Motion, Forces, and Newton s Laws

Transcription

1 Chaper 2 Moion, Forces, and Newon s Laws OUTLINE 2.1 ARISTOTLE S MECHANICS 2.2 WHAT IS MOTION? 2.3 THE PRINCIPLE OF INERTIA 2.4 NEWTON S LAWS OF MOTION 2.5 WHY DID IT TAKE NEWTON TO DISCOVER NEWTON S LAWS? 2.6 THINKING ABOUT THE LAWS OF NATURE The laws of mechanics describe he moion of objecs such as his snowboarder. ( Comsock Images/Jupierimages) We begin our sudy of physics wih he field known as mechanics. This area of physics is concerned wih he moion of objecs such as rocks, balloons, cars, waer, planes, and snowboarders. To undersand mechanics, we mus be able o answer wo quesions. Firs, wha causes moion? Second, gien a paricular siuaion, how will an objec moe? The laws of physics ha deal wih he moion of erresrial objecs were deeloped oer he course of many cenuries, culminaing wih he work of Isaac Newon and he formulaion of wha are known as Newon s laws of moion. Newon s laws are a cornersone of physics and are he basis for nearly eeryhing we do in he firs par of his book. Een so, i is useful o begin wih a brief discussion of some of he ideas ha dominaed physics before Newon s ime. Those ideas originaed many cenuries ago wih he Greek philosopher scienis Arisole. 1 While Arisole was a leading scienis of his era and was cerainly a ery deep hinker, we now know ha many of his ideas abou how objecs moe were no 1 Arisole, who lied during he fourh cenury BC, was also well-known as a eacher. One of his mos famous sudens was he man we now know as Aleander he Grea _2_c2_p26-54.indd 26 4/22/8 5:28:22 PM

2 correc. Why, hen, should we now be concerned wih such hisorical (and incorrec) noions abou moion? We eamine Arisole s ideas because many sudens begin his course wih some of he same incorrec noions. A good way o reach a horough undersanding of he physics of moion is o consider he origins of Arisole s ideas and idenify where hey go wrong. 2.1 ARISTOTLE S MECHANICS Arisole began by idenifying wo general ypes of moion: (1) celesial moion, he moion of hings like he planes, he Moon, and he sars; and (2) erresrial moion, he moion of eeryday objecs such as rocks and arrows. He belieed ha celesial moion is fundamenally differen from he moion of rocks and arrows. We all know ha while rocks and oher erresrial objecs can moe in a ariey of ways, hey usually come o res eenually. On he oher hand, i appears ha celesial objecs neer come o a sop. According o Arisole, erresrial objecs moe only when somehing acs on hem direcly. A rock moes because some oher objec, such as a person s hand, acs on i o make i moe. In conras, here does no seem (a leas o Arisole) o be anyhing acing on he Moon o cause is moion. Arisole hus raised a key poin: he moions of celesial and erresrial objecs look ery differen. A heory of mechanics mus eplain why. Le s firs focus on he moion of erresrial objecs. Arisole assered ha he naural sae for a erresrial objec is a sae of res and ha his is why erresrial objecs moe only when aced on by anoher objec. In modern erminology, Arisole claimed ha an objec only moes when aced on by a force. A force is simply a push or a pull, and Arisole belieed ha a push or a pull on one objec is always produced by a second objec. Furhermore, he claimed ha a force could only eis if he wo objecs are in direc conac. Hence, Arisole belieed ha (1) moion is caused by forces, and (2) forces are produced by conac wih oher objecs. These ideas seem quie reasonable when we consider somehing like a refrigeraor being pushed along a leel floor as in Figure 2.1. Our eeryday eperience is ha he refrigeraor only moes while someone is pushing on i. If he person sops pushing, he refrigeraor quickly comes o a sop. The person pushing on he refrigeraor eers a force direcly on he refrigeraor, and he person is in conac wih he refrigeraor. So far, so good. A key aspec of mechanics is he noion of force. We hae already noed ha a force is simply a push or a pull on an objec. A force has boh a magniude and a direcion, so force is a ecor quaniy, ofen denoed by F S. The magniude of a force is he srengh of he push or he pull, while he direcion of his ecor is he direcion of he push or he pull. We ne need o consider how o describe moion. One quaniy we normally associae wih moion is elociy. Velociy is also a ecor quaniy and is usually represened by he symbol S. A careful mahemaical definiion of elociy is gien in he ne secion. For our discussion of Arisole s ideas, i is enough o know ha he magniude of S is he disance ha an objec raels (as migh be measured in meers) per second. The direcion of he ecor S is he direcion of moion (Fig. 2.2). Arisole proposed ha force and elociy are direcly conneced. In he mahemaical language of oday he would hae wrien Figure 2.1 This person is eering a force, denoed by F S, on he refrigeraor. Force is a ecor quaniy. I has a magniude and a direcion. F S 5 FS R (INCORRECT!) (2.1) In words, his relaionship says ha he elociy S of an objec is proporional o he force F S ha acs on i, wih he consan of proporionaliy being relaed o a quaniy R, which is he resisance o moion. We ll refer o his relaionship as Arisole s law of moion. We also cauion ha his relaionship, Equaion 2.1, 2.1 ARISTOTLE'S MECHANICS _2_c2_p26-54.indd 27 4/22/8 5:28:26 PM

3 S plane S rain Feli Clouzo/Phoographer's Choice/Gey Images AP Phoo/Daid J. Phillip Figure 2.2 The elociies of an airplane and a rain are ecors, haing boh magniude and direcion. Figure 2.3 Afer i leaes he picher s hand, he only forces acing on he baseball are a force from graiy and a force from he air hrough which i moes. This second force is called air drag. is incorrec. I is no a correc law of physics. I does, howeer, seem o eplain he moion of he refrigeraor in Figure 2.1. If no force is eered on he refrigeraor, hen S F 5. According o Equaion 2.1, he elociy S is hen also zero and hence he refrigeraor does no moe. When a person pushes on he refrigeraor, S F is nonzero and Arisole s heory (Eq. 2.1) predics ha S is herefore also nonzero. Since S and S F are boh ecors, Equaion 2.1 assers ha he elociy and he force are always in he same direcion. Figure 2.4 An archer shoos an arrow sraigh upward. While he arrow is raeling on he upward par of is rajecory, he forces acing on i from graiy and from air drag are boh direced downward. Hence, hese forces are no parallel o he elociy of he arrow while i is raeling upward. The Failures of Arisole s Ideas abou Mechanics Arisole s law of moion Equaion 2.1 seems o eplain he moion of he refrigeraor in Figure 2.1, bu i does no work as well in oher siuaions. Consider he moion of a hrown baseball (Fig. 2.3). We know ha afer a baseball leaes he hrower s hand i coninues o moe unil i his he ground, is sruck by a ba, or is caugh by someone. This ery familiar moion is no consisen wih Equaion 2.1. Recall Arisole s asserion ha forces are always caused by direc conac wih anoher objec. While he ball is raeling hrough he air, here does no appear o be anoher objec in conac wih i, so according o his reasoning he force on he baseball is zero afer i leaes he hrower. If he force is zero, Equaion 2.1 says ha he ball s elociy should also be zero. Howeer, we know from eperience ha a hrown baseball coninues o moe! One way o aemp an escape from his dilemma is o recognize ha he ball also eperiences a force due o graiy. Unforunaely (a leas for Arisole), he graiaional force does no fi ino Arisole s ideas abou forces because here is nohing in direc conac wih he baseball o produce his force. Een if we oerlook his difficuly, we sill hae o consider ha he graiaional force on he ball is direced downward (oward he surface of he Earh) and hence he graiaional force and he elociy are no in he same direcion for he ball in Figure 2.3. Figures 2.3 and 2.4 boh illusrae siuaions ha conradic Equaion 2.1, which predics ha force and elociy are always parallel. I is paricularly puzzling (a leas for Arisole) ha objecs such as baseballs and arrows can be raeling upward when he forces acing on hem are direced downward. More problems wih Arisole s law of moion can be seen when we consider falling objecs. If someone simply drops a baseball, i will fall o he ground (or whaeer is below). There is nohing in direc conac wih he ball o cause i o fall. Wha, hen, makes i moe from he person s hand? Because he case of falling objecs is quie common, Arisole gae i special aenion. He proposed ha in his case here is a special force ha does no require conac wih anyhing. This force is he weigh of he objec, which Arisole belieed was equialen o wha we now call mass. 2 Arisole also hough ha he resisance facor R in Equaion 2.1 is a 2 We ll see in he ne chaper ha mass and weigh are no he same. Recall ha mass is a fundamenal physical quaniy, as eplained in Chaper 1. Weigh is defined in Chaper CHAPTER 2 MOTION, FORCES, AND NEWTON'S LAWS 24716_2_c2_p26-54.indd 28 4/22/8 5:28:27 PM

4 propery of he subsance hrough which he objec moes. For an objec dropped near he Earh s surface, his subsance is air. This reasoning leads one o epec ha if Equaion 2.1 is correc, a heay objec will fall faser han a ligh one when boh are dropped in he same medium. Thanks o eperimens performed by Galileo (Fig. 2.5), we know ha his is no he case. Galileo showed ha ligh objecs fall a he same rae as heay objecs. Despie all hese difficulies, Arisole s ideas hae a cerain appeal. They are based on he noion ha an objec s elociy is direcly proporional o he force acing on he objec, which a firs glance seems plausible. Howeer, we hae seen a number of ery simple siuaions in which his noion fails badly. In he ne few secions, we discuss he connecion beween force and moion in more deail and arrie a he laws of moion discoered by Isaac Newon. We ll hen see how Newon s laws oercome he difficulies ha frusraed Arisole. Alhough we mus rejec Arisole s heory of moion, i is sill ery insrucie o undersand where his heory goes wrong and why. Some of he eeryday ideas ha you hae brough o his course may be similar o hose of Arisole. Undersanding precisely when and why hose ideas fail will help you undersand and appreciae he correc way o describe moion. CONCEPT CHECK 2.1 Force and Moion Do he following eamples of moion appear o be consisen or inconsisen wih Arisole s law of moion? (a) A hockey puck sliding along an icy horizonal surface (b) A car coasing along a leel road (c) A piano ha is pushed across a room 2.2 WHAT IS MOTION? We hae used he erm moion quie a bi. I is now ime o consider how moion is described and measured in a precise, mahemaical sense. The concep of moion is more complicaed han you migh firs guess, and we ll need seeral quaniies posiion, elociy (which we hae already encounered), and acceleraion o describe i fully. Our ne job is o gie careful definiions for hese quaniies. Le s firs consider moion along a sraigh line, called one-dimensional moion. A hockey puck sliding on a horizonal icy surface is a good eample of his ype of moion, and he op porion of Figure 2.6A shows wha would happen if we ook a muliple-eposure phoo of such a hockey puck. These eposures are capured a eenly spaced ime inerals, and we can use hem o consruc he graph of he puck s posiion as a funcion of ime shown in Figure 2.6B. Here we measure posiion as he disance from he origin on he ais o he cener of he hockey puck. For one-dimensional moion, he alue of his disance, which we denoe by, compleely specifies he posiion of he objec. Figure 2.6A is someimes called a moion diagram, a muliple-eposure phoograph or similar skech ha shows he locaion of an objec a regularly spaced insans in ime. Noice in Figure 2.6B ha he ais is now erical as we plo he posiion () as a funcion of ime (). In such a posiion ime plo, i is conenional o plo ime along he horizonal ais. Velociy and Speed The disance beween adjacen dos on he ais in Figure 2.6A shows how far he puck has moed during each ime ineral. We hae already menioned ha elociy S is a ecor quaniy. The magniude of S is called he speed, he disance raeled per uni of ime, while he direcion of he elociy ecor gies he direcion of he moion (in his eample, S is direced o he righ). Speed is a scalar quaniy; i does no hae a direcion. In SI unis, posiion is measured in meers and ime is Figure 2.5 Galileo is repored o hae used he Leaning Tower of Pisa in Ialy in his sudies of falling objecs. Alhough he sory may no be sricly accurae, Galileo cerainly did conduc eperimens showing ha ligh and heay objecs fall a he same rae. A B C S Each do corresponds o an image of he puck in A. Figure 2.6 A Muliple images of a hockey puck raeling across an icy surface. B Plo of he posiion of he puck as a funcion of ime. The dos correspond o he images of he puck in A. C Velociy of he puck as a funcion of ime. 2.2 WHAT IS MOTION? _2_c2_p26-54.indd 29 4/22/8 5:28:29 PM

5 Insigh 2.1 SPEED AND VELOCITY Speed is equal o he magniude of he elociy. For moion in wo or hree dimensions, elociy is a ecor. In hese cases, speed is he magniude of he elociy ecor S. measured in seconds, so elociy and speed are boh measured in meers per second, or simply m/s. Speed and elociy are relaed quaniies, bu hey are no he same. Speed ells how fas an objec is moing; here i is he disance raeled per second, which is always a posiie quaniy (or perhaps zero). The elociy conains his informaion and in addiion ells he direcion of moion. For he hockey puck in Figure 2.6A, he direcion may be posiie (moion o he righ, oward larger or more posiie alues of ) or negaie (moion o he lef, oward smaller or more negaie alues of ). If we recall ha elociy is a ecor, we should recognize ha for onedimensional moion he direcion of he elociy ecor mus lie parallel o he ais. Thus, in his paricular eample and in oher cases inoling one-dimensional moion, we need only deal wih he componen of he elociy parallel o. This componen can be eiher posiie, negaie, or zero. In Chaper 1, we saw ha ecor quaniies are usually wrien wih arrows oerhead, so he elociy ecor is generally wrien as S. For one-dimensional siuaions, he elociy ecors hae only one componen, and we can simplify he noaion and refer o his componen as simply, wihou an arrow. The sign of (eiher posiie or negaie) hen gies he direcion of he elociy. Thus, for moion in one dimension, speed 5 1 one dimension 2 (2.2) In words, his epression says ha speed is equal o he magniude of he elociy. For wo- or hree-dimensional moion, speed 5 S 1wo or hree dimensions2 (2.3) where here he erical bars again indicae ha he speed is equal o he magniude of he elociy ecor. In eiher case (Eq. 2.2 or 2.3), speed is always a posiie quaniy or zero, bu neer negaie. The hockey puck in Figure 2.6A is sliding a a consan speed, so is elociy also has a consan alue as shown in he elociy ime ( ) graph in Figure 2.6C. In his case, he elociy is posiie, which means ha he direcion of moion is oward increasing alues of (i.e., o he righ). 2 1 The aerage elociy is he slope of his line ae. D D 1 2 D ae 2 1 D 2 1 Figure 2.7 Hypoheical plo of an objec s posiion as a funcion of ime. The aerage elociy during he ime ineral from 1 o 2 is he slope of he line connecing he wo corresponding poins on he cure. How Is an Objec s Velociy Relaed o Is Posiion? Velociy is he change in posiion per uni ime. Since ime is measured in seconds, i is naural o hink abou he posiion a 1-s ime inerals as suggesed by he dos in Figure 2.6B. Alernaiely, we could consider a paricular ime ineral ha begins a ime 1 and ends a ime 2 so ha he size of he ime ineral is 2 1. The change in posiion during a paricular ime ineral is called he displacemen. For one-dimensional moion, displacemen is denoed by 2 1, where 1 is he iniial posiion, he posiion a he beginning of he ime ineral ( 1 ), and 2 is he final posiion, he posiion a he end of he ineral ( 2 ). The aerage elociy during his ime ineral is ae 5 final 2 iniial final 2 iniial ae 5 D (2.4) D When an objec moes wih a consan speed, he aerage elociy is consan hroughou he moion and he posiion ime graph has a consan slope as in Figure 2.6B. For more general cases, he aerage elociy is he slope of he line segmen ha connecs he posiions a he beginning and end of he ime ineral. This siuaion is illusraed in he hypoheical graph in Figure 2.7. Anoher eample of moion along a line is he case of a rocke-powered car raeling on a fla road. Le s assume he car is iniially a res; here, iniially means ha he car is no moing when our clock reads zero. A, he drier urns on he rocke engine and he car begins o moe in a sraigh-line pah, along 3 CHAPTER 2 MOTION, FORCES, AND NEWTON'S LAWS 24716_2_c2_p26-54.indd 3 4/22/8 5:28:3 PM

6 a horizonal ais ha we denoe as. Figure 2.8A is a moion diagram showing he posiion of he car a eenly spaced insans in ime. The ais is along he road, and he posiion of he car a a paricular insan is he disance from he origin o he cener of he car. The corresponding posiion ime graph for he car is shown in Figure 2.8B, where we hae again used dos o mark he car s posiion a eenly spaced ime inerals. In his case, he dos are no equally spaced along he ais. Insead, heir spacing increases as he car raels. This means ha he car moes a greaer disance during each successie (and equal) ime ineral and hence he speed of he car increases wih ime. In his eample, he car moes oward increasing alues of, so he elociy is again posiie and increases smoohly wih ime as shown in Figure 2.8C. The precise shape of he cure will depend on he way he engine fires; we ll learn how o deal wih ha problem in Chaper 3. A B Aerage Velociy and Insananeous Velociy Figure 2.9 shows he posiion as a funcion of ime for a hypoheical objec moing in a sraigh line. We hae again used dos o mark he posiion a he beginning and end of a paricular ime ineral ha sars a 1. s and ends a 2. s. According o Equaion 2.4, he aerage elociy during his ime ineral is jus he displacemen during he ineral diided by he lengh of he ineral. Figure 2.9A shows ha his aerage elociy is he aerage slope of he posiion ime cure (i.e., he slope of he line connecing he sar o he end of he enire ineral). Wih his approach, hough, we lose all deails abou wha happens in he middle of he ineral. In Figure 2.9A, he slope of he cure aries considerably as we moe hrough he ineral from 1. s o 2. s. If we wan o ge a more accurae descripion of he objec s moion a a paricular insan wihin his ime ineral, say a 1.5 s, i is beer o use a smaller ineral. How small an ineral should we use? In Figure 2.9B, we consider slopes oer a succession of smaller ime inerals. Inuiiely, we epec ha using a smaller ineral will gie a beer measure of he moion a a paricular poin in ime. From Figure 2.9B, we see ha as we ake eer smaller ime inerals we are acually calculaing he slope of he posiion ime cure a he poin of ineres (here a 1.5 s). This slope of he posiion ime cure is called he insananeous elociy. For one-dimensional moion, he insananeous elociy is he slope of he posiion ime ( ) cure and is gien by 3 D 5 lim D S D For he eample shown in Figure 2.9, is no consan. Raher, i aries wih ime oer he ineral from 1. s o 2. s. (2.5) C Figure 2.8 A Mulieposure skech of a rocke-propelled car raeling along a horizonal road. B Posiion as a funcion of ime for he rocke-powered car. C Velociy of he car as a funcion of ime. Definiion of insananeous elociy (m) A aerage Slope elociy ae (s) (m) B This slope is he insananeous elociy a 1.5 s. D This slope is he aerage elociy from 1. s o 2. s (s) Figure 2.9 A The aerage elociy during a paricular ime ineral is he slope of he line connecing he sar of he ineral o he end of he ineral. B The insananeous elociy a a paricular ime is he slope of he cure a ha ime. The insananeous elociy in he middle of a ime ineral is no necessarily equal o he aerage elociy during he ineral. The insananeous elociy is defined as he limi of he slope oer a ime ineral as. 3 Here he erm lim (limi) means o ake he raio / as he quaniy approaches zero. 2.2 WHAT IS MOTION? _2_c2_p26-54.indd 31 4/22/8 5:28:31 PM

7 The difference beween he aerage and insananeous alues can be undersood in analogy wih a car s speedomeer. The speedomeer reading gies your insananeous speed, he magniude of your insananeous elociy a a paricular momen in ime. If you are aking a long drie, your aerage speed will generally be differen because he aerage alue will include periods a which you are sopped in raffic, passing oher cars, and so forh. In many cases, such as in discussions wih a police officer, he insananeous alue will be of greaes ineres. The insananeous elociy gies a mahemaically precise measure of how he posiion is changing a a paricular momen, making i much more useful han he aerage elociy. For his reason, from now on in his book we refer o he insananeous elociy as simply he elociy, and we denoe i by as in Equaion 2.5. CONCEPT CHECK 2.2 Esimaing he Insananeous Velociy Is here an insan in ime in Figure 2.9 a which he insananeous elociy is zero? If so, wha is he approimae alue of a which? EXAMPLE 2.1 Aerage Velociy of a Bicycle Consider he muliple images in Figure 2.1, showing a bicyclis moing along a leel road. Find he aerage elociy of he bicyclis during he ineral from 2. s o 3. s. START (Iniial posiion and ime) END (Final posiion and ime) 1. s 5 m 2. s 12 m 3. s 17 m 4. s Figure 2.1 Eample 2.1. RECOGNIZE THE PRINCIPLE The aerage elociy during a paricular ime ineral is he slope of he posiion ime graph during ha ineral. This means ha ae 5D/D (Eq. 2.4). The ime ineral is gien, so we need o deduce he displacemen from Figure 2.1. Insigh 2.2 ALWAYS DRAW A PICTURE The soluion o mos problems in physics sars wih a picure. Begin by drawing a picure showing all he gien informaion. The picure should show coordinae aes and any oher informaion ha seems relean o he problem. SKETCH THE PROBLEM Our firs sep is o draw a picure ha conains all he relean informaion; his is essenial for organizing our houghs and seeing connecions. The images in Figure 2.1 form he hear of he picure, bu because we wan o erac some quaniaie informaion, we hae added he ais. We hae chosen he origin of his ais o be a he bicyclis s posiion a 1. s. Using his coordinae ais, we can read off he alue of he bicycle s posiion a he imes of ineres. IDENTIFY THE RELATIONSHIPS The aerage elociy is he bicyclis s displacemen during a paricular ime ineral diided by he lengh of he ineral (Eq. 2.4). We hae herefore added arrows o our picure ha mark he posiions a he sar ( 2. s) and end ( 3. s) of he firs ineral of ineres. For each ineral, ae 5 D D 5 final 2 iniial final 2 iniial 32 CHAPTER 2 MOTION, FORCES, AND NEWTON'S LAWS 24716_2_c2_p26-54.indd 32 4/22/8 5:28:32 PM

8 SOLVE For he firs ineral, ae s s2 final 2 iniial m 2 5 m2 13. s 2 2. s2 7 m/s Wha hae we learned? Each bicycle in Figure 2.1 corresponds o a poin on he graph, wih coordinaes gien by reading off he alues of and. Once we had he alues of and, we hen found he aerage elociy hrough he relaion ae 5 D/D. In Eample 2.1, we saw how o use obseraions of he posiion as a funcion of ime o compue an objec s aerage elociy. As you migh epec, i is also ery useful o be able o deduce he insananeous elociy from he posiion ime behaior. We can do so by esimaing he slope of he posiion ime cure a differen alues of and using hese esimaes o make a qualiaie plo of he elociy ime relaion. This graphical approach is illusraed in Eample A (m) (m) (s) EXAMPLE 2.2 Compuing Velociy Using a Graphical Mehod A hypoheical objec moes according o he graph shown in Figure 2.11A. This objec is iniially (when is near 1 ) moing o he righ, in he posiie direcion. The objec reerses direcion near 2 and 3, and i is again moing o he righ a he end when is near 4. (a) Skech he qualiaie behaior of he elociy of he objec as a funcion of ime using a graphical approach. (b) Esimae he aerage elociy during he ineral beween 1 1. s and s. RECOGNIZE THE PRINCIPLE For par (a), we wan o find he elociy which means he insananeous elociy so we need o esimae he slope of he cure as a funcion of ime. For par (b), we use he fac ha he aerage elociy oer he ineral 1. s o 2.5 s is he slope of he cure during his ineral. B (s) Figure 2.11 Eample 2.2. A Hypoheical posiion ime graph. The slopes of he angen lines in B are equal o he elociy a arious insans in ime. SKETCH THE PROBLEM Figure 2.11B shows he graph again, his ime wih lines drawn angen o he cure a arious insans. The slopes of hese angen lines are he elociies a paricular imes 1, 2,... as indicaed in Figure 2.11A IDENTIFY THE RELATIONSHIPS AND SOLVE (a) A 1, he slope is large and posiie, so our resul for in Figure 2.12A is large and posiie a 1. A 2, he slope is approimaely zero and hence is near zero. A 3, he objec is moing oward smaller alues of posiion, so he slope of he cure and hence also he elociy are negaie. Finally, a 4, he objec is again moing o he righ as is increasing wih ime, so is again posiie. Afer esimaing he slope a hese places, we can consruc he smooh cure shown in Figure 2.12A. This figure shows he qualiaie behaior of he objec s elociy as a funcion of ime. (b) To esimae he aerage elociy beween 1 and 2, we refer o Figure 2.12B, which shows a line segmen ha connecs hese wo poins on he posiion ime graph. The slope of his segmen is he aerage elociy: ae 5 D D A (m) m 2 1 Slope ae m 1 B (s) Figure 2.12 Eample 2.2. A Qualiaie plo of he elociies obained from he slopes in Figure 2.11 B. B Calculaion of he aerage elociy during he ineral beween 1 1. s and s. 2.2 WHAT IS MOTION? _2_c2_p26-54.indd 33 4/22/8 5:28:33 PM

9 (1) Reading he alues of 1, 2, 1, and 2 from ha graph, we find ae m m s s2 1.3 m/s (2) Wha hae we learned? The (insananeous) elociy is he slope of he posiion ime graph. To find he qualiaie behaior of he elociy, we found approimae alues by drawing lines angen o he cure a seeral places and esimaing heir slopes. The alue of a a paricular alue of is always equal o he slope of he cure a ha ime. CONCEPT CHECK 2.3 The Relaion beween Velociy and Posiion (3) For which of he posiion ime graphs in Figure 2.13 are he following saemens rue? (a) The elociy increases wih ime. (b) The elociy decreases wih ime. (c) The elociy is consan (does no change wih ime). Figure 2.13 Concep Check 2.3. Definiion of aerage acceleraion Definiion of insananeous acceleraion Slope a D D Figure 2.14 Acceleraion is he slope of he elociy ime cure. For he case shown here, in which aries linearly wih ime, he aerage acceleraion is equal o he insananeous acceleraion. Acceleraion We hae seen ha wo quaniies posiion and elociy are imporan for describing he insananeous sae of moion of an objec, bu are hey sufficien? Are any oher quaniies needed o describe he physics of moion? One addiional quaniy, acceleraion, will play a cenral role in our heory of moion. Acceleraion is relaed o how he elociy changes wih ime. Consider again our rocke-powered car from Figure 2.8. In Figure 2.14, we reskech he plo ha we deried in Figure 2.8C and noice again ha he car s elociy increases as ime proceeds. Acceleraion is defined as he rae a which he elociy is changing. If he elociy changes by an amoun oer he ime ineral, he aerage acceleraion during his ineral is a ae 5 D (2.6) D As wih he elociy, we are usually concerned wih he acceleraion a a paricular insan in ime, which leads us o consider he acceleraion in he limi of ery small ime inerals. We hus define he insananeous acceleraion as D a 5 lim (2.7) D S D The insananeous acceleraion a equals he slope of he cure a a paricular insan in ime. The SI uni of acceleraion is m/s 2 (meers per second squared). We hae now inroduced seeral quaniies associaed wih moion, including posiion, displacemen, elociy, and acceleraion. These quaniies are conneced in a mahemaical sense hrough Equaions 2.5 and 2.7. We also showed imporan graphical relaionships: elociy is he slope of he posiion ime graph, while acceleraion is he slope of he elociy ime graph. You migh now be wondering if we ll coninue his progression and consider he slope of he acceleraion and so forh. The answer is ha acceleraion is as far as we need o go;,, and a are all we need in our formulaion of a complee heory of moion. Newon s laws of moion (Secion 2.4) will show us why. EXAMPLE 2.3 Acceleraion of a Spriner Consider a spriner (Fig. 2.15A) running a 1-m dash. Figure 2.15B shows he elociy ime graph for he spriner. Use a graphical approach o calculae he corresponding acceleraion ime graph. Wha is he approimae alue of he spriner s maimum acceleraion, and when does i occur? 34 CHAPTER 2 MOTION, FORCES, AND NEWTON'S LAWS 24716_2_c2_p26-54.indd 34 4/22/8 5:28:34 PM

10 (m/s) End of race Tara Moore/Sone/Gey Images 1 Sar 1 (s) A B Figure 2.15 Eample 2.3. The elociy ime graph of his spriner A is shown in B. RECOGNIZE THE PRINCIPLE Acceleraion is he slope of he elociy ime graph, so we mus esimae his slope a enough differen alues of o be able o make a qualiaie plo of he spriner s acceleraion as a funcion of ime. SKETCH THE PROBLEM In Figure 2.15B, we hae drawn in seeral lines angen o he cure a arious imes. The slopes of hese angen lines gie he acceleraion. IDENTIFY THE RELATIONSHIPS AND SOLVE We hae measured approimae alues of he slopes of he angen lines in Figure 2.15B, and he resuls are ploed in Figure The resuling acceleraion ime graph is only qualiaie (approimae). More accurae resuls would be possible had we sared wih a more deailed graph of he elociy. (Such eamples are waiing for you in he end-of-chaper problems.) Wha hae we learned? Acceleraion is he slope of he elociy ime graph, so gien he graph we can derie he qualiaie behaior of he acceleraion. The qualiaie shape of he acceleraion ime cure in Figure 2.16 is consisen wih epecaions. The larges alues of acceleraion (abou 1 m/s 2 ) occur a he sar of he race when he spriner is geing up o speed ; his is where he elociy ime cure is seepes and he slope is larges. A he end of he race as he runner crosses he finish line, she slows down and eenually comes o a sop wih a he far righ in Figure 2.15B. During ha ime, her elociy is decreasing wih ime so her acceleraion is negaie. a (m/s 2 ) Runner slows down afer crossing finish line. Figure 2.16 Eample 2.3. The corresponding acceleraion ime graph is gien qualiaiely. (s) The Relaion beween Velociy and Acceleraion An ineresing feaure of he graphs in Eample 2.3 is ha he maimum elociy and he maimum acceleraion do no occur a he same ime. I is emping o hink ha if he moion is large, boh and a will be large, bu his noion is incorrec. Acceleraion is he slope he rae of change of he elociy wih respec o ime. The ime a which he rae of change of elociy is greaes may no be he ime a which he elociy iself is greaes. CONCEPT CHECK 2.4 Analyzing a Posiion Time Graph The posiion ime cure of a hypoheical objec is shown in Figure Which of he following scenarios migh be described by his graph? (Noice ha we hae no shown numerical alues for or. No hins here!) Figure 2.17 Concep Check 2.4. Wha ype of moion is described by his posiion ime graph? 2.2 WHAT IS MOTION? _2_c2_p26-54.indd 35 4/22/8 5:28:35 PM

11 (a) A car saring from res when a raffic ligh urns green. (b) A car coming slowing o a sop when a raffic ligh urns red. (c) A bowling ball rolling oward he pins. (d) A runner slowing from op speed o a sop a he end of a race. A B a C Sick in conac wih puck Puck slows down as i slides o a sop. Acceleraion negaie as puck slows down Figure 2.18 Eample 2.4. Moion of a hockey puck as described by graphs of is posiion, elociy, and acceleraion ersus ime. EXAMPLE 2.4 Sliding o a Sop Consider a hockey puck ha sars from res. A, he puck is sruck by a hockey sick, which sars he puck ino moion wih a high elociy. The puck hen slides a ery long disance before coming o res. Draw qualiaie plos of he puck s posiion, elociy, and acceleraion as funcions of ime. RECOGNIZE THE PRINCIPLES We firs use our eperience wih hockey pucks and oher sliding objecs o deduce he qualiaie shape of he elociy ime ( ) cure. The puck sars from res, so iniially. The elociy hen increases quickly o a high alue when he sick is in conac wih he puck. Afer he puck leaes he sick, he elociy gradually decreases as he puck slides o a sop. From he behaior of he cure, we can ge he posiion and acceleraion as funcions of ime. SKETCH THE PROBLEM In Figure 2.18, imagine drawing par (b) firs (qualiaie behaior of elociy). The puck reaches a high elociy ery quickly, whereas a he end of he moion he elociy falls o zero. IDENTIFY THE RELATIONSHIPS As eplained aboe, he elociy as a funcion of ime is ploed in Figure 2.18B. To find he acceleraion, we mus find he slope of his cure as a funcion of ime. To ge he posiion, we mus find an cure whose slope gies his elociy ime graph. SOLVE The acceleraion ime graph in Figure 2.18C was obained from esimaes of he slope of he cure in par (b) a differen imes. Noice ha he acceleraion is posiie and large when he sick is in conac wih he puck. The acceleraion is negaie as he puck slides o a sop since he elociy is hen decreasing wih ime and he slope is negaie. To deduce he resul, we mus work backward from Figure 2.18B so as o make he slope of he posiion ime graph correspond o he cure. We can do so by noing ha he slope is greaes when is greaes and ha his slope is small when is small. Once reaches zero, he posiion no longer changes wih ime. Wha does i mean? These graphs of he posiion and elociy can also be appreciaed using he imelapsed skeches in Figure These skeches show he locaion of he puck a differen eenly spaced insans in ime, beginning a when i is sruck by he hockey sick and hen a 1, 2,... unil he puck comes o res a 4. These imes 1, 2, 3, and 4 are equally spaced along he ime ais (Fig. 2.18A), so he ime inerals 1 2 1,... are all equal. Howeer, he displacemens during hese inerals are no equal. For eample, he displacemen during he firs ineral (from o 1 ) is much larger han he displacemen during he second ineral. In words, he puck moes farher because i has a higher elociy during he firs ineral. Likewise, a he end of he ime period, he puck s elociy is smaller, so he disance raeled beween 3 and 4 is much smaller han he disance raeled beween and CHAPTER 2 MOTION, FORCES, AND NEWTON'S LAWS 24716_2_c2_p26-54.indd 36 4/22/8 5:28:54 PM

12 Person his puck here a Figure 2.19 Eample 2.4. Time-lapse skeches of he hockey puck s moion. EXAMPLE 2.5 Sopping in a Hurry A drier is in a hurry, and her car is raeling along a sraigh road a a elociy of 2 m/s (abou 4 mi/h) when she spos a problem in he road ahead and applies he brakes. The car hen slows o a sop according o he elociy ime cure in Figure 2.2. Find he aerage acceleraion during he ineral from. s o 4. s and also esimae he insananeous acceleraion a 2. s. RECOGNIZE THE PRINCIPLE The aerage acceleraion is he aerage slope of he elociy ime cure during he ime ineral of ineres. The insananeous acceleraion is he slope of he cure a he poin of ineres. SKETCH THE PROBLEM Figure 2.2 describes he problem and conains all he informaion we need o sole i. IDENTIFY THE RELATIONSHIPS From he definiion of he aerage acceleraion in Equaion 2.6, we hae a ae 5 D D We can read he alues of and from Figure 2.2. (1) (m/s) 2 1 Brakes applied here 2 m/s Figure 2.2 Eample 2.5. (s) SOLVE Applying Equaion (1) oer he ineral from. s o 4. s gies a ae 5 D s s2 5 D D Insering he alues from Figure 2.2 of (4. s) and (. s) 2 m/s along wih 4. s, we find m/s a ae m/s 4. s 2 This resul is he aerage acceleraion, so i is he aerage slope of he cure during he enire ineral shown in Figure 2.2. Since his cure is approimaely a sraigh 2.2 WHAT IS MOTION? _2_c2_p26-54.indd 37 4/22/8 5:28:55 PM

13 line, he aerage slope is approimaely equal o he insananeous slope a all imes in he ineral. The insananeous slope of he cure is he insananeous acceleraion a, so we also ge a < 5. m/s 2 Wha does i mean? The aerage and insananeous acceleraion in his eample are boh negaie because he elociy decreases during he ime ineral of ineres. The elociy is posiie (Fig. 2.2) during he enire ineral, so he elociy and acceleraion are in opposie direcions; here he elociy is posiie (along he direcion) while he acceleraion is negaie (along he direcion). 2.3 THE PRINCIPLE OF INERTIA We spen much of Secion 2.1 discussing Arisole s ideas abou moion, and you should now be coninced ha his heory of moion has some serious flaws. Neerheless, i is worhwhile o consider how Arisole would hae eplained he moion of he rocke-powered car in Figure 2.8. He probably would hae claimed ha when he rocke engine is urned on, he car moes by irue of he force eered on i by he engine, wih a elociy F/R as prediced from Equaion 2.1. If he rocke engine is hen urned off or runs ou of fuel, howeer, he force would anish (F ) and, according o he same argumen, he car would sop immediaely. Your inuiion should ell you ha his predicion is no correc; he car would insead coninue along, for a leas a shor period, afer he engine is urned off. Tha is, he car will coas for a while before coming o res. How would Arisole, or you, eplain his dilemma? The difficulies wih Arisole s ideas abou moion all come down o his belief ha force and elociy are direcly linked. This linkage is epressed in Equaion 2.1, which implies ha if here is a nonzero elociy, i mus be caused by a force. Tha is, if an objec is found o hae a nonzero elociy, hen according o Arisole here mus be a nonzero force acing on he objec a ha ime. Howeer, our eample wih a rocke-powered car shows ha force and elociy are no linked in his way because i is possible for he car o hae a nonzero elociy een when he force is zero. 4 The correc connecion beween force and moion is insead based on a direc linkage beween force and acceleraion. This connecion o acceleraion is a he hear of Newon s laws of moion. Before we can really appreciae Newon, howeer, we mus firs consider he work of Galileo. Alhough Galileo did no arrie a he correc laws of moion, his eperimens on he moion of erresrial objecs showed ha an objec can moe een if here is zero oal force acing on i. This resul led o he discoery of he principle of ineria, which is a cornersone of Newon s laws. Galileo s Eperimens on Moion Arisole was no able o eplain how an objec could moe when here did no appear o be any force acing on i. A good eample is he somewha idealized case of a hockey puck sliding on a ery smooh, horizonal, icy surface. Our inuiion ells us ha he puck will slide a ery long way before coming o res; in fac, if we could somehow make he surface perfecly icy so ha here is absoluely no fricion whasoeer, our inuiion suggess ha he puck would slide foreer. You migh 4 Here, we are concerned wih moion along a horizonal direcion (). The car s elociy along he horizonal direcion is no necessarily zero, while he force along his direcion is zero if he car s engine is urned off. 38 CHAPTER 2 MOTION, FORCES, AND NEWTON'S LAWS 24716_2_c2_p26-54.indd 38 4/22/8 5:28:56 PM

14 objec o his argumen on he grounds ha i is impossible o make a surface compleely fricionless. Een so, we can sill hink abou wha would happen in such a siuaion. In fac, hinking abou such idealized cases is a ery useful pracice in physics because hese cases can ofen help us see o he hear of a problem. Galileo did no carry ou eperimens wih a hockey puck on an icy surface. Insead, his eperimens inoled a ball rolling on an incline. If he ball is ery hard, like a billiard ball, and he surface of he incline is also ery hard and smooh, he effec of fricion on he ball s moion is ery small. Hence, his siuaion is acually quie similar o ha of our hockey puck. Galileo eperimened wih he moion of a ball on an incline as skeched in Figure He found ha when he ball is released from res, is elociy aries wih ime as shown in Figure 2.21B. This moion is anoher eample of one-dimensional moion, wih he direcion of moion denoed by an ais lying along he incline as shown. The elociy of he ball increases as he ball rolls down he incline, and Galileo found ha he magniude of he elociy increases linearly wih ime. Since acceleraion is he slope of he cure, he acceleraion is consan and posiie. Galileo hen repeaed his eperimen, bu his ime wih he ball rolling up an incline wih he same il angle (Fig. 2.22A). When he gae he ball some iniial elociy, he found ha i rolled up he incline wih he cure shown in Figure 2.22B. Again he obsered ha he elociy aries linearly wih ime, bu now he slope is negaie. In fac, he slope in his case is equal in magniude o he slope found for he down-iled incline. Hence, he acceleraion was opposie in sign bu equal in magniude o ha found when he ball rolled down he incline. Galileo performed his eperimen wih many inclines and wih many differen il angles, and he always obsered ha he elociy aried linearly wih ime, wih he slope of he graph being deermined by he il angle. Moreoer, he acceleraion (he slope of he relaion) when a ball rolled up a paricular incline was always equal in magniude, bu opposie in sign, when compared wih he acceleraion when he ball rolled down he same incline. He hen reasoned ha if he il of he incline were precisely zero (a perfecly leel surface), he slope of he line and hence he acceleraion would be zero. Galileo herefore assered ha on a leel surface he ball would roll wih a consan elociy. This resul may seem obious o you. For eample, i means ha a ball placed a res on a horizonal surface will remain in balance and hence be moionless, wih zero elociy. Such a ball has a consan elociy because is a consan! I was, howeer, he genius of Galileo o realize ha his eperimen implied anoher resul, which is skeched in Figure Here, a ball is rolling along a differen sor of ramp. The iniial porion of he ramp is sloped like he incline in Figure 2.21, bu he final porion is perfecly horizonal and eremely long, wih only par of i shown here. Galileo realized ha for his ype of ramp he ball would hae a posiie elociy when i complees he firs porion of he ramp. Then, on he second (horizonal) par of he ramp, he relaion would be a sraigh line wih a slope of zero. In oher words, he ball s elociy on he final porion of his ramp would hae he consan alue produced during he iniial par of he ramp, and he ball would mainain his elociy. Hence, for his idealized case ha ends wih a perfec horizonal ramp, Galileo proposed ha he ball would roll foreer. Galileo s eperimen demonsraes he principle of ineria. According o his principle, an objec (in his case, he ball) will mainain is sae of moion is elociy unless i is aced on by a force. On a horizonal surface, here is no force in he direcion of moion, so he elociy is consan and he ball rolls foreer. Wih his discoery of he principle of ineria, Galileo broke Arisole s link beween elociy and force. Galileo showed ha one can hae moion (a nonzero elociy) wihou a force. This discoery does no answer he quesion of how force is linked o moion, howeer. The answer o ha quesion was proided by Newon. (I is ineresing ha Newon was born in 1643, he year afer Galileo died. In a ery direc sense, Newon hus buil on he work of Galileo.) A B a C S Slope acceleraion Figure 2.21 A Ball rolling down a simple incline. B Galileo found ha he elociy of he ball increases linearly wih ime. The slope of his line is he acceleraion in C. Principle of ineria 2.3 THE PRINCIPLE OF INERTIA _2_c2_p26-54.indd 39 4/22/8 5:28:56 PM

15 2.4 NEWTON S LAWS OF MOTION Newon s laws of moion are hree separae saemens abou how hings moe. A Slope acceleraion Newon s Firs Law Newon s firs law is a careful saemen abou he principle of ineria ha we encounered in he preious secion in connecion wih Galileo s eperimens. Newon s firs law of moion: If he oal force acing on an objec is zero, he objec will mainain is elociy foreer. Figure 2.22 A Ball rolling up an incline wih he same angle as in Figure B Galileo found ha now he elociy of he ball decreases linearly wih ime. C The slope of his line he acceleraion has he same magniude, bu he opposie sign, as he acceleraion when he ball rolls down he incline (Fig C ). A B B a C Acceleraion is negaie Figure 2.23 When a ball rolls along his wo-par incline, is elociy increases linearly wih ime while on he iniial (sloped) porion, and is hen consan on he final (fla) porion. consan here According o Newon s firs law, if here is no oal force acing an objec ha is, if he oal force acing on i is zero he objec will moe wih a consan elociy. In oher words, such an objec will moe wih a consan speed along a paricular direcion and will coninue his moion wih he same speed in he same direcion foreer, or as long as he oal force acing on i is zero. 5 This is anoher way of saing he principle of ineria, and you can see ha i is essenially wha Galileo found in his eperimens wih balls moing along inclined or fla surfaces. The oal force acing on one of Galileo s balls rolling on a horizonal surface is zero, so in his ideal case he ball will roll foreer. Ineria and Mass Now ha we hae inroduced he principle of ineria, you should be curious abou he erm ineria and precisely wha i means. The ineria of an objec is a measure of is resisance o changes in moion. This resisance o change depends on he objec s mass. I is no possible o gie a firs principles definiion of he erm mass, bu your inuiie noion is ery useful. The mass of an objec is a measure of he amoun of maer i conains. Objecs ha conain a large amoun of maer hae a larger mass and a greaer ineria han objecs conaining a small amoun of maer. The SI uni of mass is he kilogram (kg). Mass is an inrinsic propery of an objec. For eample, he mass m of an objec is independen of is locaion; m is he same on he Earh s surface as on he Moon and in disan space. In addiion, he mass does no depend on an objec s elociy or acceleraion. Alhough an objec s mass does no appear eplicily in Newon s firs law, i has an essenial role in Newon s second law. Newon s firs law may seem surprising o you. Afer all, erresrial objecs always come o res eenually. The difference here, and a key o appreciaing Newon s firs law, is ha an objec will moe wih a consan elociy only in he ideal case ha he oal force is precisely zero. For erresrial objecs, i is ery difficul o find such ideal cases because i is difficul o compleely eliminae all forces, especially fricional forces. Hence, erresrial objecs come o res because of he forces ha ac on hem. Two oher hypoheical cases may help make Newon s firs law fi wih your inuiion. Case 1: Imagine a frozen lake wih an eremely smooh surface. A hockey puck sliding on such a perfecly fla and icy surface eperiences a ery iny fricional force and will herefore slide for a ery long disance before coming o res. If his fricional force could be made o anish, he puck would slide foreer (if he lake were large enough). Of course, acual icy surfaces are no perfec; here will always be a small amoun of fricion, and his small amoun of fricion would make he puck eenually come o res. Case 2: Imagine a spaceship ha is coasing (i.e., wih is engines urned off) someplace in he unierse ery far from any sars or planes. Such a spaceship would eperience only a ery small graiaional force (from he neares sars and planes). If he neares sars and planes were ery far 5 This resul may be surprising because i may appear o conradic your inuiion. This resul only holds if he oal force on an objec is precisely zero, howeer, and ha can be hard o achiee in pracice. See below. 4 CHAPTER 2 MOTION, FORCES, AND NEWTON'S LAWS 24716_2_c2_p26-54.indd 4 4/22/8 5:28:56 PM

16 away, his force would be negligible and he spaceship would moe wih a consan elociy, in accord wih Newon s firs law. Newon s Second Law Newon s second law of moion: In many siuaions, seeral differen forces are acing on an objec simulaneously. The oal force on he objec is he sum of hese indiidual forces, F S oal 5 g F S. The acceleraion of an objec wih mass m is hen gien by S a 5 g FS m (2.8) Newon s second law ells us how an objec will moe when aced on by a force or by a collecion of forces. This law is our link beween force and moion. The acceleraion of an objec is direcly proporional o he oal force ha acs on i. Newon s second law, Equaion 2.8, is ofen wrien in he equialen form g F S 5 ma S Keep in mind ha he erm g S F in Newon s second law is he oal force on he objec, from all sources. As you migh imagine, and as we ll see in some eamples, in mos cases here are seeral forces acing on an objec. So, we hae o add hem all up, and ha resuling ecor sum is he force g S F in Newon s second law (see Fig. 2.24). You should also recall ha ecors mus be added according o he ecor arihmeic procedures described in Chaper 1. The direcion of he acceleraion S a is herefore parallel o he direcion of he oal force g S F. In he SI sysem of unis, force is measured in unis called newons (N). We can use Newon s second law o epress his uni in erms of he primary SI unis. Mass is measured in unis of kilograms, while acceleraion has he unis meers per second squared. In erms of only he unis, Equaion 2.8 can be rearranged o read force 5 mass 3 acceleraion newons 5 kg 3 m s 2 The alue of he newon as a uni of force is herefore 1 N 5 1 kg # m/s 2 (2.9) We ll spend he ne dozen chapers or so eploring applicaions of Newon s second law. Many applicaions sar by deermining he oal force acing on an objec from all sources. The objec s acceleraion can hen be calculaed using Equaion 2.8. Acceleraion is he change in elociy per uni ime, so i is possible o use he acceleraion o deduce he elociy. In a similar manner, since elociy is he change in posiion per uni ime, one can use he elociy o find he objec s posiion as a funcion of ime. In his way, an objec s acceleraion, elociy, and posiion can all be found. S F1 S F2 S F oal S S The acceleraion a S is parallel o S F. S F Figure 2.24 When seeral forces ac on an objec, he ecor sum of hese forces g F S deermines he acceleraion according o Newon s second law. EXAMPLE 2.6 Using Newon s Second Law A single force of magniude 6. N acs on a sone of mass 1.1 kg. Find he acceleraion of he sone. RECOGNIZE THE PRINCIPLE The force on he sone and is acceleraion are relaed hrough Newon s second law, S a 5 g FS m (1) 2.4 NEWTON'S LAWS OF MOTION _2_c2_p26-54.indd 41 4/22/8 5:28:57 PM

17 Here, g F S is he oal force on he sone, which has a magniude of 6. N. SKETCH THE PROBLEM We begin by drawing a picure, showing all he forces acing on he sone (Fig. 2.25). Since here is only a single force in his eample, i is also he oal force. Figure 2.25 Eample 2.6. F IDENTIFY THE RELATIONSHIPS AND SOLVE Using Newon s second law, Equaion (1), he magniudes of he acceleraion and force are relaed by a 5 g F m 5 6. N 1.1 kg 5 6. kg # m/s m/s 1.1 kg 2 (2) The direcion of he sone s acceleraion is parallel o ha of he oal force, so he direcion of F S in Figure 2.25 gies he direcion of a S. Wha does i mean? Noice how he unis in Equaion (2) combine, as he facors of kilograms on he op and boom cancel o gie an answer wih unis of meers per second squared, as epeced for acceleraion. You should always check he unis of an answer in his way. An incorrec resul for he unis usually indicaes an error in he calculaion. y EXAMPLE 2.7 Moion of a Falling Objec Figure 2.26 Eample 2.7. Posiion (y) aboe he ground as a funcion of ime for a ball ha falls from a bridge. Figure 2.27 Eample 2.7. A ball is dropped from a bridge, saring from res ( ) a a heigh h aboe he ground. The moion of his ball is described by he y graph in Figure y A ball is dropped from a bridge ono he ground below. The heigh of he ball aboe he ground as a funcion of ime is shown in Figure Use a graphical approach o find, as funcions of ime, he qualiaie behaior of (a) he elociy of he ball, (b) he acceleraion of he ball, and (c) he oal force on he ball. RECOGNIZE THE PRINCIPLE Velociy is he slope of he posiion ime cure, so we can find from he slope of he y cure in Figure Noice ha here we use y o measure he posiion, aking he place of in preious eamples. Acceleraion is he slope of he elociy ime cure, so we can find he behaior of a once we hae he behaior of he elociy. Once we hae he acceleraion, he oal force on he ball can hen be found hrough Newon s second law, g F S 5 ma S. SKETCH THE PROBLEM Figure 2.26 shows how he posiion of he ball aries wih ime, and Figure 2.27 shows a picure of he ball as i falls from he bridge. The moion of he ball is onedimensional, falling direcly downward from he bridge o he ground, and is posiion can be measured by is heigh y aboe he ground. This noaion follows he common pracice of using o represen posiion along a horizonal direcion and y o represen posiion along a erical direcion. Figure 2.27 also shows he y coordinae ais, wih is origin a ground leel. IDENTIFY THE RELATIONSHIPS AND SOLVE (a) Velociy is he change in posiion per uni ime, so he alue of a any paricular ime is he slope of he y graph a ha insan (since in his eample y is our posiion ariable). We can obain his slope graphically by following he approach in Eample 2.2 and drawing angen lines o he y cure a arious imes in Figure 2.28A. Using esimaes for he slopes of hese lines leads o he qualiaie elociy ime graph shown alongside he y cure. Noe ha he elociy is always negaie since y 42 CHAPTER 2 MOTION, FORCES, AND NEWTON'S LAWS 24716_2_c2_p26-54.indd 42 4/22/8 5:28:58 PM

18 y a F Slope a Acceleraion is negaie A B C decreases monoonically wih ime. Also, he magniude of he elociy, which is he speed, becomes larger and larger as he ball falls. (The plo here shows as a funcion of ime up unil jus before he ball reaches he ground.) (b) Acceleraion is he slope of he cure; ha slope is consan in Figure 2.28A, leading o he qualiaie acceleraion ime graph in Figure 2.28B. The acceleraion is negaie because he alue of is decreasing (he alue of becomes more negaie wih ime). The magniude of a is approimaely consan during his period. (c) To compue he force on he ball, we use Newon s second law. According o Equaion 2.8, he oal force is proporional o he acceleraion. Een if we do no know he sources of all he forces on he ball, we can sill compue he oal force from Newon s second law. We can rearrange Newon s second law (Eq. 2.8) as g S F 5 ma S and hus arrie a he qualiaie force ime graph in Figure 2.28C. Figure 2.28 Eample 2.7. A Posiion as a funcion of ime for he falling ball from Figure The slopes of he angen lines gie he ball s elociy a hree insans in ime. The elociy of he ball as a funcion of ime is obained from ploing hese slopes. B The ball s acceleraion is consan. C The force on he ball is proporional o he ball s acceleraion. Wha hae we learned? Gien he behaior of he posiion as a funcion of ime, we can deduce (by esimaing slopes) he qualiaie behaior of boh he elociy and he acceleraion as funcions of ime. The behaior of he force can hen be found by using Newon s second law. For his falling ball, he oal force is negaie (Fig. 2.28C), meaning ha he force is direced downward, along he y direcion. This force is jus he graiaional force acing on he ball. Newon s Second Law and he Direcions of S and a S Newon s second law can be wrien as g F S 5 ma S, and we hae already menioned ha his relaion means ha he acceleraion of an objec a S is always parallel o he oal force g F S acing on he objec. We hae also seen eamples in which he acceleraion and elociy are in differen direcions, which means ha he elociy and oal force need no be in he same direcion. For eample, when an objec such as he arrow in Figure 2.29 is fired upward, is elociy is upward, along he y direcion in he figure. The oal force on his objec is due almos enirely o graiy (we ll discuss his more in laer chapers) and is downward, along y. Hence, in his case, S and g F S are in opposie direcions. A key poin is ha alhough he force and acceleraion are always parallel, he direcion of he elociy can be differen. Afer leaing he bow, is upward while he oal force on he arrow is downward. Newon s Third Law The erm g F S appearing in Newon s second law is he oal force acing on he objec whose moion we are sudying or calculaing. This force mus come from somewhere; in fac, i is always produced by oher objecs. For eample, when a baseball is sruck by a ba, he force on he ball is due o he acion of he ba. Newon s hird law is a saemen abou wha happens o he oher objec, he ba. Figure 2.29 As i raels upward, he elociy of his arrow is upward, bu he oal force on he arrow is direced downward, so S and g F S are in opposie direcions. 2.4 NEWTON'S LAWS OF MOTION _2_c2_p26-54.indd 43 4/22/8 5:28:59 PM

19 Newon s hird law of moion: When one objec eers a force on a second objec, he second objec eers a force of he same magniude and opposie direcion on he firs objec. A F on ba F on ball F on ba F on ball AP Phoo/Elaine Thompson Newon s hird law is ofen called he acion reacion principle. The forces eered beween a baseball ba and a ball (Fig. 2.3) proide an ecellen illusraion of Newon s hird law. When he ba is in conac wih he baseball, he ball eperiences a force ha leads o is acceleraion, as can be calculaed from Newon s second law. A he same ime, he ba eperiences a force acing on i ha comes from he ball. You may hae already encounered his force; i is mos noiceable when you hi he ball a lile away from he swee spo of he ba. The poin of Newon s hird law is ha forces always come in such pairs. The force eered by he ba on he ball and he force eered by he ball acing back on he ba are known as an acion reacion pair of forces. According o Newon s hird law, hese wo forces are always equal in magniude and opposie in direcion, and hey mus ac on differen objecs. B Figure 2.3 A When a ba srikes a baseball, he ba eers a large force on he ball. B According o Newon s hird law, he ball eers a reacion force a force of equal magniude and opposie direcion on he ba. Which Law Do We Use? Before we go any furher, i is worhwhile o look ahead o how we will acually use Newon s laws o calculae he moion of arious objecs. Newon s second law ells us how o calculae a paricular objec s acceleraion. For eample, if we wan o sudy he moion of a baseball, we need o calculae he oal force on he ball and hen inser his g F S ino Newon s second law (Eq. 2.8) o find he ball s acceleraion. This oal force will ofen hae conribuions from seeral sources; for a baseball, here may be a force from he impac wih a ba along wih oher forces. The main poins are ha he oal force g F S in Equaion 2.8 is he oal force on jus he ball and ha Newon s second law enables us o calculae he acceleraion of jus he ball. On he oher hand, Newon s hird law ells us ha forces always come in pairs. Hence, if we were somehow able o calculae or measure he force eered by a baseball ba on a ball, Newon s hird law ells us ha here mus be a corresponding reacion force ha acs on he ba. I is eremely useful o know abou his reacion force since i will conribue o he oal force on he ba. Newon s hird law is hus essenial for undersanding his force on he ba. Newon s hree laws of moion are he foundaion for nearly eeryhing ha we do in he firs par of his book. In his secion, we hae gien some background for each of he hree laws so ha you can appreciae wha hey ell us abou he naure of moion. We ll show in subsequen chapers ha Newon s laws conain many oher ideas and conceps, such as energy and momenum, ha are essenial o our hinking abou he physical world. EXAMPLE 2.8 Acion Reacion A person pushes a refrigeraor across he floor of a room. The person eers a force F S 1 on he refrigeraor. From Newon s hird law, we know ha F S 1 is par of an acion reacion pair of forces. Wha is he reacion force o F S 1? RECOGNIZE THE PRINCIPLE The force on he refrigeraor S F1 is caused by he acion of he person on he refrigeraor. According o Newon s hird law, he reacion force mus be equal in magniude o S F1 bu in he opposie direcion. The reacion force mus also ac on a differen objec (i canno ac on he refrigeraor). SKETCH THE PROBLEM Figure 2.31 shows he problem. 44 CHAPTER 2 MOTION, FORCES, AND NEWTON'S LAWS 24716_2_c2_p26-54.indd 44 4/22/8 5:29: PM

20 IDENTIFY THE RELATIONSHIPS AND SOLVE The reacion force S F2 is he acion of he refrigeraor back on he person as shown in Figure According o Newon s hird law, he forces in an acion reacion pair hae equal magniudes and are in opposie direcions, so S F2 52F S 1. The forces in an acion reacion pair ac on differen objecs. Wha does i mean? Forces always come in acion reacion pairs. The wo forces in an acion reacion pair always ac on differen objecs. F 2 F 1 CONCEPT CHECK 2.5 Acion Reacion Force Pairs Which of he following is no an acion reacion pair of forces? (More han one answer may be correc.) (a) The force eered by a picher on a baseball and he force eered by he ball when i his he ba (b) When you lean agains a wall, he force eered by your hands on he wall and he force eered by he wall on your hands (c) In Figure 2.3, he force eered by he ball on he ba and he force eered by he ba on he player s hands Figure 2.31 Eample 2.8. Newon s hird law, he acion reacion principle, applies o people and refrigeraors, oo. 2.5 WHY DID IT TAKE NEWTON TO DISCOVER NEWTON S LAWS? Newon s second law ells us ha he acceleraion of an objec is gien by a S 5 1 g F S 2 /m, where g F S is he oal force acing on he objec. In he simples siuaions, here may only be one or wo forces acing on an objec, and g F S is hen he sum of hese few forces. In some cases, howeer, here may be a ery large number of forces acing on an objec. Muliple forces can make hings appear o be ery complicaed, which is perhaps why he correc laws of moion Newon s laws were no discoered sooner. Forces on a Swimming Bacerium Figure 2.32 shows a phoo of he single-celled bacerium Escherichia coli, usually referred o as E. coli. An indiidual E. coli propels iself by moing hin srands of proein ha eend away from is body (raher like a ail) called flagella. Mos E. coli possess seeral flagella as in he phoo in Figure 2.32A, bu o undersand heir funcion, we firs consider he forces associaed wih a single flagellum as skeched in Figure 2.32B. A flagellum is fairly rigid, and because i has a spiral shape, one can hink of i as a small propeller. An E. coli bacerium moes abou by roaing his propeller, hereby eering a force F S w on he nearby waer. According o Newon s hird law, he waer eers a force F S E of equal magniude and opposie direcion on he E. coli, as skeched in Figure 2.32B. One migh be emped o apply Newon s second law wih he force F S E and conclude ha he E. coli will moe wih an acceleraion ha is proporional o his force. Howeer, his is incorrec because we hae no included he forces from he waer on he body of he E. coli. These forces are also indicaed in Figure 2.32B; o properly describe he oal force from he waer, we mus draw in many force ecors, pushing he E. coli in irually all direcions. A he molecular leel, we can undersand hese forces as follows. We know ha waer is composed of molecules ha are in consan moion, and hese waer molecules bombard he E. coli from all sides. Each ime a waer molecule collides wih he E. coli, he molecule eers a force on he bacerium. Collision forces are discussed more in a laer chaper; for now, we noe ha he siuaion is similar o he collision of he baseball and ba in Figure 2.3. As we saw in ha case, he wo Dr. Dennis Kunkel/Visuals Unlimied A B S F E Flagellum S F w Figure 2.32 A E. coli use he acion reacion principle o propel hemseles. An indiidual E. coli bacerium is a few micromeers in diameer (1 mm.1 m). B The flagellum eers a force S Fw on he waer, and he waer eers a force S FE 52F S w on he E. coli. There are also forces from he waer molecules (indicaed by he oher red arrows) ha ac on he cell body. 2.6 THINKING ABOUT THE LAWS OF NATURE _2_c2_p26-54.indd 45 4/22/8 5:29:1 PM