Darlene Librero pulls withone finger; Paul Dohertypulls with both hands.Question they ask of theirExploratorium class:"Who exerts more force onthe scale?"

Ifyou drop a sheet of tissue paper in front of the heavyweight boxing champion ofthe world and challenge him to hit it in midair with a force of only 50 pounds

(222 N)-sorry, the champ can't do it. In fact, his best punch couldn't even come close.Why is this? We'll see in this chapter that the tissue has insufficient inertia for a 50-poundinteraction with the champ's fist.

Forces and Interactions

FIGURE 5.1Interactive Figure ~

Youcan feel your fingers be-ing pushed by your friend'sfingers. Youalso feel thesame amount of force whenyou push on a wall and itpushes back on you. As apoint of fact, you can't pushon the wall unless it pushesback on you!

74

So far we've treated force in its simplest sense-as a push or pull. Yet no pushor pull ever occurs alone. Every force is part of an interaction between onething and another. When you push on a wall with your fingers, more is hap-pening than your pushing on the wall. You're interacting with the wall, whichalso pushes back on you. This is evident in your bent fingers, as illustrated inFigure 5.1. There is a pair of forces involved: your push on the wall and thewall's push back on you. These forces are equal in magnitude (have the samestrength) and opposite in direction, and they constitute a single interaction. Infact, you can't push on the wall unless the wall pushes back."

Consider a boxer's fist hitting a massive punching bag. The fist hits the bag(and dents it) while the bag hits back on the fist (and stops its motion). A pairof forces is involved in hitting the bag. The force pair can be quite large. Butwhat of hitting a piece of tissue paper, as discussed earlier? The boxer's fist canonly exert as much force on the tissue paper as the tissue paper can exert onthe fist. Furthermore, the fist can't exert any force at all unless what is beinghit exerts the same amount of force back. An interaction requires a pair offorces acting on two separate objects.

Other examples: You pull on a cart and it accelerates. But, in doing so, thecart pulls back on you, as evidenced perhaps by the tightening of the ropewrapped around your hand. A hammer hits a stake and drives it into the

'We tend to think that only living things are capable of pushing and pulling. But inanimate things can do thesame. So please don't be troubled about the idea of the inanimate wall pushing on you. It does, just asanother person leaning against you would.

FIGURE 5.2When you lean against awall, you exert a force on thewall. The wall simultane-ously exerts an equal andopposite force on you.Hence you don't topple over.

Forces and Interactions

Chapter 5 Newton's Third Law of Motion 75

ground. In doing so, the stake exerts an equal amount of force on the hammer,which brings the hammer to an abrupt halt. One thing interacts with another-you with the cart, or the hammer with the stake.

Which exerts the force and which receives the force? Isaac Newton'sresponse was that neither force has to be identified as "exerter" or "receiver";he concluded that both objects must be treated equally. For example, when youpull the cart, the cart pulls on you. This pair of forces, your pull on the cartand the cart's pull on you, make up the single interaction between you and thecart. In the interaction between the hammer and the stake, the hammer exertsa force against the stake but is itself brought to a halt in the process. Suchobservations led Newton to his third law of motion.

FIGURE 5.3He can hit the massive bag with considerable force. But withthe same punch he can exert only a tiny force on the tissuepaper in midair.

FIGURE 504In the interaction betweenthe hammer and the stake,each exerts the same amountof force on the other.

Newton's Third Law of Motion

..JVL) CJ ("

Push your fingerstogether and noticethat, as you pushharder, discolorationis the same for both.Aha, they both experi-ence the same magni-tude of force!

Newton's third law states:

Whenever one object exerts a force on a second object, the secondobject exerts an equal and opposite force on the first.

We can call one force the action force and the other the reaction force. Thenwe can express Newton's third law in the form:

To every action there is always an opposed equal reaction.

It doesn't matter which force we call action and which we call reaction. Theimportant thing is that they are eo- parts of a single interaction, and that nei-ther force exists without the other.

When you walk, you interact with the floor. You push against the floor, andthe floor pushes against you. The pair of forces occurs at the same time (theyare simultaneous). Likewise, the tires of a car push against the road while theroad pushes back on the tires-the tires and road simultaneously push againsteach other. In swimming, you interact with the water, pushing the water back-ward, while the water simultaneously pushes you forward-you and the waterpush against each other. The reaction forces are what account for our motionin these examples. These forces depend on friction; a person or car on ice, forexample, may be unable to exert the action force to produce the needed reac-tion force. Neither force exists without the other.

76 Part One Mechanics

;>

FIGURE S.SThe impact forces between the blue balland the yellow ball move the yellow balland stop the blue ball.

FIGURE 5.6In the interaction between the car and the truck, is the forceof impact the same on each? Is the damage the same?

CHECK YOURSELF

Does a speeding missile possess force?

FIGURE 5.7Action and reaction forces.Note that, when action is "Aexerts force on B," the reac-tion is then simply "B exertsforce on A."

Reaction: road pushes on tire

-----;)~- ---....;...J~~

Action: rocket pusheson gas Reaction; gas pushes on rocket

Action: man pulls on spring~-.

Reaction: spring pulls on man

J Action: earth pulls on ball

Reaction: ball pulls on earth

CHECK YOUR ANSWER

No, a force is not something an object has, like mass, but is part of an interactionbetween one object and another. A speeding missile may possess the capability ofexerting a force on another object when interaction occurs, but it does not possessforce as a thing in itself. As we will see in the following chapters, a speeding missilepossesses momentum and kinetic energy.

;'---------, \I I

I :II\

FIGURE 5.8Interactive Figure t..

A force acts on the orange,and the orange acceleratesto the right.

-.JVL) (/ C

A system may be astiny as an atom or aslarge as the universe.

Chapter 5 Newton's Third Law of Motion 77

Defining Your SystemAn interesting question often arises: Since action and reaction forces are equaland opposite, why don't they cancel to zero? To answer this question, we mustconsider the system involved. Consider, for example, a system consisting of asingle orange, Figure 5.8. The dashed line surrounding the orange encloses anddefines the system. The vector that pokes outside the dashed line represents anexternal force on the system. The system accelerates in accord with Newton'ssecond law. In Figure 5.9, we see that this force is provided by an apple, whichdoesn't change our analysis. The apple is outside the system. The fact that theorange simultaneously exerts a force on the apple, which is external to the sys-tem, may affect the apple (another system), but not the orange. You can't cancela force on the orange with a force on the apple. So, in this case, the action andreaction forces don't cancel.

,,--------"""" FIGURE 5.9,IIII\ ••..

Interactive Figure ~

The force on the orange,provided by the apple, is notcancelled by the reaction forceon the apple. The orange stillaccelerates .

Now let's consider a larger system, enclosing both the orange and the apple.We see the system bounded by the dashed line in Figure 5.10. Notice that the forcepair is internal to the orange-apple system. Then these forces do cancel each other.They play no role in accelerating the system. A force external to the system is neededfor acceleration. That's where friction with the floor comes into play (Figure 5.11).When the apple pushes against the floor, the floor simultaneously pushes on theapple-an external force on the system. The system accelerates to the right .

...- - - -- - - - - .- - -IIIIII

"

IIIIII

./

-~---~ "..------IIIIII

------

FIGURE 5.10 Interactive Figure •..

In the larger system of orange'~ apple, action andreaction forces are internal and cancel. If these are theonly horizontal forces, with no external force, noacceleration of the system occurs.

FIGURE 5.11 Interactive Figure ~

An external horizontal force occurs when the floor pusheson the apple (reaction to the apple's push on the floor).The orange-apple system accelerates.

Inside a football are trillions and trillions of interatomic forces at play. Theyhold the ball together, but they play no role in accelerating the ball. Althougheveryone of the inter atomic forces is part of an action-reaction pair within theball, they combine to zero, no matter how many of them there are. A force

78 Part One Mechanics

FIGURE 5.12A acts on B, and Baccelerates.

FIGURE 5.13Both A and C act on B.Theycan cancel each other, so Bdoes not accelerate.

-JVL) ('/ C

You can't push or pullon something unlessthat something simul-taneously pushes orpulls on you. That'sthe law!

external to the football, like kicking it, is needed to accelerate it. In Figure 5.12,we note a single interaction between the foot and the football.

The football in Figure 5.13, however, does not accelerate. In this case, thereare two interactions occurring-two forces acting on the football. If they aresimultaneous, equal and opposite, then the net force is zero. Do the twoopposing kicks make up an action-reaction pair? No, for they act on the sameobject, not on different objects. They may be equal and opposite, but, unlessthey act on different objects, they are not an action-reaction pair. Get it?

If this is confusing, it may be well to note that Newton had difficulties withthe third law himself. (See insightful examples of Newton's third law on pages 21and 22 in the Concept Development Practice Book.)

CHECK YOURSELF

1. On a cold, rainy day, you find yourself in a car with a dead battery. You mustpush the car to move it and get it started. Why can't you move the car byremaining comfortably inside and pushing against the dashboard?

2. Why does a book sitting on a table never accelerate "spontaneously" in responseto the trillions of interatomic forces acting within it?

3. We know that the Earth pulls on the Moon. Does it follow that the Moon alsopulls on the Earth?

4. Can you identity the action and reaction forces in the case of an object falling ina vacuum?

CHECK YOUR ANSWERS

1. In this case, the system to be accelerated is the car. If you remain inside andpush on the dashboard, the force pair you produce acts and reacts within thesystem. These forces cancel out as far as any motion of the car is concerned.To accelerate the car, there must be an interaction between the car and some-thing external-for example, you on the outside pushing against the road andon the car.

2. Everyone of these interatomic forces is part of an action-reaction pair within thebook. These forces add up to zero, no matter how many of them there are. Thisis what makes Newton's first law apply to the book. The book has zero accelera-tion unless an external force acts on it.

3. Yes, both pulls make up an action-reaction pair of forces associated with thegravitational interaction between Earth and Moon. We can say that (1) Earthpulls on Moon and (2) Moon likewise pulls on Earth; but it is more insightful tothink of this as a single interaction-both Earth and Moon simultaneously pullingon each other, each with the same amount of force.

4. To identity a pair of action-reaction forces in any situation, first identity the pairof interacting objects involved-Body A and Body B. Body A, the falling object, isinteracting (gravitationally) with Body B, the whole Earth. So the Earth pullsdownward on the object (call it action), while the object pulls upward on theEarth (reaction).

FIGURE 5.14The Earth is pulled up by theboulder with just as muchforce as the boulder is pulleddownward by the Earth.

Action and Reaction onDifferent Masses

Action and Reaction onRifle and Bullet

c

G 8d

A 0Go-

e

FIGURE 5.15Which falls toward theother, A or B? Do the accel-erations of each relate totheir relative masses?

Chapter 5 Newton's Third Law of Motion 79

Action and Reaction on Different MassesAs strange as it may first seem, a falling object pulls upward on Earth with asmuch force as Earth pulls downward on it. The resulting acceleration of thefalling object is evident, while the upward acceleration of Earth is too small todetect. So strictly speaking, when you step off a curb, the street rises ever soslightly to meet you.

We can see that the Earth accelerates slightly in response to a falling objectby considering the exaggerated examples of two planetary bodies, a through ein Figure 5.15. The forces between bodies A and B are equal in magnitude andoppositely directed in each case. If acceleration of planet A is unnoticeable ina, then it is more noticeable in b, where the difference between the masses isless extreme. In c, where both bodies have equal mass, acceleration of objectA is as evident as it is for B. Continuing, we see that the acceleration of Abecomes even more evident in d and even more so in e.

The role of different masses is evident in a fired cannon. When a cannon isfired, there is an interaction between the cannon and the cannonball (Figure 5.16).A pair of forces acts on both cannon and cannonball. The force exerted on thecannonball is as great as the reaction force exerted on the cannon; hence, thecannon recoils. Since the forces are equal in magnitude, why doesn't the can-non recoil with the same speed as the cannonball? In analyzing changes inmotion, Newton's second law reminds us that we must also consider the massesinvolved. Suppose we let F represent both the action and reaction force, m themass of the cannonball, and m the mass of the much more massive cannon.The accelerations of the cannonball and the cannon are then found by com-paring the ratio of force to mass. The accelerations are:

Cannonball: £ = amF

Cannon: - =amThis shows why the change in velocity of the cannonball is so large com-

pared with the change in velocity of the cannon. A given force exerted on asmall mass produces a large acceleration, while the same force exerted on alarge mass produces a small acceleration.

Going back to the example of the falling object, if we used similarly exagger-ated symbols to represent the acceleration of the Earth reacting to a falling object,the symbol m for the Earth's mass would be astronomical in size. The force F, theweight of the falling object, divided by this large mass would result in a micro-scopic a to represent the acceleration of the Earth toward the falling object.

FIGURE 5.16Interactive Figure ~

The force exerted against therecoiling cannon is just asgreat as the force that drivesthe cannonball inside thebarrel. Why, then, doesthe cannonball acceleratemore than the cannon?

"

80 Part One Mechanics

FIGURE 5.17The balloon recoils from theescaping air, and it movesupward.

I

FIGURE 5.18The rocket recoils from the"molecular cannonballs" itfires, and it moves upward.

We can extend the idea of a cannon recoiling from the ball it fires to under-standing rocket propulsion. Consider an inflated balloon recoiling when air isexpelled (Figure 5.17). If the air is expelled downward, the balloon acceleratesupward. The same principle applies to a rocket, which continually "recoils"from the ejected exhaust gas. Each molecule of exhaust gas is like a tiny can-nonball shot from the rocket (Figure 5.18).

A common misconception is that a rocket is propelled by the impact ofexhaust gases against the atmosphere. In fact, before the advent of rockets, itwas generally thought that sending a rocket to the Moon was impossible. Why?Because there is no air above Earth's atmosphere for the rocket to push against.But this is like saying a cannon wouldn't recoil unless the cannonball had airto push against. Not true! Both the rocket and recoiling cannon acceleratebecause of the reaction forces exerted by the material they fire-not because ofany pushes on the air. In fact, a rocket operates better above the atmospherewhere there is no air resistance.

Using Newton's third law, we can understand how a helicopter gets its lift-ing force. The whirling blades are shaped to force air particles down (action),and the air forces the blades up (reaction). This upward reaction force is calledlift. When lift equals the weight of the aircraft, the helicopter hovers in midair.When lift is greater, the helicopter climbs upward.

This is true for birds and airplanes. Birds fly by pushing air downward. Theair in turn pushes the bird upward. When the bird is soaring, the wings mustbe shaped so that moving air particles are deflected downward. Slightly tiltedwings that deflect oncoming air downward produce lift on an airplane. Air thatis pushed downward continuously maintains lift. This supply of air is obtainedby the forward motion of the aircraft, which results from propellers or jets thatpush air backward. The air, in turn, pushes the propellers or jets forward. Wewill learn in Chapter 14 that the curved surface of a wing is an airfoil, whichenhances the lifting force.

CHECK YOURSELF

1. A car accelerates along a road. Identify the force that moves the car.2. A high-speed bus and an innocent bug have a head-on collision. The force of

impact splatters the poor bug over the windshield. Is the corresponding forcethat the bug exerts against the windshield greater, less, or the same? Is theresulting deceleration of the bus greater than, less than, or the same as that ofthe bug?

FIGURE 5.19Geese fly in a V formation because airpushed downward at the tips of their wingsswirls upward, creating an updraft that isstrongest off to the side of the bird. A trail-ing bird gets added lift by positioning itselfin this updraft, pushes air downward, andcreates another updraft for the next bird,and so on. The result is a flock flyingin aV formation.

Chapter 5 Newton's Third Law of Motion 81

PRACTICING PHYSICS

Tug of War Interactive Figure ItPerform a tug-of-war between boys and girls.Do it on a polished floor that's somewhatslippery, with boys wearing socks and girlswearing rubber-soled shoes. Who will surelywin, and why? (Hint: Who wins a tug-of-war,those that pull harder on the rope or thosewho push harder against the Aoor?)

FIGURE 5.20You cannot touch withoutbeing touched-Newton'sthird law.

We see Newton's third law at work everywhere. A fish pushes the waterbackward with its fins, and the water pushes the fish forward. When the windpushes against the branches of a tree and the branches push back on the wind,we have whistling sounds. Forces are interactions between different things.Every contact requires at least a twoness; there is no way that an object canexert a force on nothing. Forces, whether large shoves or slight nudges, alwaysoccur in pairs, each of which is opposite to the other. Thus, we cannot touchwithout being touched.

I

CHEcrK YOUR ANSWERS

1. It is the road that pushes the car along. Really! Only the road provides the hori-zontal force to move the car forward. How does it do this? The rotating tires ofthe car push back on the road (action). The road simultaneously pushes forwardon the tires (reaction). How about that!

2. The magnitudes of both forces are the same, for they constitute an action-reactionforce pair that makes up the interaction between the bus and the bug. Theaccelerations, however, are very different because the masses are different. Thebug undergoes an enormous and lethal deceleration, while the bus undergoes avery tiny deceleration-so tiny that the very slight slowing of the bus is unnoticedby its passengers. But if the bug were more massive-as massive as another bus,for example-the slowing down would unfortunately be very apparent. (Can yousee the wonder of physics here? Although so much is different for the bug andthe bus, the amount of force each encounters is the same. Amazing!)

82 Part One Mechanics

Summary of Newton's Three lawsNewton's first law, the law of inertia: An object at rest tends to remain at rest;an object in motion tends to remain in motion at constant speed along astraight-line path. This property of objects to resist change in motion is calledinertia. Mass is a measure of inertia. Objects will undergo changes in motiononly in the presence of a net force.

Newton's second law, the law of acceleration: When a net force acts on anobject, the object will accelerate. The acceleration is directly proportional to the netforce and inversely proportional to the mass. Symbolically, a = F/m. Accelerationis always in the direction of the net force. When objects fall in a vacuum, the netforce is simply the weight, and the acceleration is g (the symbol g denotes thatacceleration is due to gravity alone). When objects fall in air, the net force is equalto the weight minus the force of air resistance, and the acceleration is less than g.If and when the force of air resistance equals the weight of a falling object, accel-eration terminates, and the object falls at constant speed (called terminal speed).

Newton's third law, the law of action-reaction: Whenever one object exertsa force on a second object, the second object exerts an equal and opposite forceon the first. Forces occur in pairs, one action and the other reaction, whichtogether constitute the interaction between one object and the other. Action andreaction always occur simultaneously and act on different objects. Neither forceexists without the other.

Isaac Newton's three laws of motion are rules of nature that enable us tosee how beautifully so many things connect with one another. We see these rulesat play in our everyday environment.

Vectors Vectors

FIGURE 5.21This vector, scaled so that1 cm equals 20 N, representsa force of 60 N to the right.

--JVL) ('7 C

The valentine vectorsays, "I was only ascalar until you camealong and gave medirection."

• We have learned that any quantity that requires both magnitude and directionfor a complete description is a vector quantity. Examples of vector quantitiesinclude force, velocity, and acceleration. By contrast, a quantity that can bedescribed by magnitude only, not involving direction, is called a scalar quantity.Mass, volume, and speed are scalar quantities.

A vector quantity is nicely represented by an arrow. When the length ofthe arrow is scaled to represent the quantity's magnitude, and the directionof the arrow shows the direction of the quantity, we refer to the arrow as avector.

Adding vectors that act along parallel directions is simple enough: If theyact in the same direction, they add; if they act in opposite directions, they sub-tract. The sum of two or more vectors is called their resultant. To find theresultant of two vectors that don't act in exactly the same or opposite direc-tion, we use the parallelogram rule? Construct a parallelogram wherein the

2A parallelogram is a four-sided figure with opposite sides parallel to each other. Usually, you determine thelength of the diagonal by measurement; bnt, in the special case in which the two vectors X and Y are~ __perpendicular, you can apply the Pythagorean Theorem, R2 = X2 + y2, ro find the resultant: R = VeX2 + y2).

/ I

Chapter 5 Newton's Third Law of Motion 83

II-----------------

-) =)

FIGURE 5.23When a pair of equal-lengthvectors at right angles toeach other are added, theyform a square. The diagonalofthe square is the resultant,V2 times the length ofeither side.

FIGURE 5.24The resultant of the 30-Nand 40-N forces is 50 N.

FIGURE 5.22 Interactive Figure ~

The pair of vectors at right angles to each other make two sides of a rectangle, the diagonalof which is their resultant.

two vectors are adjacent sides-the diagonal of the parallelogram shows theresultant. In Figure 5.22, the parallelograms are rectangles.

In the special case of two vectors that are equal in magnitude and perpen-dicular to each other, the parallelogram is a square (Figure 5.23). Since for anysquare the length of a diagonal is \12, or 1.41, times one of the sides, the result-ant is \12 times one of the vectors. For example, the resultant of two equalvectors of magnitude 100 acting at a right angle to each other is 141.

Force VectorsFigure 5.24 shows a pair of forces acting on a box. One is 30 newtons and theother is 40 newtons. Simple measurement shows the resultant of this pair offorces is 50 newtons.

Figure 5.25 shows Nellie Newton hanging at rest from a clothesline. Notethat the clothesline acts like a pair of ropes that make different angles with thevertical. Which side has the greater tension? Investigation will show there arethree forces acting on Nellie: her weight, a tension in the left-hand side of therope, and a tension in the right-hand side of the rope. Because of the differentangles, different rope tensions will occur in each side. Figure 5.25 shows a step-by-step solution. Because Nellie hangs in equilibrium, her weight must be sup-ported by two rope tensions, which must add vectorially to be equal and oppo-site to her weight. The parallelogram rule shows that the tension in theright-hand rope is greater than the tension in the left-hand rope. If you meas-ure the vectors, you'll see that tension in the right rope is about twice the ten-sion in the left rope. Both rope tensions combine to support her weight.

More about force vectors can be found in Appendix D at the end of thisbook and on pages 23-30 in the Practicing Physics book.

Velocity VectorsRecall, from Chapter 3, the difference between speed and velocity-speed is ameasure of "how fast"; velocity is a measure of both how fast and "in whichdirection." If the speedometer in a car reads 100 kilo meters per hour, you

84 Part One Mechanics

FIGURE 5.25Nellie Newton hangs motion-less by one hand from aclothesline. Ifthe line is onthe verge of breaking, whichside is most likely to break?

Vector Representation: Howto Add and Subtract

VectorsGeometric Addition

of Vectors

...JVL) C') C

The pair of 6-unit and8-unit vectors at rightangles to each othersay, "We may be a sixand an eight, buttogether we're a per-fect ten."

FIGURE 5.26 Interactive Figure ~

(a) Nellie's weight is shown by the downward vertical vector. An equal and opposite vectoris needed for equilibrium, shown by the dashed vector. (b) This dashed vector is thediagonal of a parallelogram defined by the dotted lines. (c) Both rope tensions are shownby the constructed vectors. Tension is greater in the right rope, the one most likely to break.

80 km!nFIGURE 5.27The 60-km/h crosswind blowsthe 80-km/h aircraft off course at100 km/h.

lOO km/nIIIII

I

Resultant

(Scale: 1Cm' 20 km/h) 60 km/n

know your speed. If there is also a compass on the dashboard, indicating thatthe car is moving due north, for example, you know your velocity-lOO kilo-meters per hour north. To know your velocity is to know your speed and yourdirection.

Consider an airplane flying due north at 80 kilometers per hour relative tothe surrounding air. Suppose that the plane is caught in a 60-kilometer-per-hourcrosswind (wind blowing at right angles to the direction of the airplane) thatblows it off its intended course. This example is represented with vectors inFigure 5.27 with velocity vectors scaled so that 1 centimeter represents 20 kilo-meters per hour. Thus, the 80-kilometer-per-hour velocity of the airplane isshown by the -l-centimeter vector and the 60-kilometer-per-hour crosswind isshown by the 3-centimeter vector. The diagonal of the constructed parallelo-gram (a rectangle, in this case) measures 5 cm, which represents 100 km/h. Sothe airplane moves at 100 km/h relative to the ground, in a direction betweennorth and northeast.

Chapter 5 Newton's Third Law of Motion 85

PRACTICING PHYSICS

Here we see a top view of an airplanebeing blown off course by wind invarious directions. Using a pencil andthe parallelogram rule, sketch thevectors that show the resulting veloc-ities for each case. In which casedoes the airplane travel fastest acrossthe ground? Slowest?

CHECK YOURSELF

Consider a motorboat that normally travels 10 krn/h in still water. If the boat headsdirectly across the river, which also flows at a rate of 10 km/h, what will be its veloc-ity relative to the shore?

PRACTICING PHYSICS

Here we see top views ofthree motorboats crossing ariver. All have the same speed relative to the water, and allexperience the same water flow. Construct resultantvectors showing the speed and direction of the boats.Then answer these questions:

(a) Which boat takes the shortest path to the oppositeshore?

(b) Which boat reaches the opposite shore first?

(c) Which boat provides the fastest ride?

¥:'):)I - ,!

,a4 ., I

( . I ,',i \ \'l! I )

bl~.; !

I !" I It J \ \

CI~)

f

!I

Components of VectorsJust as two vectors at right angles can be combined into one resultant vector,any vector can be resolved into two component vectors perpendicular to eachother. These two vectors are known as the components of the given vector theyreplace. The process of determining the components of a vector is called

CHECK YOUR ANSWER

When the boat heads cross-stream (at right angles to the river flow), its velocity is14.1 km/h, 45 degrees downstream (in accord with the diagram in Figure 5.23).

86 Part One Mechanics

FIGURE 5.28The horizontal and verticalcomponents of a ball'svelocity.

FIGURE 5.29Construction of the verticaland horizontal componentsofa vector.

Velocity of stone.(

Verticalcomponent ofstone's velocity'-

...••.

"-'\

\

\

,,----- <,

.::11' / '"Horizontal component~J of stone's velocity

~-----VI

III

III

VI V~----------I

I : XI

I

I

II____ ...J __

II

y

resolution. Any vector drawn on a piece of paper can be resolved into a verti-cal and a horizontal component.

Vector resolution is illustrated in Figure 5.29. A vector V is drawn in theproper direction to represent a vector quantity. Then vertical and horizontallines (axes) are drawn at the tail of the vector. Next, a rectangle is drawn thathas V as its diagonal. The sides of this rectangle are the desired components,vectors X and Y. In reverse, note that the vector sum of vectors X and Y is V.

We'll return to vector components when we treat projectile motion inChapter 10.

EXERCISE

With a ruler, draw the horizontal and vertical components of the two vectors shown.Measure the components and compare your findings with the answers given at thebottom of the page.

ANSWERS

Left vector: The horizontal component is 2 cm; the vertical component is 2.6 cm. Rightvector: The horizontal component is 3.8 cm; the vertical component is 2.6 cm.

Chapter 5 Newton's Third Law of Motion 87

Summary of TermsNewton's third law Whenever one object exerts a force on a

second object, the second object exerts an equal and op-posite force on the first.

Vector quantity A quantity that has both magnitude anddirection. Examples are force, velocity, and acceleration.

Scalar quantity A quantity that has magnitude but notdirection. Examples are mass, volume, and speed.

Vector An arrow drawn to scale used to represent a vectorquantity.

Resultant The net result ofa combination of two or morevectors.

Review QuestionsForces and Interactions

1. When you push against a wall with your fingers, theybend because they experience a force. Identity thisforce.

2. A boxer can hit a heavy bag with great force. Whycan't he hit a piece of tissue paper in midair with thesame amount offorce?

3. How many forces are required for an interaction?

Newton's Third Law of Motion4. State Newton's third law of motion.

S. Consider hitting a baseball with a bat. If we call theforce on the bat against the ball the action force,identity the reaction force.

6. Consider the apple and the orange (Figure 5.9). Ifthesystem is considered to be only the orange, is there anet force on the system when the apple pulls?

7. If the system is considered to be the apple and theorange together (Figure 5.10), is there a net force onthe system when the apple pulls (ignoring frictionwith the floor)?

8. To produce a net force on a system, must there be anexternally applied net force?

9. Consider the system of a single football. If you kick it,is there a net force to accelerate the system? If a friendkicks it at the same time with an equal and oppositeforce, is there a net force to accelerate the system?

Action and Reaction on Different Masses10. The Earth pulls down on you with a gravitational

force that you call your weight. Do you pull up onthe Earth with the same amount of force?

11. If the forces that act on a cannonball and the recoil-ing cannon from which it is fired are equal in magni-

tude, why do the cannonball and cannon have verydifferent accelerations?

12. Identity the force that propels a rocket.

13. How does a helicopter get its lifting force?

14. Can you physically touch a person without thatperson touching you with the same amount of force?

Summary of Newton's Three Laws1 S. Fill in the blanks: Newton's first law is often called

the law of ; Newton's second law is the lawof ; and Newton's third law is the law of____ and _

16. Which ofthe three laws defines the concept offorceinteraction?

Vectors17. Cite three examples of a vector quantity and three ex-

amples of a scalar quantity.

18. Why is speed considered a scalar and velocity a vector?

19. According to the parallelogram rule, what quantity isrepresented by the diagonal of a constructed paral-lelogram?

20. Consider Nellie hanging at rest in Figure 5.25. If theropes were vertical, with no angle involved, whatwould be the tension in each rope?

21. When Nellie's ropes make an angle, what quantitymust be equal and opposite to her weight?

22. When a pair of vectors are at right angles, is theresultant always greater than either of the vectorsseparately?

ProjectHold your hand like a flat wing outside the window of amoving automobile. Then slightly tilt the front edge up-ward and notice the lifting effect. Can you see Newton'slaws at work here?

One-Step Calculations1. Calculate the resultant of the pair of velocities

100 krn/h north and 75 krn/h south. Calculate theresultant if both of the velocities are directed north.

Resultant of Two Vectors at Right Angles

to Each Other; R = V(X2 + r)

2. Calculate the magnitude of the resultant ofa pair of1Ou-krn/h velocity vectors that are at right angles toeach other.

88 Part One Mechanics

3. Calculate the resultant of a horizontal vector with amagnitude of 4 units and a vertical vector with amagnitude of3 units.

4. Calculate the resultant velocity of an airplane thatnormally flies at 200 krn/h ifit encounters a SO-km/hwind from the side (at a right angle to the airplane).

Exercises1. Reconcile the fact that friction acts in a direction to

oppose motion even though you rely on friction topropel you forward when walking.

2. The photo shows Steve Hewitt and daughterGretchen. Is Gretchen touching her dad, or is dadtouching her? Explain.

3. When you rub your hands together, can you pushharder on one hand than the other?

4. For each of the following interactions, identity actionand reaction forces. (a) A hammer hits a nail.(b) Earth gravity pulls down on a book. (c) Ahelicopter blade pushes air downward.

5. You hold an apple over your head. (a) Identity all theforces acting on the apple and their reaction forces.(b) When you drop the apple, identity all the forcesacting on it as it falls and the corresponding reactionforces. Neglect air drag.

6. Identity the action-reaction pairs of forces for the fol-lowing situations: (a) You step off a curb. (b) You patyour tutor on the back. (c) A wave hits a rocky shore.

7. Consider a baseball player batting a ball. (a) Identitythe action-reaction pairs when the ball is being hit,and (b) while the ball is in flight.

8. What physics is involved for a passenger feelingpushed backward into the seat of an airplane when itaccelerates along the runway in takeoff?

9. If you drop a rubber ball on the floor, it bouncesback up. What force acts on the ball to provide thebounce?

10. When you kick a football, what action and reactionforces are involved? Which force, ifany, is greater?

11. Is it true that, when you drop from a branch to theground below, you pull upward on Earth? lf so, thenwhy is the acceleration of Earth not noticed?

12. Within a book on a table, there are billions of forcespushing and pulling on all the molecules. Why is itthat these forces never by chance add up to a netforce in one direction, causing the book to accelerate"spontaneously" across the table?

13. Two 1OO-N weights are attached to a spring scale asshown. Does the scale read 0, 100, or 200 N, or doesit give some other reading? (Hint: Would it read anydifferently if one of the ropes were tied to the wall in-stead of to the hanging 1OO-N weight?)

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14. When the athlete holds the barbelloverhead, the reaction force is theweight of the barbell on his hand.How does this force vary for thecase in which the barbell is acceler-ated upward? Downward?

15. Consider the two forces acting onthe person who stands still-namely, the downward pull of gravityand the upward support ofthe floor.Are these forces equal and opposite?Do they form an action-reaction pair?Why or why not?

16. Why can you exert greater force on the pedals of abicycle if you pull up on the handlebars?

17. Does a baseball bat slow down when it hits a ball?Defend your answer.

18. Why does a rope climber pull downward on the ropeto move upward?

19. You push a heavy car by hand. The car, in turn,pushes back with an opposite but equal force on you.Doesn't this mean that the forces cancel one another,making acceleration impossible? Why or why not?

20. A farmer urges his horse to pull a wagon. The horserefuses, saying that to try would be futile, for it wouldflout Newton's third law. The horse concludes thatshe can't exert a greater force on the wagon than thewagon exerts on her, and, therefore, that she won'tbe able to accelerate the wagon. What is your expla-nation to convince the horse to pull?

21. The strong man will push the two initially stationaryfreight cars of equal mass apart before he himselfdrops straight to the ground. Is it possible for him togive either of the cars a greater speed than the other?Why or why not?

22. Suppose that two carts, one twice as massive as theother, Ay apart when the compressed spring thatjoins them is released. How fast does the heavier cartroll compared with the lighter cart?

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•• ••23. If you exert a horizontal force of 200 N to slide a

crate across a factory floor at constant velocity, howmuch friction is exerted by the floor on the crate? Isthe force offriction equal and oppositely directed toyour 200-N push? If the force offriction isn't thereaction force to your push, what is?

24. If a Mack truck and Honda Civic have a head-on col-lision, upon which vehicle is the impact force greater?Which vehicle experiences the greater deceleration?Explain your answers.

25. Ken and Joanne are astronauts Aoating some dis-tance apart in space. They are joined by a safety cordwhose ends are tied around their waists. If Ken startspulling on the cord, will he pull Joanne toward him,or will he pull himself toward Joanne, or will both as-tronauts move? Explain.

26. Which team wins in a tug-of-war-the team that pullsharder on the rope, or the team that pushes harderagainst the ground? Explain.

27. In a tug-of-war between Sam and Maddy, eachpulls on the rope with a force of250 N. What is thetension in the rope? If both remain motionless, whathorizontal force does each exert against the ground?

28. Consider a tug-of-war on a smooth floor betweenboys wearing socks and girls wearing rubber-soledshoes. Why do the girls win?

29. Two people of equal mass attempt a tug-of-war witha 12-m rope while standing on frictionless ice. When

Chapter 5 Newton's Third Law of Motion 89

they pull on the rope, each of them slides toward theother. How do their accelerations compare, and howfar does each person slide before they meet?

30. What aspect of physics was not known by the writerof this newspaper editorial that ridiculed early experi-ments by Robert H. Goddard on rocket propulsionabove the Earth's atmosphere? "Professor Goddard... does not know the relation of action to reaction,and of the need to have something better than avacuum against which to react ... seems to lackthe knowledge ladled out daily in high schools."

31. Which ofthe following are scalar quantities, which arevector quantities, and which are neither? (a) velocity;(b) age; (c) speed; (d) acceleration; (e) temperature.

32. What can you correctly say about two vectors thatadd together to equal zero?

33. Can a pair of vectors with unequal magnitudes everadd to zero? Can three unequal vectors add to zero?Defend your answers.

34. When can a nonzero vector have a zero horizontalcomponent?

35. When, if ever, can a vector quantity be added to ascalar quantity?

36. Which is more likely to break-a hammock stretchedtightly between a pair of trees or one that sags morewhen you sit on it?

37. A heavy bird sits on a clothesline. Will the tension inthe clothesline be greater if the line sags a lot or ifitsags a little?

38. The rope supports a lantern that weighs 50 N. Isthe tension in the rope less than, equal to, or morethan 50 N? Use the parallelogram rule to defendyour answer.

39. The rope is repositioned as shown, and still supportsthe 50-N lantern. Is the tension in the rope less than,equal to, or more than 50 N? Use the parallelogramrule to defend your answer.

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90 Part One Mechanics

40. Why does vertically falling rain make slanted streakson the side windows of a moving automobile? If thestreaks make an angle of 450

, what does this tellyou about the relative speed of the car and thefalling rain?

41. A balloon floats motionless in the air. A balloonistbegins climbing the supporting cable. In which direc-tion does the balloon move as the balloonist climbs?Defend your answer.

42. Consider a stone at rest on the ground. There aretwo interactions that involve the stone. One isbetween the stone and the Earth: Earth pulls downon the stone (its weight) and the stone pulls up onthe Earth. What is the other interaction?

43. A stone is shown at reston the ground. (a) Thevector shows the weightof the stone. Completethe vector diagram show-ing another vector thatresults in zero net force on the stone. (b) What is theconventional name of the vector you have drawn?

44. Here a stone is suspended atrest by a string. (a) Draw forcevectors for all the forces thatact on the stone. (b) Shouldyour vectors have a zero resul-tant? (c) Why, or why not?

45. Here the same stone is beingaccelerated vertically upward. (a) Drawforce vectors to some suitable scaleshowing relative forces acting on thestone. (b) Which is the longer vector,and why?

46. Suppose the string in the precedingexercise breaks and the stone slows inits upward motion. Draw a force vectordiagram of the stone when it reaches the top of itspath.

47. What is the acceleration of the stone of Exercise 46at the top of its path?

48. Here the stone is sliding down a friction-free incline.(a) ldentify the forces that act on it, and drawappropriate force vectors. (b) By the parallelogramrule, construct theresultant force on thestone (carefullyshowing that it has adirection parallel tothe incline-the samedirection as the stone'sacceleration ).

49. Here the stone is atrest, interacting withboth the surface ofthe incline and theblock. (a) ldentify allthe forces that act onthe stone, and drawappropriate force vectors. (b) Show that the netforce on the stone is zero. (Hint 1: There are two nor-mal forces on the stone. Hint 2: Be sure the vectorsyou draw are for forces that act on the stone, not bythe stone on the surfaces.)

SO. In drawing a diagram offorces that act on a sprinter,which of these should not be drawn: Weight, mg;force the sprinter exerts on the ground; tension in thesprinter's lower legs?

Problems1. A boxer punches a sheet of paper in midair and

brings it from rest up to a speed of25 m/s in 0.05 s.If the mass of the paper is 0.003 kg, what force doesthe boxer exert on it?

2. If you stand next to a wall on a frictionless skate-board and push the wall with a force of 40 N, howhard does the wall push on you? If your mass is 80 kg,what's your acceleration?

3. Ifraindrops fall vertically at a speed of3 m/s and youare running at 4 rn/s, how fast do they hit your face?

4. Forces of3.0 Nand 4.0 N act at right angles on ablock of mass 2.0 kg. How much acceleration occurs?

5. Consider an airplane that normally has an airspeedof 100 km/h in a 1Ofl-krn/h crosswind blowing fromwest to east. Calculate its ground velocity when itsnose is pointed north in the crosswind.

6. A canoe is paddled at 4 krn/h directly across a riverthat flows at 3 krn/h, as shown in the figure. (a) Whatis the resultant speed of the canoe relative to theshore? (b) In approximately what direction shouldthe canoe be paddled to reach a destination directlyacross the river?

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