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Chapter 4
Force and Motion
Recommended class days: 2
When asked to draw a force diagram for some simple situation, most students emerging from any level
of introductory physics course are likely to draw objects which look like a porcupine shot by an Indian
hunting party—the number and direction of pointed entities being essentially stochastic.
Arnold Arons (1979)
Background Information
The chapter on Physics Education Research has already provided much of the background
information on student difficulties with force and motion. The major reference is Halloun and
Hestenes (1985b). Other reference are cited in Reddish (1994). This section will summarize the
earlier information.
There are three basic issues, all of which cause serious difficulties for students:
• What is a force?
• What is the connection between force and motion?
• How are forces between different object related?
What is a force? Students don’t have a clear idea of just what a force is. They tend not to
distinguish between force, inertia, energy, power, or even velocity, often using these terms
interchangeably. In addition:
• Some students believe that only animate objects can exert forces. They don’t believe that a table
exerts an upward force on an object; the table simply “gets in the way of the object wanting to
fall.”
• Forces recognized by physicists are often seen by students as simply influences on an object’s
motion, not as forces. Thus friction is not a force but merely “what makes it stop.” Gravity is not
a force but simply “what makes it fall.”
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• A majority of students believe in an impetus theory of motion. In throwing a ball, the hand
imparts a “force of the throw” to the ball. This is a property of the ball (an inherent force) and
travels with the ball “to keep it moving.” Typically 75% or more of students beginning calculusbased physics thinking that a ball tossed upward has an upward “force of the throw” or “force of
your hand” on it after it leaves your hand.
• Students tend to view forces from the perspective of the applier of force rather than from the
perspective of the object experiencing force. That is, they recognize the pushes and pulls they
must apply to move an object, but they don’t recognize that the object may experience
additional inanimate forces of friction, gravity, and so on. This is one of the major reasons they
think that motion requires a force.
How are force and motion connected? The prevailing student belief is that motion requires a
force. This belief is based on much common-sense evidence, and it is a belief that is highly resistant
to change. More specifically, the “student version” of the laws of motion is:
• If there’s no force on an object, the object is at rest or will immediately come to rest.
• The converse is not true. An object at rest does not automatically imply no net force.
• Motion requires a force or, alternatively, force causes motion.
• In general, force is proportional to velocity.
These are the most common student beliefs, but not every student necessarily holds every belief. It
is important to note that students are not at all consistent in their application of these beliefs; for
example, they may apply some beliefs to vertical motion but not to horizontal motion. Even so,
recognition of these prevailing alternative conceptions will give instructors better insight into
student responses and the difficulties students face.
How are forces between two different objects related? Newton’s third law is profound,
much more so than the first two. For students, the third law is a subtle and difficult topic, and they
will find this to be a challenging subject. However, if you devote ample time to the underlying
principles and provide students with ample opportunities for practice and feedback, most will end
up finding this a rewarding chapter where the concepts of force and motion suddenly begin to
“make sense.”
Suppose a large truck and a compact car have a head-on collision. During the collision, is the
force of the truck on the car larger, smaller, or equal to the force of the car on the truck? This
question, and several similar questions on the Force Concept Inventory, is initially missed by 70%
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to 80% of students in typical calculus-based physics courses. Presumably students entering an
algebra-based course would fare considerably worse.
Halloun and Hestenes (1985b) have characterized student beliefs about interactions in terms of a
dominance principle: The larger (or faster or more active) object exerts a larger force than the
smaller (or slower or less active) object. Students tend to view an interaction as a “conflict” in
which the stronger wins. It’s not hard to understand how this common-sense view comes about.
After all, the effect of the collision on the compact car is much larger than its effect on the truck.
Different effects would seem to require different causes, hence different amounts of force. The
difference in the masses does not appear to students as a significant factor in drawing conclusions
about forces. This basic misconception about interaction forces is likely the most persistent and hard
to change of all the student misconceptions in mechanics.
Some of the more specific difficulties students have with Newton’s third law and with
interacting systems are:
• Students don’t believe Newton’s third law. It’s too contrary to common sense.
• Students have difficulty identifying action/reaction force pairs:
They match two forces on the same object.
They place forces on the wrong objects.
They don’t believe that long-range forces (e.g., gravity) have reaction forces.
• Students confuse equal force with equal acceleration.
• Students don’t understand tension:
They think that tension is the sum of the forces exerted at the two ends of a string.
They think that tension exerts a force only in the direction of motion.
They think that tension can pass through an object to another string on the other side.
• Students often don’t recognize that objects connected by an inextensible string must have
accelerations of equal magnitude.
Interacting-system problems are difficult and frustrating for students. There are no magic
formulas to search for, so students must correctly apply Newton’s second law to each object. But
even if they realize this to be the proper approach, they cannot succeed without first identifying all
the interaction forces and then making proper use of the third law.
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Student Learning Objectives
In covering the material of this chapter, students will
• Recognize what does and does not constitute a force.
• Identify the specific forces acting on an object.
• Draw an accurate free-body diagram of an object.
• Begin the process of understanding the connection between force and motion.
• Understand the relationship between forces between two different objects.
• Begin learning how to explain an observation on the basis of physical principles.
Pedagogical Approach
Chapter 4 is a qualitative overview to dynamics, a prelude to the quantitative analysis of
Chapters 5–8. This is analogous to the qualitative introduction to motion in Chapter 1, followed
by quantitative kinematics in Chapter 2.
This is an extremely important chapter that should not be rushed. Successful learning of
Newtonian mechanics requires a major reorientation of thinking for most students. This chapter lays
the conceptual groundwork that tends to be overlooked in the typical rush to start using Newton’s
laws to solve problems. As the chapter objectives note, students must recognize what a force is,
correctly identify the forces on an object, and be able to express their knowledge about forces on a
free-body diagram before they can successfully solve dynamics problems that go beyond simple
plug-and-chug. Most of the chapter, the Student Workbook exercises, and the end-of-chapter
problems are devoted to these issues.
Chapter 4 introduces another new idea—that of experimental evidence. The fact that
acceleration is proportional to net force is an observation about how the world works. It is
important, through lecture demonstrations and laboratory activities, to present students with clear
and compelling evidence that force determines an object’s acceleration, not its velocity. This
chapter seeks to establish the connection between force and acceleration, then Chapter 5 will follow
up on this idea with actual problem solving.
Newton’s third law is only introduced at the end of the chapter. Although we ultimately want
students to understand forces as interactions, the conceptual hurdles that students face are high enough
already, and the topic of interactions adds yet more. The authors have found that students do much
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better if instruction concentrates first on how a single object responds to forces. Only after these ideas
begin to seem comfortable can students deal with the issue of how two objects interact with each other.
Chapter 4 introduces the idea of dividing a problem into the system, which is the object or
objects of interest, and the environment, which is everything else. For most of the chapter, the
system is a single object and all forces originate in the environment. When the third law is
introduced, the system may consist of two or more interacting objects. This system/environment
division is a pedagogical device that will be expanded and elaborated on in many chapters to come.
Instructors may want to look ahead in the textbook to Figures 10.4 and 11.15 to see where these
ideas are headed.
Suggested Lecture Outlines
DAY 1: We have found it effective to focus the first class day on learning to identify forces. The
connection between force and motion then enters on Day 2.
This is a chapter for which instructors are urged to have a wide variety of simple props
available—springs, ropes, sticks, masses, blocks, rubber balls, etc. An effective tactic is to first
apply a force to a student, asking him or her to verify from sensations that there is a push or a pull.
Then apply the same force to an inanimate object, such as a block, and ask if the block also
experiences the same force.
If you pull on a student’s arm (gently!), he will agree that he feels a pulling force. If you now
hand him one end of a rope, pull on the other end, and ask what he experiences, more than likely
he’ll reply, “You’re pulling on me.” This is an opportunity to distinguish the immediate cause at a
point of contact, namely the tension in the rope, from the ultimate cause of whatever pulls the other
end of the rope.
Hand a student one end of a spring and pull the other end. Have her agree that she feels a pulling
force as the spring stretches. Then hang a mass from a spring. Because the spring stretches, most
will now agree that the spring exerts a force on the mass. Then hang the same mass from a string.
They don’t see the string stretch, so does it exert a force? You can ask students to imagine if they
were hanging from a rope tied around their waist—would they feel a force from the rope? Once
they agree that the string does exert a force on the block, you have a good opportunity to talk briefly
about molecular bonds as atomic-level springs; the string does stretch ever so slightly as the
molecular bonds stretch, and that’s what tension is.
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These activities are good preparation for the critical demonstration of Day 1. Place a block in
the center of the lecture table and ask students what force or forces are acting on the block. You’ll
get lots of responses of “gravity.” If you inquire about other forces, a few will say “the normal
force.” Reply, “OK, so you learned in high school about this thing called the normal force, but how
many of you really believe that the table is exerting an upward force on the block?” My experience
is that less than one-third of the students will raise their hands. While their high school physics
books and teachers may have told them about normal forces, their doubts arise because they don’t
see any mechanism by which the table exerts a force on the block. They need to be convinced, with
evidence, that such a force is really there.
To begin a series of demonstrations, hand out several compression springs (fairly stiff ones) and
ask students to squeeze them. They’ll agree that the spring pushes back when it’s squeezed. Stand a
fairly soft spring on the table and set the block on it, giving a very visible compression. (You’ll
need some way to stabilize the block on top the spring.) They’ll agree that the spring exerts an
upward force on the block. Then switch to a stiffer spring that barely compresses. This leads to the
conclusion that the amount of compression is not the issue. Now place the block on a thin board (or
meter stick) that is supported at the ends, causing the board to sag. They’ll agree that the board is
also springy and exerts an upward force. (Ask them to imagine how their finger would feel if they
pressed the board down and held it.) Finally, return to the block sitting on the table. No discernible
sag or compression, so is there a force?
The coup de grace is to place a mirror flat on the table, then reflect a laser beam from it at
grazing incidence (laser beam almost parallel to table surface) with the reflected laser spot striking
the side wall of the classroom. Ask a student to place a pencil tip next to the laser spot, then climb
up and stand on the lecture table! Virtually all lecture tables will flex enough under the weight of
the instructor to deflect the laser spot up a couple of millimeters. (It’s good to try this in advance to
make sure it’s going to work. Note that many lecture tables have cross bracing underneath, and you
don’t want to stand right over one of the stiffer braces.) Voila! The table really does compress, even
if only a microscopic amount, and it exerts the same upward force as a spring. (Those molecular
bonds again, as it’s good to note.)
This sequence of demonstrations takes some time, but it’s well worth it. They make a
memorable impression on students. We have received numerous unsolicited comments on
evaluation forms that this demonstration was what really convinced students about the reality of
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forces exerted by inanimate objects. (Note that appealing to Fnet  0 for static equilibrium to infer
the existence of a normal force is not in the least convincing to most students. Many don’t yet
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accept that an object at rest must have Fnet  0. )
You can consolidate what they’ve seen about force to this point: forces are a push or pull, they
occur at a point of contact between the object and some identifiable agent that exerts the force, and
forces can be exerted by either animate or inanimate agents. This is a good opening to consider
friction. Ask them to imagine dragging their hand across a very rough surface. Is there a force?
What direction? Then what about a box sliding across the surface?
Then turn to gravity. Drop a ball—why does it fall? This is a different type of force, a longrange force. You might want to have some magnets to demonstrate other long-range forces, but
emphasize that gravity is the only long-range force that will be considered for quite some time.
Every other force must be a contact force.
Finally, toss a ball straight up into the air and inquire about the force or forces on the ball after it
leaves your hand but before it reaches the top. If you had opened class with this question, a majority
of the students would likely assign an upward “force of the throw” to the ball. Many may be
doubting this answer after the sequence of demonstrations you’ve been through, but many others
will still want an upward “force of the throw.” You can play devil’s advocate, first getting them to
agree that there’s no contact with anything in the environment, then asking “But how can it go up
unless there’s an upward force?” Don’t answer! This is where you really want them to be
confronted by difficulties with their alternative conceptions of force and motion. Ask them to think
about it, and promise to resolve the issue during the next class.
With the time remaining, go over Tactics Box 4.2 for identifying forces and work through
several simple examples. You can start out by carefully working through the steps of the Tactics
Box with the following example:
Example: A block being dragged up a hill by a rope.
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Then let them practice a few themselves. Two possibilities include:
Example: A block A hangs from the ceiling by a rope. Another block B hangs from A. Identify the
forces acting on A.
Example:
A note about notation: There is no happy solution to the issue of what symbols to use for forces.
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T for tension, N for normal force, and W for weight run into conflicts with other uses of T, N, and
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W. Fstring, Fgrav, and other subscripted uses of F are accurate, but they’re tedious and students
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have a tendency to forget the subscripts. We’ve elected to keep T for tension but use lower case n
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and w for normal force and weight. Regardless of the choices made, remind students that the
symbols are a language and that they can’t communicate with you or other students if they try to
invent their own language. Insist that students learn and use the notation of the text and the
classroom.
DAY 2: The objective of Day 2 is to “discover” Newton’s first and second laws. What are the
consequences of a force on an object? You’ll want to use whatever demonstrations you have
available to show that an object continues to move at constant velocity in the absence of a net force
and that force causes an object to accelerate. Convincing demonstrations here go a long way toward
changing the common student beliefs that motion requires a force and that force is proportional to
velocity.
For demonstrating the first law, gliders on an air track are particularly effective here, as they can
be made to move quite slowly, without much change in their speed. Also effective are dry ice pucks
that slide down a long sheet of glass on a cushion of CO2 vapor. Other interesting demonstrations
that illustrate the first law include:
Demonstration: Show how to get ketchup out of a bottle, as in the Try it Yourself on page 107 of
the text. The green gel used to ease sunburn pain works well here, and requires no refrigeration.
First hit down on the bottom of the inverted bottle (which is pointing up), the “standard” way of
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getting ketchup out. You’ll see that the ketchup tends to move toward the bottle’s bottom. But if
you hit sharply up near the neck, the bottle will jerk upward while the ketchup remains stationary,
leading to a net forward motion of the ketchup.
Demonstration: A classic demonstration is to pull a tablecloth out from under some dishes. The
key is to pull rapidly with a small downward component to the pull. You will need to justify a bit
why you can neglect the force of the tablecloth on the dishes; it’s mostly because it acts for only a
very short time.
Demonstration: Show some test-dummy car crash videos, if available. Such videos vividly
illustrating the role of the first law in “throwing” passengers from a car.
Computer-based force probes and motion sensors are especially effective for demonstrating
Newton’s second law. Attach a force probe to a low-friction cart on a track, then use a string and
pulley to connect the force probe to a hanging weight. When the weight is released, the force probe
measures the string tension while the motion detector measures the motion. This gives excellent
results, showing that a constant force produces a linearly increasing velocity and a constant
acceleration. Doubling the tension force doubles the acceleration, while doubling the cart mass
halves the acceleration. These results can all be shown in just a few minutes with software tools.
Priscilla Laws (1997) has developed an alternative approach that can be shown in a large lecture
hall, using volunteers, though it is better suited to lab. One student sits on a cart with low-friction
wheels (a Kinesthetics Cart is available from PASCO) and holds one end of a spring scale. Another
students pulls the cart by pulling the other end of the spring scale, endeavoring to keep the reading
as constant as possible. Nearly all students expect, before trying this, that they’re going to walk or
run at a steady speed; they are very surprised to find that they must accelerate to keep the force
constant. Once the cart is moving, they can let the spring scale reading drop to zero but the cart
keeps moving!
These demonstrations lead to the introduction of Newton’s second law in the operational form
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a  Fnet / m. Although F  ma looks more elegant, it conveys to many students the wrong
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impression that “ m a is a force.” The operational form conveys a better sense of cause and effect
and helps students to develop better ability to reason with Newtonian concepts.
The summary at the end of these demonstrations is that “motion does not need a cause.” The
question is not “Why does an object move?” but “Why does it change its motion?” Remind students
that this is easy to say, but it takes effort and practice to begin to think this way. They shouldn’t be
surprised if it all seems confusing at first, but promise them that they will “get it” if they persevere.
Several examples are worthwhile at this point, depending on how much time is left.
Example 1: An elevator is going up at a steady speed. First have students identify tension and
weight as the only two forces. Then ask: “Is T greater than, equal to, or less than w? Or is there not
enough information to tell?” Many will answer “greater” because “motion requires a force.”
Example 2: Push a block across the table at steady speed. Since you’re exerting a force on it, why
isn’t it accelerating? Ask students to identify all the forces and to draw a free-body diagram.
Finally, ask them to compare the size of the pushing force and the size of the friction force.
Example 3: Push the same block fairly quickly, then release it so that it slides some distance
before stopping. Have students analyze the forces and reach the conclusion that the acceleration
vector points backward. Remind them of what they learned in kinematics about situations in which
the acceleration vector is opposite the velocity vector. When done, congratulate them on having
performed a Newtonian analysis to explain why a block coasts to a stop after you release it.
Example 4: Return to the issue of the ball tossed straight up—where you left students hanging on
Day 1 as to how it moves up without an upward force. Now you can complete the analysis, using
Example 3 as a horizontal analogy. Remind them that no cause is needed for the ball to move
upward—inertia takes care of that. The proper question to ask is not “Why does it move upward?”
but “Why does it slow down and eventually fall?”
As you work through these examples, point out that you are giving an explanation of an event in
terms of physical principles (Newton’s laws) and logical inference. They’ll be asked to give similar
explanations in some end-of-chapter problems (especially problems such as 64 and 65). Students
find this kind of reasoning very hard to do, so suggest that they use the just-worked examples as a
template.
Free-body diagrams have been skipped over thus far in class, although you likely will have
drawn some simple ones in the process of explaining things. If students understand how to identify
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forces, they’ve already overcome the largest hurdle to drawing free-body diagrams. They can
practice on their own, following examples in the text, and you can give more explicit examples in
the next chapter. One issue to watch for, and perhaps note in simple examples, is that free-body
diagrams show only the forces acting on the object. Many students will try to include forces exerted
by the object on other systems. If you have a few minutes left, working through the free-body
diagram for an object sliding on an inclined plane, with friction, is an excellent example.
DAY 3: On Day 3 you can turn to interacting objects and Newton’s third law. Most students readily
accept that if A pushes/pulls B, then B pushes/pulls back on A. If you ask a student to stretch a
spring, she can “feel” that the spring pulls on her hand at the same time she pulls on the spring.
Long-range forces are more troublesome because students don’t yet understand the role of mass in
the “outcome” of an interaction. The earth clearly pulls down on a ball that is dropped, but there’s
little evidence of the ball exerting a force on the earth. You can make this idea plausible by
demonstrations with magnets. If a student holds two fairly strong magnets, he can feel that each is
pulling (or pushing) on the other. If you attach magnets in the repulsive orientation to two gliders on
an air track, both gliders move apart when released. More importantly, if you now weight the
gliders differently, students can see that there is a mass effect and that the lighter glider does most
of the motion. Now it’s a much smaller step to accepting that the ball really does exert a reaction
force on the earth.
With this as a starting point, it’s good to spend an entire class asking students to follow the steps
in Tactics Box 4.4 for identifying and labeling action/reaction pairs and for drawing free-body
diagrams. It’s worth starting with a simple example, such as a block sitting at rest on a table. Have
students identify the objects that make up the system—the block and the table—and identify all
forces acting on the objects. They should be able to identify the external forces, which act from
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outside the system, and give these labels like nT and wB .
Then they should identify the forces between objects that make up the system. The latter are the
forces that make up the action/reaction pairs of Newton’s third law, and should be labeled using the
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FA on B notation, and connected with dotted lines. Even this simple situation will rapidly lead to
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conflict for many students who want w and n to be an action/reaction pair.
Then place a second block on top of the first. The lower block now experiences two normal
forces, one from above and one from below. Again, this apparently trivial situation is initially
difficult for many students. Fortunately, most will “get it” after just a few such examples.
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The important idea of propulsion forces is often difficult for students to grasp. You can convey
the idea by asking them to imagine what would happen on a frictionless surface. Also ask them to
imagine what direction loose gravel would be “kicked” as they sprint forward or a car accelerates
forward. The force of the tire on the gravel must be directed backward, so the force of the gravel on
the tire—the propulsion force—must be directed forward.
These exercises show that the members of an action/reaction pair act in opposite directions. But
what about the magnitudes of these forces? Unfortunately, it isn’t easy to provide a demonstration
of the equal magnitudes. By far the most convincing demonstration we know of is the collidingforce-probe experiment comes from Thornton, Sokoloff, and Law’s RealTime Physics. Two lowfriction carts of very different masses are pushed toward each other such that the collision occurs
between the tips of their respective force probes. This gives instant and dramatic confirmation that
the forces between two colliding carts are always equal in magnitude, regardless of the masses or
the initial velocities of the carts. A simpler demonstration is to have two students of different size
push against each other with bathroom scales, each calling out the reading on “his/her” scale as they
move forward or backward. An interesting variant is for one student to stand on a skateboard. Even
though this student begins to accelerate backward, “losing” the battle, both students will still call
out the same forces.
Clicker Question: 10-year-old Sarah stands on a skateboard. Her older brother Jack starts
pushing her backward and she starts speeding up. The force of Jack on Sarah is
A. Greater than the force of Sarah on Jack.
B. Equal to the force of Sarah on Jack.
C. Less than the force of Sarah on Jack.
Other Resources
In addition to the specific suggestions made above in the daily lecture outlines, here are some other
suggestions for demonstrations and questions you could weave into your class time.
Suggested Demonstrations
Tug of War. Select two teams—the boys vs. the girls—for a tug of war. Choose them so that the
boys’ team is the stronger. The have a gentle tug of war, which the boys will presumably win. Ask
which team pulled harder. Now make the boys wear plastic grocery bags on their feet. The girls will
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now win. You can then point out the important role of friction in determining who wins; it is net
force that determines acceleration.
Sample Reading Quiz Questions
1. What is a “net force?”
2. List at least three of the steps used to identify the forces acting on an object.
3. Which of these is not a force discussed in this chapter?
A. The tension force.
C. The orthogonal force.
B. The normal force.
D. The thrust force.
4. An action/reaction pair of forces
A. points in the same direction.
B. acts on the same object.
C. are always long-range forces.
D. acts on two different objects.
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