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Chapter 5
Applying Newton’s Laws
Recommended class days: 4
Background Information
Chapter 5 is a continuation of Chapter 4, with the focus now shifting to quantitative problem
solving. Learning to recognize and identify forces carefully and accurately is one of the largest
hurdles students face. If they’ve begun to get a good handle on this from Chapter 4, then the present
chapter becomes more a matter of consolidating and using that information.
Chapter 4 introduced free-body diagrams, but likely you didn’t get too much opportunity to
have students practice drawing free-body diagrams in class. That needs to be an important focus of
the present chapter. Student difficulties with free-body diagrams include:
• Identifying a force correctly, but not knowing which way it points.
• Including forces exerted by the object, not just forces exerted on the object.
• Placing the tips, rather than the tails, of the vectors at the origin, which makes it hard to
determine the x- and y-components of the forces.
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• Including “ ma ” as a separate force. After all, Newton’s second law says F  ma. This error is
likely a continued belief in the need for some kind of “force of motion.”
• Not using a coordinate system, or using an inappropriate one.
• Making errors in determining the component of the force vectors.
Fortunately, these are mostly technical errors rather than conceptual errors. Most can be overcome
fairly quickly with focused practice and feedback on drawing free-body diagrams.
A conceptual issue does arise with mass, weight, and apparent weight. Daily life makes no
distinction between mass and weight, so students are accustomed to using the terms
interchangeably. They need an opportunity to practice using the distinctions.
The authors have found the concept of apparent weight to be pedagogically useful. Students
associate weight with what a scale reads. They begin to exhibit confusion and uncertainty if you
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direct their attention to what a scale would read in a rocket taking off, in a roller coaster going over
the top, or in an elevator in free fall after the cable breaks. Some will assert that weight changes in
these circumstances, but they’re not sure how to reconcile that with weight as a fixed and
unchangeable force of the earth’s gravity on the object. It takes quite a bit of discussion for most to
accept that they cannot sense the long-range weight force itself, that the sensation they call weight
is actually the sensation of the normal force of a surface pressing against them. The magnitude n of
the normal force is then defined to be the apparent weight. Its value depends on the acceleration,
and students can then recognize that their apparent weight will be more or less than their true weight
in an elevator accelerating up or down.
This understanding seems to be a significant conceptual step upward for many students. Perhaps
this is because the exercises force them to focus carefully on forces and motion, rather than falling
back upon equations or upon the alternative conceptions they brought to class with them. The idea
of apparent weight is later used to understand how water stays in a bucket swung over your head,
whether or not the string stays taut for a ball swung in a vertical circle, and why astronauts are
“weightless.” (A significant number of students believe astronauts are weightless because there’s no
gravity in space.)
Student Learning Objectives
In covering the material of this chapter, students will
• Draw and make effective use of free-body diagrams.
• Recognize and solve simple equilibrium problems.
• Distinguish mass, weight, and apparent weight.
• Learn and use simple models of friction and drag.
• Apply the full strategy for force and motion problems to problems in single-particle dynamics.
• Understand how to apply Newton’s laws for systems of interacting particles.
Pedagogical Approach
After four chapters of preparation, Chapter 5 finally focuses on solving dynamics problems. Even
so, much of the emphasis remains on analyzing problem statements and on Newtonian reasoning.
All of the analysis tools—pictorial representation, motion diagrams, and free-body diagrams—are
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now gathered into the full problem-solving strategy on page 142. The worked examples in this
chapter demonstrate all the steps of the problem-solving strategy and explain the underlying
reasoning, so they are longer and much more detailed than the examples in most other texts. They
provide a model for students to follow when working end-of-chapter problems on Dynamics
Worksheets from the Student Workbook.
An important goal is for students to recognize that they can “read” the necessary information
about forces off their free-body diagrams. Thus there is a strong emphasis on identifying forces and
drawing free-body diagrams. Instructors will want to emphasis that this part of the analysis is the
physics, and that the subsequent solution for the acceleration and the motion is “merely”
mathematics. Memorizing every equation in the book will be of little use to students if they can’t
identify forces and draw a correct free-body diagram.
Many students, especially those who are new to vectors, will still be having difficulty using
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vector components or recognizing that F  ma is a shorthand way to write two simultaneous
equations involving the components. They will get the components mixed up, perhaps use
magnitudes instead of components, and have difficulty reassembling components into a magnitude
and direction. Instructors need to demonstrate these steps in considerable detail.
Students must learn to recognize equilibrium situations, then set the acceleration to zero or use
the first law rather than the second law. Many typical single-particle problems are, effectively, “in
equilibrium” along one axis but not the other. For example, an accelerating car is in equilibrium
along the vertical axis. This is another issue so obvious to instructors that they may fail to call
attention to it. Nonetheless, students don’t always recognize this, so they may have an ay in their
equations that they don’t know what to do with. Instructors should point this out during examples.
Also point out the wisdom of choosing a coordinate system with a  0 along one axis.
Friction is introduced as a model, but many students don’t distinguish between a model and a
physical law, such as Newton’s second law. This is a good point for a short digression about the
role of simplifying assumptions in problem solving. Kinetic and rolling friction are
straightforward, but most students have a harder time with static friction. They don’t understand
how there can be a force without a well-specified value, and they often have difficulty knowing
which way the static friction force points. It’s good to ask them what would happen to the object
if there were no friction, then note that the static friction force responds as needed—both size
and direction—to prevent slipping. Because there’s an upper limit to the size of the static friction
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force, the object will slip if the limit is exceeded. Note that some students will try to apply
fs max  s mg indiscriminately to any situation. Instructors should call attention to this by using
some examples in which n  mg.
The drag force due to motion through a medium is a complicated force. The moving objects in
an introductory physics course (balls, people, cars, etc.) are best characterized by a quadratic drag
force D  1 C Av 2, where  is the density of the medium and C is the drag coefficient. C depends
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not only on the object’s shape but also depends weakly on the speed v. Even so, for objects of
modest size (balls, cars) moving through air at modest speeds, it turns out that
1
C
2
 1 kg/m 3. This
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is the basis for the model of drag D  ( 1 Av 2, direction opposite the motion) used in the textbook.
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Although oversimplified, it has the virtue of allowing students to see some of the consequences of
drag without complications that would distract them from the main ideas of the chapter. Because of
its importance in biology, this text also discusses drag in liquids, using Stokes’s law as a prototype
for the drag force in liquids.
The “big picture” you want students to see in Chapter 5 is:
individual forces  net force  acceleration  kinematics
You can encourage this awareness by
• Insisting that students use all the steps in the problem-solving strategy, preferably on a
Dynamics Worksheet.
• Insisting that they begin the mathematical representation of each problem with a full statement
of Newton’s first or second law.
• Insisting that they use the free-body diagram to obtain the force components for Newton’s
laws.
This is the point at which you’re likely to begin getting resistance from students who want to use an
equation-hunting strategy. It’s worth a few minutes to say you realize their frustration but to
reiterate your reasons for insisting they follow the problem-solving strategy. By the time you get
through Chapter 8, in another couple of weeks, you’ll have nearly everyone convinced that they are
successfully solving problems they could not possibly have solved by hunting for equations.
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Suggested Lecture Outlines
To an experienced physicist, this chapter says little more than F  ma. It would seem to be a
chapter that could be covered easily in two days, maybe even one. However, this is the first chapter
in which students are asked to solve serious physics problems. They have their alternative
conceptions of force and motion to deal with, they’re struggling to identify forces correctly and
draw free-body diagrams, and many will still be having trouble using vector components. Patience
in this chapter, with lots of example problem solving, will pay off in the long run.
DAY 1: Equilibrium and Newton’s second law. A reasonable approach is to spend a good
part of the first day on equilibrium problems. This gives you an opportunity to emphasize
identifying forces by drawing free-body diagrams, using the problem-solving strategy, and solving
two simultaneous equations. Since you’ve not yet introduced mass, you should specify actual
weights in newtons. You could also ask how much friction force would be needed to prevent an
object from sliding on an incline, but it’s best not to get into coefficients of friction at this time.
Work through as many examples as possible, making points within the context of a problem rather
than lecturing about them. You’ll want to strike a balance between work that you do, as a
demonstration, and work that you ask students to do. We usually ask students to do one part at a
time—such as “draw the free-body diagram”—then wait for feedback and discussion before going
on to the next part.
You can then transition into simple second-law problems where they don’t need to use w  mg
(specify the actual weight if you need to) or the friction model. A gradual build-up of complexity,
adding just one new feature at a time, allows the students to keep up with you. The following
example is good because many students incorrectly believe that T  w for a block moving upward
at constant speed. This gives you the opportunity to make a distinction between motion and
acceleration.
Example: A 100 kg block with a weight of 980 N hangs on a rope. Find the tension in the rope if
the block is stationary, then if it’s moving upward at a steady speed of 5 m/s, and finally if it’s
accelerating upward at 5 m/s2 .
Although this is a simple problem, students should draw a motion diagram, identify the forces,
draw a free-body diagram, then explicitly use Newton’s first or second law. For this step they will
need to write, e.g., Fx  Tx  wx  0. It’s important to emphasize that Tx and wx are the
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components of T and w, which can be positive or negative. Thus we don’t write  wx , because the
value represented by wx is already negative, as we’ll show in the next step.
Now it’s time to write the components in terms of magnitudes of vectors and angles (when
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applicable). Here’s where we write wx   w, indicating that w points down.
Example: A ball weighing 60 N is pulled back by a rope by an angle of 20. What is the tension
in the pulling rope?
Again, have them work through the steps above, giving them feedback after each step. For this
one, they’ll need to write the components in terms of magnitudes and sines/cosines of angles. To
help them work through this rather difficult example, you can write a table, such as the one in
Example 5.2 of the text, containing entries like T1x 
. It’s then their job to fill in the blanks.
DAY 2: Further applications of Newton’s second law. Once students are comfortable with
the problem-solving strategy, you can move into mass, weight, friction, and drag.
Example: To illustrate the application of Newton’s second law, start with a simple onedimensional example:
A sled with a mass of 20 kg slides along frictionless ice at 4.5 m/s. It then crosses a rough patch
of snow that exerts a friction force of 12 N. How far does it slide on the snow before coming to rest?
Again, be sure to have your students carefully work out each step in the solution: visual
overview, force identification, free-body diagram, Newton’s second law in component force, and
kinematics. Pause after each step to discuss the solution so far.
Example: Next, consider an object moving horizontally, but subject to forces that are not along the
coordinate axes: Macie pulls a 40 kg rolling trunk by a strap angled at 30 from the horizontal. She
pulls with a force of 40 N, and there is a 30 N rolling friction force acting on trunk. What is the
trunk’s acceleration?
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Example: Students often have quite a bit of difficulty resolving components of vectors in a tilted
coordinate system. Explain why we choose such systems: The kinematics is much simpler because
the y-component of the acceleration is zero. As an introductory example of tilted axes, show a mass
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m as a point on the board and draw a downward weight vector w. Then ask students to find the
components wx and wy in each of the three coordinate systems shown below.
You may first want them to prove that the angle  between the weight vector and the negative
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y-axis is the same as the “tilt angle” of the coordinate system. This decomposition of w is essential
in any inclined-plane problem, but many students will stumble over it if you haven’t explicitly
called their attention to the procedure.
Example: A 75 kg skier starts down a 50-m-high, 10° slope on frictionless skis. What is his speed
at the bottom?
In doing this example you can demonstrate that the acceleration on a frictionless incline is
a  g sin  —a fact that was asserted in Chapter 3 but did not really prove. This problem also
provides an opportunity to talk about modeling and making approximations. Is it reasonable to
treat the skis as frictionless? Is it reasonable to ignore air resistance? Is the answer exactly the
speed of a real skier, approximately the speed of a real skier, or not even close to the speed of a
real skier?
Example: Burglars are trying to push a 1000 kg safe up a 20° frictionless slope with a horizontal
force of 4000 N. What is the safe’s acceleration? If the answer is found first in symbolic form, you
can check the limiting case of   0, as well as exploring the special angle for which ax  0.
Example: A 50 kg student gets in a 1000 kg elevator at rest. As the elevator begins to move, she
has an apparent weight of 600 N for the first 3 s. How far has the elevator moved, and in which
direction, at the end of 3 s?
DAY 3: Friction. It is worthwhile to revisit some of the previous examples with friction added.
For example, you can repeat the skier example after giving the students s  0.12 and k  0.06.
You can tell them that the safe in the burglar example has s  0.8 and k  0.5, then ask them
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how much tension is needed in the rope to start the safe moving. If that tension is maintained after
the motion starts, does the safe move with constant velocity or constant acceleration? What value of
v or a?
Example: A car traveling at 20 m/s stops in a distance of 50 m. Assume that the deceleration is
constant. The coefficients of friction between a passenger and the seat are s  0.5 and k  0.3
Will a 70 kg passenger slide off the seat if not wearing a seat belt?
This is an excellent example, but one that students find very difficult. First of all, the question
doesn’t tell them what to compute. Second, the passenger’s mass is extraneous information. Third,
it confronts their difficulties with static friction. All the analysis tools that have been developed so
far are needed to understand and solve this problem, and walking students through it is a valuable
exercise. You need to establish that static friction is the only horizontal force on the passenger, that
the acceleration (as seen from a motion diagram) points backward, and so static friction is the force
that decelerates the passenger. But since static friction has a maximum possible value, there’s a
maximum possible deceleration. Finally, they have to realize that the question will be answered by
comparing the car’s deceleration to the maximum possible deceleration of the passenger. In this
case, the passenger stays on the seat.
DAY 4: Newton’s third law. Today we’ll turn our attention to Newton’s third law and interacting
systems of objects. To apply the third law in practical problems requires, in addition to what we
learned about the third law in Chapter 4, an understanding of acceleration constraints, tension, and
the role of ropes and pulleys. As for the second law, these concepts are best illustrated by working
several examples, again with active student involvement.
Example: Although a similar example is done in the text, it’s useful to start out with the simple
case of two blocks being pushed by a constant force. Students can begin by drawing correct,
separate, free-body diagrams for each block, identifying all forces and the action/reaction pairs.
Many students will have difficulty correctly identifying the “backward” force of the leading block
on the trailing one, so be sure to discuss this issue when drawing your version of the free-body
diagram. From their diagrams they should be able by now to write Newton’s second law for each
block. There are then two key points they’ll need to understand in order to continue. First, according
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to the third law, the magnitudes of the forces FA on B and FB on A must be equal. Second, there is the
acceleration constraint that aAx  aBx .
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Example: The example of two blocks arranged as shown is quite useful in getting students to draw
proper free-body diagrams and to correctly identify action/reaction pairs
(of which there are two). Students should be carefully guided through the full force
identification/free body strategy. Clicker questions, such as the following, can be helpful during
examples to keep the students actively engaged.
Clicker Question: Which pair of forces is an action/reaction pair?
A. The tension of the string and the friction force acting on A.
B. The normal force on A due to B and the weight of A.
C. The normal force on A due to B and the weight of B.
D. The friction force acting on A and the the friction force acting on B.
An important result of Chapter 5 is the massless rope approximation. You shouldn’t dwell much on
the details of this approximation, but there are two practical rules for ropes that often come up in
problem solving:
1) The tension in a rope is equal to the force exerted on it at either end. Thus if I pull on a rope
with a 100 N force, the tension in the rope will be 100 N.
2) The rope pulls on whatever it’s tied to with a force whose magnitude is the tension in the
rope. Thus the rope pulls on what it’s tied to at both ends with the same force.
The idea of tension as “pulling in both directions” is easier if the first day has called attention to the
“backward” reaction force when one object pushes a second. Start by asking students to compare
the tension in two ropes: One is tied to the wall and pulled by a student with a 100 N force, the other
is pulled by two students in a tug-of-war as each pulls with a 100 N force. Research has shown that
most students expect the tension in the tug-of-war rope to be twice as large because “it’s being
pulled twice as hard.” It’s good to demonstrate the equal tensions by using spring scales in a set-up
such as that shown in the figure. Then have students go through the force analysis, making an
imaginary division of the rope into two halves so that they can see the tension force as “holding the
rope together.”
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Example: A standard but very good example is one block free to slide on a horizontal table,
connected by a string going over a pulley to a hanging block. Again, have your students identify the
forces acting on each block and draw free-body diagrams for each. They’ll need to understand that
the tension force on each block is the same, unaffected by the pulley. The acceleration constraint
here is tricky, because of the minus sign in aAx  aBy .
Other Resources
In addition to the specific suggestions made above in the daily lecture outlines, here are some other
suggestions for demonstrations, examples, questions, and additional topics that you could weave
into your class time.
Suggested Demonstrations
A block sliding down a ramp. To demonstrate static and kinetic friction, show how a block
remains “stuck” to a board as the board is tilted upward. A free-body diagram will show that the
static friction force must be increasing as the board is further tilted. The maximum static friction
force can be identified as that friction force applied just before the block slips. Further, the fact that
the block accelerates down the slope is evidence that k  s .
Sample Reading Quiz Questions
1. What is the difference, or is there a difference, between mass and weight?
2. What is apparent weight?
3. The coefficient of static friction is
a. smaller than the coefficient of kinetic friction.
b. equal to the coefficient of kinetic friction.
c. larger than the coefficient of kinetic friction.
d. not discussed in this chapter.
4. The force of friction is described by
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A. the law of friction.
B. the theory of friction.
C. a model of friction.
D. the friction of hypothesis.
Additional Student Response System (“Clicker”) Questions
1. One end of a rod is free to slide on a frictionless surface. The other end is suspended by a
string. Which figure shows how the rod would hang from the string?
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2. What are the components of w in the coordinate system shown?
Additional Examples
A wooden box, with a mass of 22 kg, is pulled at a constant speed with a rope that makes an angle
of 25 with the wooden floor. What is the tension in the rope?
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