v
f
be larger or smaller if the wheel fell on ice
with small friction compared to a rough surface? Describe your intuition
along with the reasoning. You must draw a diagram and explain your
approach to get full credit.
SCIENCE IN THE 1920s
In the 1920s, science was enjoying a tremendous popular resurgence, and the burgeoning
mass-circulation press, aided by the advertising industry, had become propagandists for the ad-
vances of modern technology, daily trumpeting such marvels as Einsteins revolutionary theory
of relativitylocked in the atom, reported the Saturday Evening Post, was a source of power
inconceivably greater than any possible requirement of the human raceto the latest high-
powered vacuum cleaner. Einstein was front-page news, and reporters followed his every move,
documenting his self-effacing mannerisms and utterances as further evidence of his genius. He
was the worlds most celebrated scientist, noted the historian Daniel Kevles, and his cult status
not only helped enlarge the prestige of pure science, it endowed the entire profession with a
kind of awesome glamor. By 1925, the New Republic wrote that scientists were regarded as
members of an exclusive and powerful fraternity: Today the scientist sits in the seats of the
mighty. He is the president of great universities, the chairman of semi-ofcial government coun-
cils, the trusted adviser of states and even corporations.
Jennett Conant, Tuxedo Park Simon & Schuster, New York, NY, 2002, pp. 5556.
1109 1109 Am. J. Phys., Vol. 70, No. 11, November 2002 Chandralekha Singh
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