Part I Newton’s LawsChapter 1 Concepts of MotionChapter 2 Kinematics: The Mathematics

of MotionChapter 3 Vectors and Coordinate

SystemsChapter 4 Force and MotionChapter 5 Dynamics I: Motion

Along a LineChapter 6 Dynamics II: Motion in a PlaneChapter 7 Dynamics III: Motion in a CircleChapter 8 Newton’s Third Law

Part II Conservation LawsChapter 9 Impulse and MomentumChapter 10 EnergyChapter 11 Work

Part III Applications of NewtonianMechanics

Chapter 12 Newton’s Theory of GravityChapter 13 Rotation of a Rigid BodyChapter 14 OscillationsChapter 15 Fluids and Elasticity

Part IV ThermodynamicsChapter 16 Solids, Liquids, and GasesChapter 17 Work, Heat, and the First Law

of ThermodynamicsChapter 18 The Micro/Macro ConnectionChapter 19 Heat Engines and Refrigerators

v

ContentsPhysics for Scientists and Engineers: A Strategic Approach

Part V Waves and OpticsChapter 20 Traveling WavesChapter 21 SuperpositionChapter 22 Wave OpticsChapter 23 Ray OpticsChapter 24 Modern Optics and Matter

Waves

Part VI Electricity and MagnetismChapter 25 Electric Charges and ForcesChapter 26 The Electric FieldChapter 27 Gauss’s LawChapter 28 Current and ConductivityChapter 29 The Electric PotentialChapter 30 Potential and FieldChapter 31 Fundamentals of CircuitsChapter 32 The Magnetic FieldChapter 33 Electromagnetic InductionChapter 34 Electromagnetic Fields and

WavesChapter 35 AC Circuits

Part VII Relativity and Quantum PhysicsChapter 36 RelativityChapter 37 The End of Classical PhysicsChapter 38 QuantizationChapter 39 Wave Functions and

ProbabilitiesChapter 40 One-Dimensional Quantum

MechanicsChapter 41 Atomic PhysicsChapter 42 Nuclear Physics

The two chapters in this booklet, Chapters 4 and 5, are based on informationabout motion, vectors, and kinematics presented in Chapters 1–3. Much of theinformation in these three chapters is the same as in any physics textbook, but thisbook does stress visualization of motion more than other textbooks. In addition,the early chapters in this book begin to develop a structured approach to problemsolving. Research has shown that these two techniques help students develop abetter grasp of conceptual ideas and more robust problem-solving abilities. Thisquick synopsis cannot entirely replace Chapters 1–3, but it summarizes importantideas that may not be in your regular textbook and it defines some of the termsused in Chapters 4 and 5.

The Particle ModelFor many objects, such as runners and rockets, the motion of the object as awhole is not influenced by the “details” of the object’s size and shape. To describethe object’s motion, all we really need to keep track of is the motion of a singlepoint, such as a white dot painted on the side of the object. If we restrict our atten-tion to translational motion (i.e., not rotating), we can consider the object as if itwere just a single point, without size or shape. An object that can be representedas a mass at a single point in space is called a particle.

Treating an object as a particle is, of course, a simplification of reality. It iswhat we call a model. Models allow us to focus on the important aspects of a phe-nomenon without getting sidetracked by those aspects that play only a minor role.The particle model of motion is a simplification in which we treat a movingobject as a mass concentrated at a single point.

Motion DiagramsA motion diagram is a tool for visualizing motion. The primary purpose of amotion diagram is to determine the direction in which an object’s velocity andacceleration vectors point. To make a motion diagram, first imagine illuminatingthe object with a powerful strobe light as it moves. You would see the object atdiscrete, equally spaced intervals of time If you draw these, representing theobject as a particle, you have the beginnings of a motion diagram. Figure 1ashows an example of an object speeding up in a straight line. This might be a jetplane taking off or an Olympic sprinter.

A vector drawn from one point to the next is the displacement vectorBecause the average velocity during this interval is the averagevravg 5 Drr/Dt,

Drr.

Dt.

vii

IntroductionA Quick Synopsis of Chapters 1–3

velocity vector points in the same direction as and has the same relativelength. Thus the second step in drawing a motion diagram is to draw arrows con-necting each point to the next and label them Figure 1b shows, not surprisingly,that the velocity vectors increase in length as the object speeds up.

Average acceleration is Consequently, the average accelerationvector points in the same direction as the change in the velocity vector. Whilevisualizing is easy, visualizing is trickier. Suppose an object’s velocitychanges from to By definition,

We can turn this around to write

That is, is the vector you must to add to in order to get This idea is thebasis for finding acceleration vectors.

TACTICS BOX 1 Finding Acceleration Vectors

Notice that the acceleration vector goes beside the point, not beside the velocityvectors. This is because each acceleration vector is determined as the differencebetween the two velocity vectors on either side of a point. The length of does nothave to be exactly the length of it is the direction of that is important.

Note � Strictly speaking, the vectors displayed on a motion diagram are theaverage velocity and the average acceleration A motion diagram is aqualitative tool for visualizing motion and determining the direction of vec-tors. Later, when we are ready for quantitative analysis, we will use the instan-taneous velocity and acceleration defined in calculus. But maintaining adistinction between the two would make motion diagrams very cumbersome.Consequently, we will drop the subscript and refer to simply and when weuse motion diagrams to obtain a qualitative understanding of motion. �

We can now put the pieces together to draw a complete motion diagram.

TACTICS BOX 2 Drawing Motion DiagramsShow the object’s position at equally spaced intervals of time. Repre-sent the object as a dot (a particle). Five or six dots is usually sufficientto make the motion clear.Draw velocity vectors. Velocity vectors connect each dot to the next.Draw acceleration vectors. Tactics Box 1 showed how accelerationvectors are found by determining the direction of The accelerationvector is drawn beside the dot between the two velocities it links.

Dvr.3

2

1

arvr

aravg .vravg

arDvr;ar

Part of amotion diagram.

We want to find the accelerationvector at this point.

Draw and label an accelerationvector a beside the midpointbetween the two velocity vectors.

a

v1

v1

Draw v1 and v2 withtheir tails together.

Draw Dv from the tip of v1 to the tip of v2.The acceleration vector a points this direction.

Dv

v2

v2

r r

r

r

r r

r r

r

r

rr

r

1

2

3

vr2 .vr1Dvrvr2 5 vr1 1 Dvr

Dvr 5 vr2 2 vr1

vr2 .vr1

DvrDrrDvr,

aravg 5 Dvr/Dt.

vr.

Drr

viii Introduction

The object is represented bydots at equal intervals of time.

DrDisplacement vector

Direction of motion

(a)

r

v

(b)

Label shows that this is a row ofvelocity vectors. Their lengthsincrease as the object speeds up.

r

Figure 1 The first steps in drawing amotion diagram.

Figure 2 shows a complete motion diagram with all the acceleration vectors added.The acceleration vectors may be obvious for an object speeding up to the right.

They are less obvious for objects that slow down or that move along curved tra-jectories, and that is where motion diagrams become more powerful.

Introduction ix

va

Velocity vectors

Acceleration vectors

r

r

Figure 2 A complete motion diagram.

EXAMPLE 1 Drawing a Motion DiagramDraw a motion diagram of a ball that is tossed straight upinto the air and then falls straight back down. Begin thediagram just after you release the ball.

Figure 3 shows the ball as a particle at equal intervalsof time. The upward and downward motion have beendisplaced for clarity.The green velocity vectors connect each dot to the next. The ovals show how to find the acceleration vector atthree points in the motion. The acceleration vectorpoints downward as the ball rises. The accelerationvector also points downward as the ball falls. Both“speeding up” and “slowing down” occur with thesame acceleration vector! This is an important conclu-sion, one worth pausing to think about.

What is the acceleration at the highest point? Theinset shows how the acceleration is determined fromthe last upward velocity before the top point and thefirst downward velocity. We see that the accelerationat the top points downward, just as it does elsewherein the motion.

Note � It seems hard to understand how the ball can beaccelerating at the top point. Many people expect theacceleration at the top to be zero. But the velocity atthe top point is changing—from “up” to “down.” Forthe velocity to change, there must be an acceleration.In order to change the velocity vector from up todown, the acceleration must point downward. Here wehave used a motion diagram to discover an accelera-tion that is not so obvious. �

3

2

1

Figure 3 The motion diagram of a ball tossed straight up.

v

Dv

Dv

aat top

aa

v

DvFinding awhilegoing up

Finding awhilegoing down

Last velocityupward

First downwardvelocity

This is the samepoint shown twice.

The upward anddownward motion aredisplaced for clarity.They really occuralong the same line.

r

r

rr

r r

r

rr

r

Motion diagrams lead to an important conclusion for motion along a straight line:

An object is speeding up if and only if and point the same direction.

An object is slowing down if and only if and point in opposite directions.

An object’s velocity is constant if and only if

Pictorial RepresentationsYou’ve probably been told to “draw a picture” as part of solving a problem, butperhaps you didn’t know what to draw. There really is a method for drawing pic-tures, one that helps you be a better problem solver. It is called the pictorial rep-resentation of the problem:

ar 5 0.

arvrarvr

TACTICS BOX 3 Drawing a Pictorial RepresentationSketch the situation. Not just any sketch. Show the object at the begin-ning of the motion, at the end, and at any point where the character of themotion changes. Very simple drawings are adequate.Establish a coordinate system. Select your axes and origin to matchthe motion.Define symbols. Use the sketch to define symbols representing quanti-ties such as position, velocity, acceleration, and time. Every variableused later in the mathematical solution should be defined on the sketch.Some will have known values, others are initially unknown, but allshould be given symbolic names.List known information. Make a table of the quantities whose valuesyou can determine from the problem statement or that can be foundquickly with simple geometry or unit conversions. Some quantities areimplied by the problem, rather than explicitly given. Others are deter-mined by your choice of coordinate system.Identify the desired unknowns. What quantity or quantities will allowyou to answer the question? These should have been defined as symbolsin step 3. Don’t list every unknown; only the one or two needed toanswer the question.

It’s not an overstatement to say that a well-done pictorial representation of theproblem will take you halfway to the solution.

5

4

3

2

1

x Introduction

Figure 4 A pictorial representation.

y

x

a0

x0, v0, t0 x1, v1, t1 x2, v2, t2

a1

SketchCoordinate system

Define symbols

List known information

Identify desired unknown

Known:

Find:

x0 5 v0 5 t0 5 0

x2

a0 5 50 m/s2

a1 5 0 m/s2t1 5 5 s

t2 5 t1 1 3 s 5 8 s

12

3

4

5

EXAMPLE 2 Drawing a Pictorial RepresentationDraw a pictorial representation for the following problem:

A rocket sled accelerates at for 5 seconds,then coasts for 3 seconds. What is the total distancetraveled?

Solution: The motion has a beginning, an end, and apoint where the nature of the motion changes from accel-erating to coasting. These are the three points sketched inFigure 4. A coordinate system has been chosen with theorigin at the starting point. The quantities x, v, and t are

50 m/s2

needed at each of three points, so these have been definedon the sketch and distinguished by subscripts. Accelera-tions are associated with intervals between the points, soonly two accelerations are defined. Values for three quan-tities are given in the problem statement. Others, such as

and are inferred from our choice ofcoordinate system. The value is part of ourinterpretation of the problem. Finally, we identify asthe quantity that will answer the question. We now under-stand quite a bit about the problem and are ready to start aquantitative analysis.

x2

v0 5 0 m/st0 5 0 s,x0 5 0 m

Problem Solving in MechanicsPhysics is not mathematics. Physics is about the world around us, and the ques-tions we ask in physics are ones such as, “How much fuel is needed to get this

rocket into orbit?” or “What happens when I close the switch in this circuit?”Mathematics is just one of many tools needed to answer these questions.

Part of the process of answering physics questions is evaluating the informa-tion. Is the information sufficient to answer the question, or is some item miss-ing? Is all of the information relevant? A second step in the process, after theinformation is evaluated, is to translate the words into symbols that can be manip-ulated, calculated, and graphed. This translation from words to symbols is theheart of problem solving in physics. It is the point where ambiguous words andphrases must be clarified, where the imprecise must be made precise, and whereyou arrive at an understanding of exactly what the question is asking.

One of the goals of this textbook is to help you learn a strategy for solvingphysics problems. An important aspect of an expert-like problem-solving strategyis to represent your knowledge of the situation in many different ways. The use ofrepresentations simplifies the critical task of translating from words to symbols,and they form the basis for a problem-solving strategy:

Problem-Solving Strategy

Model: It’s impossible to treat every detail of a situation. Simplify the sit-uation with a model that captures the essential features. For example, theobject in a mechanics problem is usually represented as a particle.

Visualize: This is the step where expert problem solvers put most of theireffort.

■ Draw a pictorial representation. This helps you assess the informa-tion you are given and starts the process of translating the problem intosymbols.

■ Draw a physical representation. This helps you visualize importantaspects of the physics. Motion diagrams are part of the physical representa-tion. Chapter 4 will introduce free-body diagrams to display informationabout forces.

■ Use a graphical representation if it is appropriate for the problem.

Solve: Only after modeling and visualizing are complete is it time todevelop a mathematical representation with specific equations that mustbe solved. All symbols used here should have been defined in the pictorialrepresentation.

Assess: Is your result believable? Does it have proper units? Does it makesense?

This strategy is both flexible and robust. Very simple problems may notrequire all of the steps, and with experience you will soon recognize which mod-els and representations are important for different kinds of problems. But when-ever you are faced with a new problem, the four-step Model, Visualize, Solve,and Assess strategy will give you the tool to find the answer.

This strategy is useful in all areas of physics, but we’ll close this synopsis bylooking at some issues specific to mechanics. An object’s motion is described byits position velocity and acceleration These are vectors. The quantities x(or y), v, and a used in one-dimensional-motion problems are actually vector

ar.vr,rr,

Introduction xi

Building a house requires careful planning.The architect’s visualization and drawingshave to be complete before the detailedprocedures of construction get underway.The same is true for solving problems inphysics.

components along the axis of motion. For motion along the x-axis, for example, x,v, and a are actually and Their vector nature appears through their signs:

■ v is positive if the velocity vector points to the right (or up). It is negative ifthe velocity vector points to the left (or down).

■ a is positive if the acceleration vector points to the right (or up). It is negativeif the acceleration vector points to the left (or down).

The appropriate sign for v is usually clear, although remembering to use the appro-priate sign may be a problem. Determining the sign of a is more difficult, and thisis where a motion diagram can help.

It is tempting to think that positive a represents “speeding up” and that nega-tive a is “slowing down.” But this isn’t true. The sign of a is solely a function ofwhich way the vector points. For the ball in Example 1 (Figure 3), the accelera-tion a is negative during the entire motion because the vector points down theentire time. The ball slows during the first half of the motion, where and pointin opposite directions. But it speeds up during the second half, even though a isnegative, because and point in the same direction.

The final example demonstrates the full problem-solving strategy.vrar

vrarar

arar

vrvr

ax .vx ,rx ,

xii Introduction

Figure 5 Pictorial and physical representations of acannonball.

Known:

Find:

y0 5 t0 5 0

y1

v0 5 100 m/s2

v1 5 0a0 5 2g 5 29.80 m/s2

x

y

y1, v1, t1

y0, v0, t0

a0

Pictorial representation Physical representation

a

v

r

r

EXAMPLE 3 Using the Problem-Solving StrategyA cannonball is shot straight up with an initial speed of

How high does it go?

Model: The cannonball will be represented as a particle.We’ll assume it is fired at t 5 0 s.

100 m/s.

quantity a must be negative. We write this as where the acceleration due to gravity is apositive number.

A critical aspect of setting up this problem is knowingwhere it ends. How do we put “how high” into symbols?The very top point of the trajectory is a turning point, apoint where the instantaneous velocity is Thusthe highest point is where This was notstated but is part of our interpretation of the problem.

Solve: This is a free-fall problem because the cannonball,after being launched, moves entirely under the influenceof gravity. We need a relationship between distance andvelocity that doesn’t require knowing the time interval.One of the standard equations of free-fall kinematics is

where Using and we have

Solving for we find that the cannonball reaches height

Assess: Is this answer reasonable? A speed of is—pretty fast! The calculated height is

While we cannot say that 510 m is nec-essarily better than 400 m or 600 m, it seems not unrea-sonable. The point of the assessment is not to prove thatthe answer has to be right, but to recognize answers thatare obviously wrong.

510 m < 1500 ft.<200 mph

100 m/s

y1 5v0

2

2g5

1100 m/s 2 2

2 19.80 m/s2 25 510 m

y1 ,

0 5 v0

2 2 2gy1

v1 5 0 m/s,y0 5 0 mDy 5 y1 2 y0 .

v1

2 5 v0

2 2 2g Dy

v1 5 0 m/s.y1

v 5 0 m/s.

g 5 9.80 m/s2a 5 2g,

Visualize: Figure 5 shows a pictorial representation andthe motion diagram of a physical representation. Themotion diagram shows that points downward, so thear

This drag racer can cover a quarter mile from a standing start in less than 5 sec-onds. Not bad! We could use kinematics to describe the car’s motion with pictures,graphs, and equations. By defining position, velocity, and acceleration and dress-ing them in mathematical clothing, kinematics provides a language to describehow something moves. But kinematics would tell us nothing about why the caraccelerates so quickly. For the more fundamental task of understanding the causeof motion, we turn our attention to dynamics. Dynamics joins with kinematics toform mechanics, the general science of motion. We study dynamics qualitativelyin this chapter, then develop it quantitatively in the next four chapters.

The theory of mechanics originated in the mid-1600s when Sir Isaac Newtonformulated his laws of motion. These fundamental principles of mechanicsexplain how motion occurs as a consequence of forces. Newton’s laws are morethan 300 years old, but they still form the basis for our contemporary understand-ing of motion.

A challenge in learning physics is that a textbook is not an experiment. Thebook can assert that an experiment will have a certain outcome, but you may notbe convinced unless you see or do the experiment yourself. Newton’s laws arefrequently contrary to our intuition, and a lack of familiarity with the evidence forNewton’s laws is a source of difficulty for many people. You will have an oppor-tunity through lecture demonstrations and in the laboratory to see for yourself theevidence supporting Newton’s laws. Physics is not an arbitrary collection of defi-nitions and formulas, but a consistent theory as to how the universe really works.It is only with experience and evidence that we learn to separate physical factfrom fantasy.

97

� Looking AheadThe goal of Chapter 4 is toestablish a connection betweenforce and motion. In this chapteryou will learn to:

■ Recognize what a force is and is not.

■ Identify the specific forcesacting on an object.

■ Draw free-body diagrams.■ Understand the connection

between force and motion.

� Looking BackTo master the material introducedin this chapter, you mustunderstand how acceleration isdetermined and how vectors areused. Please review:

■ Section 1.5 Acceleration.■ Section 3.2 Properties of

vectors.

Force and Motion4A drag race is a

memorableexample of the

connectionbetween force

and motion.

4.1 ForceIf you kick a ball, it rolls across the floor. If you pull on a door handle, the dooropens. You know, from many years of experience, that some sort of force isrequired to move these objects. Our goal is to understand why motion occurs, andthe observation that force and motion are related is a good place to start.

The two major issues that this chapter will examine are:

■ What is a force?■ What is the connection between force and motion?

We begin with the first of these questions in the table below.

98 C H A P T E R 4 . Force and Motion

What is a force?

A force is a push or a pull.

Our commonsense idea of a force is that it is a push or a pull. We will refine this idea as we goalong, but it is an adequate starting point. Notice our careful choice of words: We refer to “a force,”rather than simply “force.” We want to think of a force as a very specific action, so that we can talkabout a single force or perhaps about two or three individual forces that we can clearly distinguish.Hence the concrete idea of “a force” acting on an object.

A force acts on an object.

Implicit in our concept of force is that a force acts on an object. In other words, pushes and pulls areapplied to something—an object. From the object’s perspective, it has a force exerted on it. Forcesdo not exist in isolation from the object that experiences them.

A force requires an agent.

Every force has an agent, something that acts or exerts power. That is, a force has a specific,identifiable cause. As you throw a ball, it is your hand, while in contact with the ball, that is theagent or the cause of the force exerted on the ball. If a force is being exerted on an object, you mustbe able to identify a specific cause (i.e., the agent) of that force. Conversely, a force is not exerted on an object unless you can identify a specific cause or agent. Although this idea may seem to bestating the obvious, you will find it to be a powerful tool for avoiding some common misconceptionsabout what is and is not a force.

A force is a vector.

If you push an object, you can push either gently or very hard. Similarly, you can push either left orright, up or down. To quantify a push, we need to specify both a magnitude and a direction. It shouldthus come as no surprise that a force is a vector quantity. The symbol for a force is the vector symbol

The size or strength of a force is its magnitude

A force can be either a contact force . . .

There are two basic classes of forces, depending on whether the agent touches the object or not.Contact forces are forces that act on an object by touching it at a point of contact. The bat musttouch the ball to hit it. A string must be tied to an object to pull it. The majority of forces that we will examine are contact forces.

. . . or a long-range force.

Long-range forces are forces that act on an object without physical contact. Magnetism is anexample of a long-range force. You have undoubtedly held a magnet over a paper clip and seen the paper clip leap up to the magnet. A coffee cup released from your hand is pulled to the earth by the long-range force of gravity.

F.Fr

.

Object

Agent

Let’s summarize these ideas as our definition of force:

■ A force is a push or a pull on an object.■ A force is a vector. It has both a magnitude and a direction.■ A force requires an agent. Something does the pushing or pulling.■ A force is either a contact force or a long-range force. Gravity is the only long-

range force we will deal with until much later in the book.

There’s one more important aspect of forces. If you push against a door (theobject) to close it, the door pushes back against your hand (the agent). If a towrope pulls on a car (the object), the car pulls back on the rope (the agent). Thus, ingeneral, if an agent exerts a force on an object, the object exerts a force on theagent. We really need to think of a force as an interaction between two objects.Although the interaction perspective is a more exact way to view forces, it addscomplications that we would like to avoid for now. Our approach will be to startby focusing on how a single object responds to forces exerted on it. Then, inChapter 8, we’ll return to the larger issue of how two or more objects interactwith each other.

NOTE � In the particle model, objects cannot exert forces on themselves. Aforce on an object will always have an agent or cause external to the object.Now, there are certainly objects that have internal forces (think of all theforces inside the engine of your car!), but the particle model is not valid if youneed to consider those internal forces. If you are going to treat your car as aparticle and look only at the overall motion of the car as a whole, that motionwill be a consequence of external forces acting on the car. �

Force VectorsWe can use a simple diagram to visualize how forces are exerted on objects.Because we are using the particle model, in which objects are treated as points,the process of drawing a force vector is straightforward. Here is how it goes:

TACTICS BOX 4.1 Drawing force vectors

Step 2 may seem contrary to what a “push” should do, but recall that moving avector does not change it as long as the length and angle do not change. The vec-tor is the same regardless of whether the tail or the tip is placed on the particle.Our reason for using the tail will become clear when we consider how to combineseveral forces.

Figure 4.1 on the next page shows three examples of force vectors. One is apush, one a pull, and one a long-range force, but in all three the tail of the forcevector is placed on the particle representing the object.

Fr

F

Represent the object as a particle.

Draw the force vector as an arrow pointingin the proper direction and with a lengthproportional to the size of the force.

Give the vector an appropriate label.

Place the tail of the force vectoron the particle.

r

1

2

3

4

4.1 . Force 99

Combining ForcesFigure 4.2a shows a box being pulled by two ropes, each exerting a force on thebox. How will the box respond? Experimentally, we find that when several indi-vidual forces . . . are exerted on an object, they combine to form a netforce given by the vector sum of the individual forces:

(4.1)

Recall that is the symbol meaning “is defined as.” Mathematically, this sum-mation is called a superposition of forces. The net force is sometimes called theresultant force. Figure 4.2b shows the net force on the box.

Two forces are exerted on an object. What third forcewould make the net force point to the left?

4.2 A Short Catalog of ForcesThere are many forces we will deal with over and over. This section will intro-duce you to some of them. Many of these forces have special symbols. As youlearn the major forces, be sure to learn the symbol for each.

WeightA falling rock is pulled toward the earth by the long-range force of gravity. Grav-ity is what keeps you in your chair, keeps the planets in their orbits around thesun, and shapes the large-scale structure of the universe. We’ll have a thoroughlook at gravity in Chapter 12. For now we’ll concentrate on objects on or near thesurface of the earth (or other planet).

The gravitational pull of the earth on an object on or near the surface of theearth is called weight. The symbol for weight is Weight is the only long-rangeforce we will encounter in the next few chapters. The agent for the weight force is

wr .

(d)

F3

rF3

(c)

r

F3

(b)

r

F3

(a)

rF1

F2

r

r

Two forcesexerted onan object

STOP TO THINK 4.1

;

Fr

net ; aN

i51Fr

i 5 Fr

1 1 Fr

2 1 c1 Fr

N

Fr

3,Fr

2,Fr

1,

100 C H A P T E R 4 . Force and Motion

Find the netforce on the box.

Pulling forceof the ropes

Fnet 5 F1 1 F2

F1

F2

(b) r

r r r

r

FIGURE 4.2 Two forces applied to a box.

Top viewof box

(a)

Ground

The weight forcepulls the box down.

wr

FIGURE 4.3 Weight.

The rope is the agent.

Pulling force of rope

Box

The spring is the agent.

Pushing force of spring

Box

Earth is the agent.

Long-rangeforce ofgravity

Box

FIGURE 4.1 Three force vectors.

the entire earth pulling on an object. Weight acts on an object whether the objectis moving or at rest. The weight vector always points vertically downward, asshown in Figure 4.3 on the previous page.

NOTE � We often refer to “the weight” of an object. This is an informalexpression for the magnitude of the weight force exerted on the object. Note that weight is not the same thing as mass. We will briefly examine mass later in the chapter and explore the connection between weight and mass inChapter 5. �

Spring ForceSprings exert one of the most common contact forces. A spring can either push(when compressed) or pull (when stretched). Figure 4.4 shows the spring force. In both cases, pushing and pulling, the tail of the force vector is placed on theparticle in the force diagram. There is no special symbol for a spring force, so wesimply use a subscript label: F

r

sp.

w,

4.2 . A Short Catalog of Forces 101

A stretched spring exerts a force on an object.

(b)

A stretched spring exertsa pulling force on an object.

Fsp

r

(a)

A compressed spring exertsa pushing force on an object.

Fsp

r

FIGURE 4.4 The spring force.

T

The rope exerts a tensionforce on the sled.

r

FIGURE 4.5 Tension.

T

Molecularbonds

Atoms

Rope

Atomic-levelview of a rope,with molecularbonds as springs

Trr

FIGURE 4.6 An atomic-level view of tension.

Although you may think of a spring as a metal coil that can be stretched orcompressed, this is only one type of spring. Hold a ruler, or any other thin piece of wood or metal, by the ends and bend it slightly. It flexes. When you let go, it “springs” back to its original shape. This is just as much a spring as is a metal coil.

Tension ForceWhen a string or rope or wire pulls on an object, it exerts a contact force that wecall the tension force, represented by a capital The direction of the tensionforce is always in the direction of the string or rope, as you can see in Figure 4.5.The commonplace reference to “the tension” in a string is an informal expressionfor the size or magnitude of the tension force.

If you were to use a very powerful microscope to look inside a rope, youwould “see” that it is made of atoms joined together by molecular bonds. Molecu-lar bonds are not rigid connections between the atoms. They are more accuratelythought of as tiny springs holding the atoms together, as in Figure 4.6. These arevery stiff springs, to be sure, but pulling on the ends of a string or rope stretchesthe molecular springs ever so slightly. The tension within a rope and the tensionexperienced by an object at the end of the rope are really the net spring forcebeing exerted by billions and billions of microscopic springs.

This atomic-level view of tension introduces a new idea: a microscopicatomic model for understanding the behavior and properties of macroscopicobjects. We will frequently use an atomic model to obtain a deeper understandingof our observations.

T ,

T

r

.

Normal ForceIf you sit on a bed, the springs in the mattress compress and, as a consequenceof the compression, exert an upward force on you. Stiffer springs would showless compression but still exert an upward force. The compression of extremelystiff springs might be measurable only by sensitive instruments. Nonetheless,the springs would compress ever so slightly and exert an upward spring forceon you.

Figure 4.7 shows an object resting on top of a sturdy table. The table may notvisibly flex or sag, but—just as you do to the bed—the object compresses themolecular springs in the table. The size of the compression is very small becausemolecular springs are so stiff, but it is not zero. As a consequence, the com-pressed molecular springs push upward on the object. We say that “the table”exerts the upward force, but it is important to understand that the pushing is reallydone by molecular springs. Similarly, an object resting on the ground compressesthe molecular springs holding the ground together and, as a consequence, theground pushes up on the object.

We can extend this idea. Suppose you place your hand on a wall and leanagainst it, as shown in Figure 4.8. Does the wall exert a force on your hand? Asyou lean, you compress the molecular springs in the wall and, as a consequence,they push outward against your hand. So the answer is “yes,” the wall does exert aforce on you.

The force the table surface exerts is vertical, the force the wall exerts is hori-zontal. But in all cases, the force exerted on an object that is pressing against asurface is in a direction perpendicular to the surface. Mathematicians refer to aline that is perpendicular to a surface as being normal to the surface. In keepingwith this terminology, we define the normal force as the force exerted by a sur-face (the agent) against an object that is pressing against the surface. The symbolfor the normal force is

We’re not using the word normal to imply that the force is an “ordinary” forceor to distinguish it from an “abnormal force.” A surface exerts a force perpen-dicular (i.e., normal) to itself as the molecular springs press outward. Figure 4.9shows an object on an inclined surface, a common situation. Notice how the nor-mal force is perpendicular to the surface.

We have spent a lot of time describing the normal force because many peoplehave a difficult time understanding it. The normal force is a very real force arisingfrom the very real compression of molecular bonds. It is in essence just a springforce, but one exerted by a vast number of microscopic springs acting at once.The normal force is responsible for the “solidness” of solids. It is what preventsyou from passing right through the chair you are sitting in and what causes thepain and the lump if you bang your head into a door. Your head can then tell youthat the force exerted on it by the door was very real!

FrictionYou’ve certainly observed that a rolling or sliding object, if not pushed or pro-pelled, slows down and eventually stops. You’ve probably discovered that youcan slide better across a sheet of ice than across asphalt. And you also know thatmost objects stay in place on a table without sliding off even if the table isn’tabsolutely level. The force responsible for these sorts of behavior is friction. Thesymbol for friction is a lower case

Friction, like the normal force, is exerted by a surface. On a microscopic level,friction arises as atoms from the object and atoms on the surface run into eachother. The rougher the surface is, the more these atoms are forced into close prox-imity and, as a result, the larger the friction force. We will develop a simple model

f r

.

nr

nr.

102 C H A P T E R 4 . Force and Motion

The compressedmolecular springsin the wall pressoutward againsther hand.

FIGURE 4.8 The wall pushes outwardagainst your hand.

nr

The surface pushes outwardagainst the bottom of the frog.The push is perpendicularto the surface.

FIGURE 4.9 The normal force.

The compressedmolecular springs pushupward on the object.

Molecular bonds

Atoms

FIGURE 4.7 Atomic-level view of the forceexerted by a table.

of friction in the next chapter that will be sufficient for our needs. For now, it isuseful to distinguish between two kinds of friction:

■ Kinetic friction, denoted appears as an object slides across a surface. This isa force that “opposes the motion,” meaning that the friction force vector points in a direction opposite the velocity vector (i.e., “the motion”).

■ Static friction, denoted is the force that keeps an object “stuck” on a surfaceand prevents its motion. Finding the direction of is a little trickier than find-ing it for Static friction points opposite the direction in which the objectwould move if there were no friction. That is, it points in the direction neces-sary to prevent motion.

Figure 4.10 shows examples of kinetic and static friction.

f r

k.f r

s

f r

s,v

r

f r

k

f r

k,

4.2 . A Short Catalog of Forces 103

fk

v

Kinetic frictionopposes themotion.

r

r

Static friction actsin the direction thatprevents slipping.

fs

r

FIGURE 4.10 Kinetic and static friction.

Air resistance is a significant forceon falling leaves. It points oppositethe direction of motion.

DD

D

r

r

r

FIGURE 4.11 Air resistance is an exampleof drag.

Thrust force is exertedon a rocket by exhaustgases.

Fthrust

r

FIGURE 4.12 Thrust force on a rocket.

NOTE � A surface exerts a kinetic friction force when an object moves rela-tive to the surface. A package on a conveyor belt is in motion, but it does notexperience a kinetic friction force because it is not moving relative to the belt.So to be precise, we should say that the kinetic friction force points opposite toan object’s motion relative to a surface. �

DragFriction at a surface is one example of a resistive force, a force that opposes orresists motion. Resistive forces are also experienced by objects moving throughfluids—gases and liquids. The resistive force of a fluid is called drag and is sym-bolized as Drag, like kinetic friction, points opposite the direction of motion.Figure 4.11 shows an example of drag.

Drag can be a large force for objects moving at high speeds or in dense fluids.Hold your arm out the window as you ride in a car and feel how the air resistanceagainst it increases rapidly as the car’s speed increases. Drop a lightweight objectinto a beaker of water and watch how slowly it settles to the bottom. In both casesthe drag force is very significant.

For objects that are heavy and compact, that move in air, and whose speed isnot too great, the drag force of air resistance is fairly small. To keep things as sim-ple as possible, you can neglect air resistance in all problems unless a problemexplicitly asks you to include it. The error introduced into calculations by thisapproximation is generally pretty small. This textbook will not consider objectsmoving in liquids.

ThrustA jet airplane obviously has a force that propels it forward during takeoff. Like-wise for the rocket being launched in Figure 4.12. This force, called thrust,occurs when a jet or rocket engine expels gas molecules at high speed. Thrust is a

Dr

.

contact force, with the exhaust gas being the agent that pushes on the engine. Theprocess by which thrust is generated is rather subtle, and we will postpone a fulldiscussion until we introduce Newton’s third law in Chapter 8. For now, we willtreat thrust as a force opposite the direction in which the exhaust gas is expelled.There’s no special symbol for thrust, so we will call it

Electric and Magnetic ForcesElectricity and magnetism, like gravity, exert long-range forces. The forces ofelectricity and magnetism act on charged particles. We will study electric andmagnetic forces in detail in Part VI of this textbook. For now, it is worth noting thatthe forces holding molecules together—the molecular bonds—are not actuallytiny springs. Atoms and molecules are made of charged particles—electrons andprotons—and what we call a molecular bond is really an attractive electric forcebetween these particles. So when we say that the normal force and the tensionforce are due to “molecular springs,” or that friction is due to atoms running intoeach other, what we’re really saying is that these forces, at the most fundamentallevel, are actually electric forces between the charged particles in the atoms.

4.3 Identifying ForcesForce and motion problems generally have two basic steps:

1. Identify all of the forces acting on an object.2. Use Newton’s laws and kinematics to determine the motion.

Understanding the first step is the primary goal of this chapter. We’ll turn ourattention to step 2 in the next chapter.

A typical physics problem describes an object that is being pushed and pulledin various directions. Some forces are given explicitly, others are only implied. Inorder to proceed, it is necessary to determine all the forces that act on the object.It is also necessary to avoid including forces that do not really exist. Now that youhave learned the properties of forces and seen a catalog of typical forces, we candevelop a step-by-step method for identifying each force in a problem. Thisprocedure for identifying forces is part of the physical representation of theproblem.

TACTICS BOX 4.2 Identifying forces

●1 Identify “the system” and “the environment.” The system is the objectwhose motion you wish to study; the environment is everything else.

●2 Draw a picture of the situation. Show the object—the system—andeverything in the environment that touches the system. Ropes, springs,and surfaces are all parts of the environment.

●3 Draw a closed curve around the system. Only the object is inside thecurve; everything else is outside.

●4 Locate every point on the boundary of this curve where the environ-ment touches the system. These are the points where the environmentexerts contact forces on the object.

●5 Name and label each contact force acting on the object. There is atleast one force at each point of contact; there may be more than one.When necessary, use subscripts to distinguish forces of the same type.

●6 Name and label each long-range force acting on the object. For now,the only long-range force is weight.

Fr

thrust.

104 C H A P T E R 4 . Force and Motion

Force Notation

General force

Weight

Spring force

Tension

Normal force

Static friction

Kinetic friction

Drag

Thrust Fr

thrust

Dr

f r

k

f r

s

nrT

r

Fr

sp

wrFr

NOTE � You might have expected two friction forces and two normalforces in Example 4.2, one on each ski. Keep in mind, however, that we’reworking within the particle model, which represents the skier by a singlepoint. A particle has only one contact with the ground, so there is a single nor-mal force and a single friction force. The particle model is valid if we want toanalyze the translational motion of the skier as a whole, but we would have togo beyond the particle model to find out what happens to each ski. �

Now that you’re getting the hang of this, the next example is meant to lookmuch more like a sketch you should make when asked to identify forces in ahomework problem.

4.3 . Identifying Forces 105

EXAMPLE 4.1 Forces on a bungee jumperA bungee jumper has leapt off a bridge and is nearing the bottom of her fall. What forces are being exerted on the bungeejumper?

Tension T

Weight w

Identify the system. Here the system isthe bungee jumper.

Draw a picture of the situation.

Draw a closed curve around the system.

Locate the points where the environment touchesthe system. Here the only point of contact is wherethe cord attaches to her ankles.

Name and label each contact force. The forceexerted by the cord is a tension force.

Name and label long-range forces. Weightis the only one.

r

r

1

4

5

6

2

3

FIGURE 4.13 Forces on a bungee jumper.

EXAMPLE 4.2 Forces on a skierA skier is being towed up a snow-covered hill by a tow rope. What forces are being exerted on the skier?

Weight w

Tension T

Normal force nKinetic friction fk

Identify the system. Herethe system is the skier.

Draw a picture ofthe situation.

Draw a closed curvearound the system.

Locate the points where the environmenttouches the system. Here the rope and theground touch the skier.

Name and label each contact force. The ropeexerts a tension force and the ground exertsboth a normal and a kinetic friction force.

Name and label long-range forces. Weightis the only one.

r

r

rr

1

2

3

4

5

6

FIGURE 4.14 Forces on a skier.

4.4 What Do Forces Do? A Virtual ExperimentThe fundamental question is: How does an object move when a force is exerted onit? The only way to answer this question is to do experiments. To do experiments,however, we need a way to reproduce the same amount of force again and again.

Let’s conduct a “virtual experiment,” one you can easily visualize. Imagineusing your fingers to stretch a rubber band to a certain length—say 10 centimeters—that you can measure with a ruler. We’ll call this the standard length. Figure 4.16shows the idea. You know that a stretched rubber band exerts a force because yourfingers feel the pull. Furthermore, this is a reproducible force. The rubber bandexerts the same force every time you stretch it to the standard length. We’ll call thisthe standard force F.

What happens to the force if you put two identical rubber bands around yourfingers and stretch both to the standard length? If you are not sure, find a few rub-ber bands and try this. You will discover that two rubber bands exert a largerpulling force than one rubber band. This is not surprising. If two rubber bands areeach pulling equally hard, the net pull is twice that of one rubber band: N side-by-side rubber bands, each pulled to the standard length, will exert N timesthe standard force:

Now we’re ready to start the virtual experiment. Imagine an object to whichyou can attach rubber bands, such as a block of wood with a hook. If you attach arubber band and stretch it to the standard length, the object experiences the sameforce F as did your finger. N rubber bands attached to the object will exert N timesthe force of one rubber band. The rubber bands give us a way of applying aknown and reproducible force to an object.

Our next task is to measure the object’s motion in response to these forces.Imagine using the rubber bands to pull the object across a horizontal table. Fric-tion between the object and the surface might affect our results, so let’s just elimi-nate friction. This is, after all, a virtual experiment! (In practice you could nearly

Fnet 5 NF.

Fnet 5 2F.

106 C H A P T E R 4 . Force and Motion

Standard length 2F

Two rubber bandsstretched the standardlength exert twicethe standard force.

r

Slack rubber band

Standard length Fr

One rubber bandstretched the standardlength exerts thestandard force F.

FIGURE 4.16 A reproducible force.

EXAMPLE 4.3 Forces on a rocketA rocket is being launched to place a new satellitein orbit. Air resistance is not negligible. What forcesare being exerted on the rocket?

Air

Exhaust

Thrust Fthrust

Weight w

Drag Dr

r

r

FIGURE 4.15 Forces on a rocket.

You’ve just kicked a rock, and it is now sliding across theground about 2 meters in front of you. Which of these forces act on the rock? Listall that apply.

a. Gravity, acting downward.b. The normal force, acting upward.c. The force of the kick, acting in the direction of motion.d. Friction, acting opposite the direction of motion.e. Air resistance, acting opposite the direction of motion.

STOP TO THINK 4.2

eliminate friction by pulling a smooth block over a smooth sheet of ice or by sup-porting the object on a cushion of air.) To make measurements, lay a meter stickalong the edge of the table and hang a movie camera over the table to record the motion.

Our experiment will be easiest to interpret if the force is constant throughoutthe object’s motion. If you stretch the rubber band and then release the object, itmoves toward your hand. But as it does so, the rubber band gets shorter and thepulling force decreases. To keep the pulling force constant, you must move yourhand at just the right speed to keep the length of the rubber band from changing!Figure 4.17 shows the experiment being carried out. Once the motion is complete,you can use motion diagrams (made from the movie frames) and kinematics toanalyze the object’s motion.

4.4 . What Do Forces Do? A Virtual Experiment 107

va

PullRubber band

Maintain constant stretch.

Frictionlesssurface

Motion diagram

Camera

r

r

FIGURE 4.17 Measuring the motion of an object that is pulled with a constant force.

TABLE 4.1 Acceleration due to anincreasing force

Rubber bands Force Acceleration

1 F

2 2F

3 3F

N NF aN 5 Na1

(((

a3 5 3a1

a2 5 2a1

a1

5a1

4a1

3a1

2a1

1a1

0

Acc

eler

atio

n (i

n m

ultip

les

of a

1)

0 1 2 3

Force (number of rubber bands)

4 5

Acceleration is directlyproportional to force.The slope of the line is c.

FIGURE 4.18 Graph of acceleration versusforce.

The first important finding of this experiment is that an object pulled with aconstant force moves with a constant acceleration. This finding could not havebeen anticipated in advance. It’s conceivable that the object would speed up for awhile, then move with a steady speed. Or that it would continue to speed up, but that the rate of increase, the acceleration, would steadily decline. These are conceivable motions, but they’re not what happens. Instead, the object con-tinues to accelerate with a constant acceleration for as long as you pull it with aconstant force.

The next question is: What happens if you increase the force by using severalrubber bands? To find out, use 2 rubber bands. Stretch both to the standardlength to double the force, then measure the acceleration. Then measure theacceleration due to 3 rubber bands, then 4, and so on. Table 4.1 shows the resultsof this experiment. You can see that doubling the force causes twice the accelera-tion, tripling the force causes three times the acceleration, and so on.

Figure 4.18 is a graph of the data. Force is the independent variable, the oneyou can control, so we’ve placed force on the horizontal axis to make an accelera-tion-versus-force graph. The graph shows that the acceleration is directly pro-portional to the force. This is our second important finding. Recall thatproportionality indicates a linear relationship whose graph passes through theorigin (y-intercept of zero). This result can be written

(4.2)

where c is called the proportionality constant. The proportionality constant c isthe slope of the graph.

The final question for our virtual experiment is: How does the accelerationdepend on the size of the object? (The “size” of an object is somewhat ambigu-ous. We’ll be more precise below.) To find out, glue the original object and anidentical copy together, and then, applying the same force as you applied to the

a 5 cF

original, single object, measure the acceleration of this new object. Three objectsglued together make an object three times the size of the original. Doing severalsuch experiments, applying the same force to each object, would give you theresults shown in Table 4.2. An object twice the size of the original has only halfthe acceleration of the original object when both are subjected to the same force.An object three times the size of the original has one-third the acceleration.

Figure 4.19 shows these results added to the graph of Figure 4.18. You can seethat the proportionality constant c between acceleration and force—the slope ofthe line—changes with the size of the object. The graph for an object twice thesize of the original is a line with half the slope. It may seem surprising that largerobjects have smaller slopes, so you’ll want to think about this carefully.

MassNow, “twice the size” is a little vague; we could mean the object’s externaldimensions or some other measure. Although mass is a common word, we’veavoided the term so far because we first need to define what mass is. Because wemade the larger objects in our experiment from the same material as the originalobject, an object twice the size has twice as many atoms—twice the amount ofmatter—as the original. Thus it should come as no surprise that it has twice themass as the original. Loosely speaking, an object’s mass is a measure of theamount of matter it contains. This is certainly our everyday meaning of mass,but it is not yet a precise definition.

Figure 4.19 showed that an object with twice the amount of matter as the origi-nal accelerates only half as quickly if both experience the same force. An objectwith N times as much matter has only of the original acceleration. The morematter an object has, the more it resists accelerating in response to a force. You’refamiliar with this idea: Your car is much harder to push than your bicycle. Thetendency of an object to resist a change in its velocity (i.e., to resist acceleration)is called inertia. Figure 4.19 tells us that larger objects have more inertia thansmaller objects of the same material.

We can make this idea precise by defining the inertial mass m of an object to be

We usually refer to the inertial mass as simply “the mass.” Mass is an intrinsicproperty of an object. It is the property that determines how an object acceleratesin response to an applied force.

Two rubber bands stretched to the standard length cause anobject to accelerate at Suppose another object with twice the mass ispulled by four rubber bands stretched to the standard length. The acceleration ofthis second object is

a. b. c. d. e.

4.5 Newton’s Second LawWe can now summarize the results of our experiment. Figure 4.18 showed that theacceleration is directly proportional to the force, a conclusion that we wrote inEquation 4.2 with the unspecified proportionality constant c. Now we see that c, theslope of the acceleration-versus-force graph, is the inverse of the inertial mass m.

16 m/s2.8 m/s2.4 m/s2.2 m/s2.1 m/s2.

2 m/s2.STOP TO THINK 4.3

m ;1

slope of the acceleration-versus-force graph5

Fa

1N

108 C H A P T E R 4 . Force and Motion

TABLE 4.2 Acceleration with differentnumbers of objects

Number of objects Acceleration

1

2

3

N aN 5 1N a1

((

a3 5 13 a1

a2 5 12 a1

a1

Acc

eler

atio

n

Force

1 objectSlope 5 1

2 objectsSlope 5 3 objectsSlope 5

4 objectsSlope 5

The slope is inverselyproportional to thenumber of objectsbeing accelerated.

12

13

14

FIGURE 4.19 Acceleration-versus-forcegraphs for objects of different size.

Thus we’ve found that a force of magnitude causes an object of mass m toaccelerate with

Equation 4.3 answers the question with which we started: How does an objectmove when a force is exerted on it? A force causes an object to accelerate! Fur-thermore, the size of the acceleration is directly proportional to the size of theforce and inversely proportional to the object’s mass.

This is an important finding, but our experiment was limited to looking at anobject’s response to a single applied force. Realistically, an object is likely to besubjected to several distinct forces . . . that may point in differentdirections. What happens then? In that case, it is found experimentally that theacceleration is determined by the net force.

Newton was the first to recognize the connection between force and motion.This relationship is known today as Newton’s second law.

Newton’s second law An object of mass m subjected to forces . . . will undergo an acceleration given by

(4.3)

where the net force is the vector sum of the indi-vidual forces. The acceleration vector points in the same direction as thenet force vector

It may seem puzzling that we’ve skipped over Newton’s first law. The reasons forthis will become clear as we continue our discussion of dynamics. For now, thecritical idea is that an object’s acceleration vector points in the same direc-tion as the net force vector

The significance of Newton’s second law cannot be overstated. There was noreason to suspect that there should be any simple relationship between force andacceleration. Yet there it is, a simple but exceedingly powerful equation relatingthe two. Newton’s work, preceded to some extent by Galileo’s, marks the begin-ning of a highly successful period in the history of science during which it waslearned that the behavior of physical objects can often be described and predictedby mathematical relationships. While some relationships are found to apply onlyin special circumstances, others seem to have universal applicability. Those equa-tions that appear to apply at all times and under all conditions have come to becalled “laws of nature.” Newton’s second law is a law of nature; you will meetothers as we go through this book.

We can rewrite Newton’s second law in the form

(4.4)

which is how you’ll see it presented in many textbooks. Equations 4.3 and 4.4 aremathematically equivalent, but Equation 4.3 better describes the central idea ofNewtonian mechanics: A force applied to an object causes the object to accelerate.

Be careful not to think that one force “overcomes” the others to determine themotion. Forces are not in competition with each other! It is the sum of all theforces, that determines the acceleration

As an example, Figure 4.20a shows a box being pulled by two ropes. Theropes exert tension forces and on the box. Figure 4.20b represents the boxas a particle, shows the forces acting on the box, and adds them graphically to

T

r

2T

r

1

ar.Fr

net ,

Fr

net 5 mar

Fr

net .ar

Fr

net .ar

Fr

net 5 Fr

1 1 Fr

2 1 Fr

3 1 c

ar 5Fr

net

m

arFr

3,Fr

2,Fr

1,

Fr

3,Fr

2,Fr

1,

a 5Fm

F

4.5 . Newton’s Second Law 109

Top viewof box

Two ropes exertingtension forces on a box

T2

T1

(a)

r

r

FIGURE 4.20 Acceleration of a pulled box.

The accelerationis in the directionof Fnet.

T2

Fnet

a

T1

(b)

r

r

r

r

r

find the net force The box will accelerate in the direction of with an accel-eration of magnitude

NOTE � The acceleration is not You must add the forces as vec-tors, not merely add their magnitudes as scalars. �

Units of ForceBecause the units of force must be mass units multiplied by accelera-tion units. We’ve previously specified the SI unit of mass as the kilogram. We cannow define the basic unit of force as “the force that causes a 1 kg mass to acceler-ate at From the second law, this force is

This basic unit of force is called a newton:

One newton is the force that causes a 1 kg mass to accelerate at Theabbreviation for newton is N. Mathematically,

The newton is a secondary unit, meaning that it is defined in terms of the pri-mary units of kilograms, meters, and seconds. We will introduce other secondaryunits as needed.

It is important to develop a feeling for what the size of forces should be.Table 4.3 shows some typical forces. As you can see, “typical” forces on “typical”objects are likely to be in the range 0.01–10,000 N. Forces less than 0.01 N aretoo small to consider unless you are dealing with very small objects. Forcesgreater than 10,000 N would make sense only if applied to very massive objects.

The unit of force in the English system is the pound (abbreviated lb). Althoughthe definition of the pound has varied throughout history, it is now defined interms of the newton:

You very likely associate pounds with kilograms rather than with newtons. Every-day language often confuses the ideas of mass and weight, but we’re going toneed to make a clear distinction between them. More on this in the next chapter.

Three forces act on an object. In which direction does theobject accelerate?

4.6 Newton’s First LawAs we remarked earlier, Aristotle and his contemporaries in the world of ancientGreece were very interested in motion. One question they asked was: What is the“natural state” of an object if left to itself? It does not take an expensive research

(a)

aa a

aa

F3

F2

F1

(e)(d)(c)(b)

In which directiondoes the objectaccelerate?

r

r

r

r

rr

r

r

STOP TO THINK 4.4

1 pound 5 1 lb ; 4.45 N

1 N 5 1 kg m/s2.1 m/s2.

1 basic unit of force ; 1 kg 3 1

m

s25 1

kg m

s2

1 m/s2.”

Fr

net 5 mar,

( T1 1 T2 ) /m.

ar 5Fr

net

m5

Tr

1 1 Tr

2

m

Fr

net Fr

net .

110 C H A P T E R 4 . Force and Motion

TABLE 4.3 Approximate magnitude of some typical forces

Approximatemagnitude

Force (newtons)

Weight of a U.S. quarter 0.05

Weight of a 1-pound object 5

Weight of a 110-pound person 500

Propulsion force of a car 5,000

Thrust force of a rocket motor 5,000,000

program to see that every moving object on earth, if left to itself, eventuallycomes to rest. Aristotle concluded that the natural state of an earthly object is tobe at rest. An object at rest requires no explanation; it is doing precisely whatcomes naturally to it. A moving object, though, is not in its natural state and thusrequires an explanation: Why is this object moving? What keeps it going and pre-vents it from being in its natural state?

Galileo reopened the question of the “natural state” of objects. He suggestedfocusing on the limiting case in which resistance to the motion (e.g., friction or airresistance) is zero. This is an idealization that may not be realizable in practice,but Galileo had asserted previously, with great success, that the idealized case canestablish a general principle. Many careful experiments in which he minimizedthe influence of friction led Galileo to a conclusion that was in sharp contrast toAristotle’s belief that rest is an object’s natural state.

Galileo found that an external influence (i.e., a force) is needed to make anobject accelerate—to change its velocity. In particular, a force is needed to put anobject in motion. But, in the absence of friction or air resistance, a moving objectcontinues to move along a straight line forever with no loss of speed. In otherwords, the natural state of an object—its behavior if free of external influences—is uniform motion with constant velocity! This does not happen in practicebecause friction or air resistance prevents the object from being left alone. “Atrest” has no special significance in Galileo’s view of motion; it is simply uniformmotion that happens to have

Galileo’s experiments were limited to motion along horizontal surfaces. It wasleft to Newton to generalize this result, and today we call it Newton’s first law of motion.

Newton’s first law An object that is at rest will remain at rest, or an objectthat is moving will continue to move in a straight line with constant velocity,if and only if the net force acting on the object is zero.

Newton’s first law is also known as the law of inertia. If an object is at rest, it hasa tendency to stay at rest. If it is moving, it has a tendency to continue movingwith the same velocity.

NOTE � The first law refers to net force. An object can remain at rest, or canmove in a straight line with constant velocity, even though forces are exertedon it as long as the net force is zero. �

Notice the “if and only if” aspect of Newton’s first law. If an object is at rest ormoves with constant velocity, then we can conclude that there is no net force actingon it. Conversely, if no net force is acting on it, we can conclude that the object willhave constant velocity, not just constant speed. The direction remains constant, too!

An object on which the net force is zero, is said to be in mechanicalequilibrium. According to Newton’s first law, there are two distinct forms ofmechanical equilibrium:

1. The object is at rest. This is static equilibrium.2. The object is moving in a straight line with constant velocity. This is

dynamic equilibrium.

Two examples of mechanical equilibrium are shown in Figure 4.21. Both sharethe common feature that the acceleration is zero:

What Good Is Newton’s First Law?The first law completes our definition of force. It answers the question: What is aforce? If an “influence” on an object causes the object’s velocity to change, theinfluence is a force.

ar 5 0r

.

Fr

net 5 0r

vr 5 0r

.

4.6 . Newton’s First Law 111

v 5 0a 5 0

r r

r r

An object at rest isin static equilibrium:Fnet 5 0.r r

FIGURE 4.21 Two examples of mechanicalequilibrium.

An object moving in a straight line at constantvelocity is in dynamic equilibrium: Fnet 5 0.

a 5 0

r r

rr

rv

Newton’s first law changes the question the ancient Greeks were trying toanswer: What causes an object to move? Newton’s first law says no cause isneeded for an object to move! Uniform motion is the object’s natural state.Nothing at all is required for it to remain in that state. The proper question,according to Newton, is: What causes an object to change its velocity? Newton,with Galileo’s help, also gave us the answer. A force is what causes an object tochange its velocity.

The preceding paragraph contains the essence of Newtonian mechanics. Thisnew perspective on motion, however, is often contrary to our common experi-ence. We all know perfectly well that you must keep pushing an object—exertinga force on it—to keep it moving. Newton is asking us to change our point of viewand to consider motion from the object’s perspective rather than from our personalperspective. As far as the object is concerned, our push is just one of severalforces acting on it. Others might include friction, air resistance, or gravity. Onlyby knowing the net force can we determine the object’s motion.

Newton’s first law may seem to be merely a special case of Newton’s sec-ond law. After all, the equation tells us that an object moving withconstant velocity ( ) has The difficulty is that the second lawassumes that we already know what force is. The purpose of the first law is toidentify a force as something that disturbs a state of equilibrium. The secondlaw then describes how the object responds to this force. Thus from a logicalperspective, the first law really is a separate statement that must precede thesecond law. But this is a rather formal distinction. From a pedagogical per-spective it is better—as we have done—to use a commonsense understandingof force and start with Newton’s second law.

Inertial Reference FramesIf a car stops suddenly, you may be “thrown” into the windshield if you’re notwearing your seat belt. You have a very real forward acceleration relative to thecar, but is there a force pushing you forward? A force is a push or a pull caused byan identifiable agent in contact with the object. Although you seem to be pushedforward, there’s no agent to do the pushing.

The difficulty—an acceleration without an apparent force—comes from usingan inappropriate coordinate system. Your acceleration measured in a coordinatesystem attached to the car is not the same as your acceleration measured in a coor-dinate system attached to the ground. Newton’s second law says Butwhich Measured in which coordinate system?

We define an inertial reference frame as a coordinate system in which New-ton’s laws are valid. The first law provides a convenient way to test whether acoordinate system is an inertial reference frame. If (an object is at rest ormoving with constant velocity) only when then the coordinate system inwhich is measured is an inertial reference frame.

Not all coordinate systems are inertial reference frames. Figure 4.22a shows aphysics student cruising at constant velocity in an airplane. If the student places aball on the floor, it stays there. There are no horizontal forces, and the ballremains at rest relative to the airplane. That is, in the airplane’s coordinatesystem when Newton’s first law is satisfied, so this airplane is an inertialreference frame.

The physics student in Figure 4.22b conducts the same experiment duringtakeoff. She carefully places the ball on the floor just as the airplane starts toaccelerate down the runway. You can imagine what happens. The ball rolls to theback of the plane as the passengers are being pressed back into their seats. Noth-ing exerts a horizontal contact force on the ball, yet the ball accelerates in theplane’s coordinate system. This violates Newton’s first law, so the plane is not aninertial reference frame during takeoff.

Fr

net 5 0r

.ar 5 0

r

arFr

net 5 0r

,ar 5 0

r

ar?Fr

net 5 mar.

Fr

net 5 0r

.ar 5 0r

Fr

net 5 mar

112 C H A P T E R 4 . Force and Motion

a 5 0

The ball stays in place.

A ball with no horizontal forces stays at restin an airplane cruising at constant velocity.The airplane is an inertial reference frame.

r r

(a)

FIGURE 4.22 Reference frames.

The ball rolls to the back.

The ball rolls to the back of the plane duringtakeoff. An accelerating plane is not aninertial reference frame.

(b)

Accelerating

This guy thinks there’s a force hurling himinto the windshield.What a dummy!

In the first example, the plane is traveling with constant velocity. In the sec-ond, the plane is accelerating. Accelerating reference frames are not inertialreference frames. Consequently, Newton’s laws are not valid in a coordinate sys-tem attached to an accelerating object.

But accelerating with respect to what? The plane accelerated with respect to the earth. But the earth is accelerating as it rotates on its axis and revolvesaround the sun. The entire solar system is accelerating as our Milky Way galaxyrotates.

This is a subtle and difficult question, one that Einstein grappled with as hedeveloped his theory of relativity. We cannot give a complete answer here, butsuffice it to say that an inertial reference frame is a reference frame that is notaccelerating with respect to the distant stars. Thus the earth is not exactly an iner-tial reference frame. However, the earth’s acceleration with respect to the distantstars is so small that violations of Newton’s laws can be measured only inextremely high-precision experiments. We will treat the earth and laboratoriesattached to the earth as inertial reference frames, an approximation that is exceed-ingly well justified.

We will prove in Chapter 6 that a coordinate system moving with constantvelocity relative to an inertial reference frame is also an inertial reference frame, areference frame in which Newton’s laws are valid. Because the earth is an inertialreference frame, the airplane of Figure 4.22a is also an inertial reference frame.But a car braking to a stop is not, so you cannot use Newton’s laws in the car’sreference frame.

To understand the motion of objects in the car, such as the passengers, you needto measure velocities and accelerations relative to the ground. From the perspec-tive of an observer on the ground, the body of a passenger in a braking car tries tocontinue moving forward with constant velocity, exactly as we would expect onthe basis of Newton’s first law, while his immediate surroundings are decelerat-ing. The passenger is not “thrown” into the windshield. Instead, the windshieldruns into the passenger!

Common Misconceptions About ForceIt is important to identify correctly all the forces acting on an object. It is equallyimportant not to include forces that do not really exist. We have established anumber of criteria for identifying forces; the two critical ones are:

■ A force has an agent. Something tangible and identifiable causes the force.■ Forces exist at the point of contact between the agent and the object experi-

encing the force (except for the few special cases of long-range forces).

We all have had many experiences suggesting that a force is necessary to keepsomething moving. Consider a bowling ball rolling along on a smooth floor. It isvery tempting to think that a horizontal “force of motion” keeps it moving in theforward direction. But if we draw a closed curve around the ball, nothing contactsit except the floor. No agent is giving the ball a forward push. According to ourdefinition, then, there is no forward “force of motion” acting on the ball. So whatkeeps it going? Recall our discussion of the first law: no cause is needed to keep anobject moving at constant velocity. It continues to move forward simply becauseof its inertia.

One reason for wanting to include a “force of motion” is that we tend to viewthe problem from our perspective as one of the agents of force. You certainly haveto keep pushing to shove a box across the floor at constant velocity. If you stop, itstops. Newton’s laws, though, require that we adopt the object’s perspective. Thebox experiences your pushing force in one direction and a friction force in theopposite direction. The box moves at constant velocity if the net force is zero.This will be true as long as your pushing force exactly balances the friction force.

4.6 . Newton’s First Law 113

When you stop pushing, the friction force causes an acceleration that slows andstops the box.

A related problem occurs if you throw a ball. A pushing force was indeedrequired to accelerate the ball as it was thrown. But that force disappears theinstant the ball loses contact with your hand. The force does not stick with the ballas the ball travels through the air. Once the ball has acquired a velocity, nothing isneeded to keep it moving with that velocity.

A final difficulty worth noting is the force due to air pressure. You may havelearned in an earlier science class that air, like any fluid, exerts forces on objects.Perhaps you learned this idea as “the air presses down with a weight of 15 poundson every square inch.” There is only one error here, but it is a serious one: the worddown. Air pressure, at sea level, does indeed exert a force of 15 pounds per squareinch, but in all directions. It presses down on the top of an object, but also inwardon the sides and upward on the bottom. For most purposes, the net force due to airpressure is zero! The only way to experience an air pressure force is to form a sealaround one side of the object and then remove the air, creating a vacuum. Whenyou press a suction cup against the wall, you press the air out and the rubber formsa seal that prevents the air from returning. Now the air pressure does hold the suc-tion cup in place! We do not need to be concerned with air pressure until Part III ofthis book.

4.7 Free-Body DiagramsHaving discussed at length what is and is not a force, we are ready to assembleour knowledge about force and motion into a single diagram called a free-bodydiagram. You will learn in the next chapter how to write the equations of motiondirectly from the free-body diagram. Solution of the equations is a mathematicalexercise—possibly a difficult one, but nonetheless an exercise that could be doneby a computer. The physics of the problem, as distinct from the purely calcula-tional aspects, are the steps that lead to the free-body diagram.

A free-body diagram represents the object as a particle and shows all of theforces acting on the object. The free-body diagram joins with the motion diagramand force identification to form the full physical representation of a problem.

TACTICS BOX 4.3 Drawing a free-body diagram

●1 Identify all forces acting on the object. This step was described in Tac-tics Box 4.2.

●2 Draw a coordinate system. Use the axes defined in your pictorial repre-sentation. If those axes are tilted, for motion along an incline, then theaxes of the free-body diagram should be similarly tilted.

●3 Represent the object as a dot at the origin of the coordinate axes.This is the particle model.

●4 Draw vectors representing each of the identified forces. This wasdescribed in Tactics Box 4.1. Be sure to label each force vector.

●5 Draw and label the net force vector Draw this vector beside thediagram, not on the particle. Or, if appropriate, write . Thencheck that points in the same direction as the acceleration vector on your motion diagram.

arFr

net

Fr

net 5 0r

Fr

net.

114 C H A P T E R 4 . Force and Motion

4.7 . Free-Body Diagrams 115

EXAMPLE 4.5 An ice block shoots across a frozen lakeBobby straps a small model rocket to a block of ice and shootsit across the smooth surface of a frozen lake. Friction is negligi-ble. Draw a full physical representation of the block of ice.

MODEL Treat the block of ice as a particle. The full physicalrepresentation consists of a motion diagram to determine aforce identification picture, and a free-body diagram. The state-ment of the situation implies that friction is negligible.

ar ,

The vectors have been drawn accordingly, andthis leaves the net force vector pointing toward the right, inagreement with from the motion diagram.ar

( wy 5 2ny ) .

EXAMPLE 4.4 An elevator accelerates upwardAn elevator, suspended by a cable, speeds up as it moves upward from the groundfloor. Draw a free-body diagram of the elevator.

MODEL Treat the elevator as a particle.

VISUALIZE

FIGURE 4.23 Free-body diagram of an elevator accelerating upward.

Force identification

Tension T

Weight wIdentify allforces actingon the object.

r

r1

Free-body diagram Draw a coordinatesystem.

Represent theobject as a dotat the origin.

Draw and labelFnet beside thediagram.

y

x

T

Fnet

Draw vectors forthe identified forces.

w

r

r

r

r

2

3

45

ASSESS The coordinate axes, with a vertical y-axis, are the ones we would use in apictorial representation of the motion. The elevator is accelerating upward, so must point upward. For this to be true, the magnitude of must be larger than themagnitude of The diagram has been drawn accordingly.wr .

T

rFr

net

ASSESS The motion diagram tells us that the acceleration is inthe According to the rules of vector addition, thiscan be true only if the upward-pointing and the downward-pointing are equal in magnitude and thus cancel each otherwr

nr1x-direction.

FIGURE 4.24 Physical representation for a block of ice shooting across a frictionless frozen lake.

a

v

y

x

n

w

Fthrust

Fnet

Start

Check that Fnet points in the same direction as a.

Thrust force Fthrust

Weight w

Normal force n

r

r

r

r

r

r

r

r

r r

r

Force identificationMotion diagram Free-body diagram

VISUALIZE

Free-body diagrams will be our major tool for the next several chapters. Care-ful practice with the workbook exercises and homework in this chapter will payimmediate benefits in the next chapter. Indeed, it is not too much to assert that aproblem is half solved, or even more, when you complete the free-body diagram.

An elevator suspended by a cable is moving upward andslowing to a stop. Which free-body diagram is correct?

y

x

T

w

y

x

T

w

y

x

T

Fnet 5 0

Felevator

w

y

x

T

w

(a) (b) (c) (d) (e)

r

r

r

r

r

r r r

r

r

STOP TO THINK 4.5

116 C H A P T E R 4 . Force and Motion

EXAMPLE 4.6 A skier is pulled up a hillA tow rope pulls a skier up a snow-covered hill at a constantspeed. Draw a full physical representation of the skier.

MODEL This is Example 4.2 again with the additional informa-tion that the skier is moving at constant speed. The skier will be

treated as a particle in dynamic equilibrium. If we were doing a kinematics problem, the pictorial representation would use a tilted coordinate system with the x-axis parallel to the slope,so we use these same tilted coordinate axes for the free-bodydiagram.

the incline above the horizontal. The skier moves in a straightline with constant speed, so and, from Newton’s firstlaw, Thus we have drawn the vectors such that the y-component of is equal in magnitude to Similarly, mustbe large enough to match the negative x-components of both and wr .

f r

k

T

r

nr.wrFr

net 5 0r

.ar 5 0

rASSESS We have shown pulling parallel to the slope and which opposes the direction of motion, pointing down the slope.

is perpendicular to the surface and thus along the y-axis.Finally, and this is important, the weight is vertically down-ward, not along the negative y-axis. In fact, you should convinceyourself from the geometry that the angle between the vector and the negative y-axis is the same as the angle of u

wru

wrnr

f r

k,T

r

FIGURE 4.25 Physical representation for a skier being towed at a constant speed.

Weight w

Tension T

Normal force nFriction fk

w

y

xn

fk

Fnet 5 0

T

��v

Force identification Free-body diagramMotion diagram

a 5 0

Check that Fnet matches a.

Notice that the angle betweenthe w and the negative y-axisis the same as the angle ofthe incline.

r

r r

r

r

r

r

r

r

r

r

r

r rr

r

VISUALIZE

S U M M A R Y

The goal of Chapter 4 has been to learn how force and motion are connected.

Summary 117

TERMS AND NOTATION

dynamicsmechanicsforce, agentcontact forcelong-range forcenet force,

superposition of forcesweight, tension force, atomic modelnormal force, friction, or drag,

thrust, inertiainertial mass, mNewton’s second lawnewton, NNewton’s first lawmechanical equilibrium

static equilibriumdynamic equilibriuminertial reference framefree-body diagram

Fr

thrust

Dr

f r

sf r

k

nr

T

rwr

Fr

net

Fr

GENERAL PRINCIPLES

Newton’s First LawAn object at rest will remain at rest, or anobject that is moving will continue to movein a straight line with constant velocity, ifand only if the net force on the object iszero.

The first law tells us that no “cause” isneeded for motion. Uniform motion is the“natural state” of an object.

Newton’s Second LawAn object with mass m will undergo acceleration

where is the vectorsum of all the individual forces acting on the object.

The second law tells us that a net force causes an object to accelerate. This is the connectionbetween force and motion that we are seeking.

Fr

net 5 Fr

1 1 Fr

2 1 Fr

3 1 c

ar 51m

Fr

net

a 5 0

Fnet

5 0r

rr

vr vr vr vrvr

r

Fr

vrvrvrvrvr

ar

Newton’s laws arevalid only in inertialreference frames.

IMPORTANT CONCEPTS

Acceleration is the link to kinematics.

From a, find v and x.From v and x, find a.

is the condition for equilibrium.

Static equilibrium if .Dynamic equilibrium if constant.

Equilibrium occurs if and only if Fr

net 5 0r

.

vr 5vr 5 0

r

ar 5 0r

Mass is the resistance of anobject to acceleration. It is an intrinsic property of anobject.

Force is a push or a pull on an object.

• Force is a vector, with a magnitude anda direction.

• Force requires an agent.

• Force is either a contact force or a long-range force.

KEY SKILLS

Identifying ForcesForces are identified by locating the points where the environmenttouches the system. These are pointswhere contact forces are exerted. Inaddition, objects with mass feel along-range weight force.

Free-Body DiagramsA free-body diagram represents theobject as a particle at the origin of a coordinate system. Force vectorsare drawn with their tails on theparticle. The net force vector isdrawn beside the diagram.

Thrust force Fthrust

Weight w Normal force nr

r

r

x

y

n Fthrust

Fnet

w

r r

r

r

Exercises

Section 4.1 Force

1. Write a one-paragraph essay on the topic “What Is a Force?”Explain in your own words how you recognize a force andwhat the properties of forces are.

Section 4.3 Identifying Forces

2. A mountain climber is hanging from a rope in the middle of acrevasse. The rope is vertical. Identify the forces on the moun-tain climber.

3. A baseball player is sliding into second base. Identify theforces on the baseball player.

4. A jet plane is speeding down the runway during takeoff. Airresistance is not negligible. Identify the forces on the jet.

Section 4.4 What Do Forces Do?

5. Figure Ex4.5 shows an acceleration-versus-force graph forthree objects pulled by rubber bands. The mass of object 2 is0.20 kg. What are the masses of objects 1 and 3? Explain yourreasoning.

FIGURE EX4.5

6. Two rubber bands pulling on an object cause it to accelerate at

a. What will be the object’s acceleration if it is pulled by fourrubber bands?

b. What will be the acceleration of two of these objects gluedtogether if they are pulled by two rubber bands?

Section 4.5 Newton’s Second Law

7. Write a one-paragraph essay on the topic “Force and Motion.”Explain in your own words the connection between forceand motion. Where possible, cite evidence supporting yourstatements.

8. Figure Ex4.8 shows an acceleration-versus-force graph for a500 g object. Redraw this graph and add appropriate acceler-ation values on the vertical scale.

1.2 m/s2.

5a1

4a1

3a1

2a1

1a1

00 1 2 3 4 5 6

Force (number of rubber bands)

Acc

eler

atio

n (m

ultip

les

of a

1) 12

3

118 C H A P T E R 4 . Force and Motion

FIGURE EX4.8

9. Figure Ex4.9 shows an object’s acceleration-versus-forcegraph. What is the object’s mass?

FIGURE EX4.9

10. Based on the information in Table 4.3, estimatea. The weight of this textbook.b. The propulsion force of a bicycle.

11. Based on the information in Table 4.3, estimatea. The weight of a pencil.b. The propulsion force of a sprinter.

Section 4.6 Newton’s First Law

Exercises 12 through 14 show two forces acting on an object.Redraw the diagram, then add a third force that will cause theobject to be in equilibrium. Label the new force .

12. 13. 14.

FIGURE EX4.12 FIGURE EX4.13 FIGURE EX4.14

Section 4.7 Free-Body Diagrams

Exercises 15 through 17 show a free-body diagram. For each:a. Redraw the free-body diagram.b. Write a short description of a real object for which this is the

correct free-body diagram. Use Examples 4.4, 4.5, and 4.6 asmodels of what a description should be like.

F1

r

F2

r

F1

r

F2

rF1

r

F2

r

Fr

3

4

3

2

1

00.0 0.5 1.0

a (m/s2)

F (N)

?

?

?

?

00 1 2

a (m/s2)

F (N)

E X E R C I S E S A N D P R O B L E M S

15. 16.

FIGURE EX4.15 FIGURE EX4.16

17.

FIGURE EX4.17

Exercises 18 through 21 describe a situation. For each, identify allforces acting on the object and draw a free-body diagram of theobject.18. Your car is sitting in the parking lot.19. An ice hockey puck glides across frictionless ice.20. An elevator, hanging from a cable, descends at steady speed.21. Your physics textbook is sliding across the table.

Problems

22. Redraw the two motion diagrams shownin Figure P4.22, then draw a vector besideeach one to show the direction of the netforce acting on the object. Explain yourreasoning.

FIGURE P4.22

23. Redraw the two motion diagrams shown in Figure P4.23, thendraw a vector beside each one to show the direction of the netforce acting on the object. Explain your reasoning.

FIGURE P4.23

(b)

vr

v(a) r

v

(b)

r

v

(a)

r

w

y

x

n

fk

r

r

r

Fnet

r

y

xwD

Fnet

Fthrust

rr

r

r

y

x

T

Fnet 5 0

w

r

r

r r

Exercises and Problems 119

24. A force with x-component acts on a 2.0 kg object as itmoves along the x-axis. The object’s acceleration graph (versus t) is shown in Figure P4.24. Draw a graph of versus t.

FIGURE P4.24

25. A force with x-component acts on a 500 g object as itmoves along the x-axis. The object’s acceleration graph (versus t) is shown in Figure P4.25. Draw a graph of versus t.

FIGURE P4.25

26. A force with x-component acts on a 2.0 kg object as it movesalong the x-axis. A graph of versus t is shown in Figure P4.26.Draw an acceleration graph ( versus t) for this object.

FIGURE P4.26

27. A force with x-component acts on a 500 g object as itmoves along the x-axis. A graph of versus t is shown in Fig-ure P4.27. Draw an acceleration graph ( versus t) for thisobject.

FIGURE P4.27

28. A constant force is applied to an object, causing the object toaccelerate at What will the acceleration be ifa. The force is halved?b. The object’s mass is halved?c. The force and the object’s mass are both halved?d. The force is halved and the object’s mass is doubled?

10 m/s2.

1.5

1.0

0.5

0.0

20.5

1 2 3 4

Fx (N)

t (s)

ax

Fx

Fx

3

2

1

0

21

1 2 3 4

Fx (N)

t (s)

ax

Fx

Fx

1.5

1.0

0.5

0.0

20.5

1 2 3 4

ax (m/s2)

t (s)

Fx

ax

Fx

3

2

1

0

21

1 2 3 4

ax (m/s2)

t (s)

Fx

ax

Fx

29. A constant force is applied to an object, causing the object toaccelerate at What will the acceleration be ifa. The force is doubled?b. The object’s mass is doubled?c. The force and the object’s mass are both doubled?d. The force is doubled and the object’s mass is halved?

Problems 30 through 36 show a free-body diagram. For each:a. Redraw the diagram.b. Identify the direction of the acceleration vector and show it

as a vector next to your diagram. Or, if appropriate, write

c. If possible, identify the direction of the velocity vector andshow it as a labeled vector.

d. Write a short description of a real object for which this is thecorrect free-body diagram. Use Examples 4.4, 4.5, and 4.6 asmodels of what a description should be like.

30.

FIGURE P4.30

31.

FIGURE P4.31

32.

FIGURE P4.32

33.

FIGURE P4.33

34.

FIGURE P4.34w

y

x

n

fk

Fnet

r r

r

r

w

y

x

r

w

y

x

n

Fnet

fkFsp

r

rr

r

r

w

y

x

n

Fnet 5 0

FthrustDr

r

r

r

r r

w

y

x

n

Fnet

Tfk

r

rr

r

r

v

r

ar 5 0r

.

ar

8.0 m/s2.

120 C H A P T E R 4 . Force and Motion

35.

FIGURE P4.35

36.

FIGURE P4.36

Problems 37 through 46 describe a situation. For each, draw a fullphysical representation. A full physical representation is a motiondiagram, force identification, and a free-body diagram.37. An elevator, suspended by a single cable, has just left the

tenth floor and is speeding up as it descends toward theground floor.

38. A rocket is being launched straight up. Air resistance is notnegligible.

39. A jet plane is speeding down the runway during takeoff. Airresistance is not negligible.

40. You’ve slammed on the brakes and your car is skidding to astop while going down a hill.

41. A skier is going down a slope. A horizontal headwind isblowing in the skier’s face. Friction is small, but not zero.

42. You’ve just kicked a soccer ball and it is now rolling acrossthe grass.

43. A styrofoam ball has just been shot straight up. Air resistanceis not negligible.

44. A spring-loaded gun shoots a plastic ball. The trigger has justbeen pulled and the ball is starting to move down the barrel.The barrel is horizontal.

45. A person on a bridge throws a rock straight down toward thewater. The rock has just been released.

46. A gymnast has just landed on a trampoline. She’s still movingdownward as the trampoline stretches.

Challenge Problems

47. A heavy box is in the back of a truck. The truck is acceleratingto the right. Draw a full physical representation of the box.

48. A bag of groceries is on the back seat of your car as you stopfor a stop light. The bag does not slide. Draw a full physicalrepresentation of the bag.

49. A rubber ball bounces. We’d like to understand how the ballbounces.a. A rubber ball has been dropped and is bouncing off the

floor. Draw a motion diagram of the ball during the brieftime interval that it is in contact with the floor. Show 4 or 5frames as the ball compresses, then another 4 or 5 framesas it expands. What is the direction of during each ofthese parts of the motion?

b. Draw a picture of the ball in contact with the floor andidentify all forces acting on the ball.

ar

20°20°

w

y

x

Fnet

Fthrust

r

r

r

w

y

xn

fk

Fnet

T

r

r

r

r

r

c. Draw a free-body diagram of the ball during its contactwith the ground. Is there a net force acting on the ball? Ifso, in which direction?

d. Write a paragraph in which you describe what you learnedfrom parts a to c and in which you answer the question:How does a ball bounce?

50. If a car stops suddenly, you feel “thrown forward.” We’d liketo understand what happens to the passengers as a car stops.Imagine yourself sitting on a very slippery bench inside a car.This bench has no friction, no seat back, and there’s nothingfor you to hold to.a. Draw a picture and identify all of the forces acting on you

as the car travels at a perfectly steady speed on levelground.

Exercises and Problems 121

b. Draw your free-body diagram. Is there a net force on you?If so, in which direction?

c. Repeat parts a and b with the car slowing down.d. Describe what happens to you as the car slows down.e. Use Newton’s laws to explain why you seem to be

“thrown forward” as the car stops. Is there really a forcepushing you forward?

f. Suppose now that the bench is not slippery. As the carslows down, you stay on the bench and don’t slide off.What force is responsible for your deceleration? In whichdirection does this force point? Include a free-body dia-gram as part of your answer.

STOP TO THINK ANSWERS

Stop to Think 4.1: c.

Stop to Think 4.2: a, b, and d. Friction and the normal force arethe only contact forces. Nothing is touching the rock to provide a“force of the kick.” We’ve agreed to ignore air resistance unless aproblem specifically calls for it.

Stop to Think 4.3: b. Acceleration is proportional to force, sodoubling the number of rubber bands doubles the acceleration ofthe original object from to But acceleration is alsoinversely proportional to mass. Doubling the mass cuts the acceler-ation in half, back to 2 m/s2.

4 m/s2.2 m/s2

x-component of F3 is tothe left and larger thanthe x-component of F2.

F1F2

F3 y-component of F3cancels y-component of F1.

r

r

r

r

r

r

r

Stop to Think 4.4: d.

Stop to Think 4.5: c. The acceleration vector points downward asthe elevator slows. points in the same direction as so also points down. This will be true if the tension is less than theweight: T , w.

Fr

netar,Fr

net

F1

F3

F2

F1 1 F2

F3

First addF1 and F2.

Then add F3. a is in the samedirection as Fnet.

Fnet ar

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This is Fnet.r

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