# Physics 2A Chapters 6 - Work & Energy Fall 2017

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1 Physcs A Chapters 6 - Work & Energy Fall 017 These notes are eght pages. A quck summary: The work-energy theorem s a combnaton o Chap and Chap 4 equatons. Work s dened as the product o the orce actng on an object and the dsplacement o the object. Knetc energy o an object s the energy t possesses that s assocated wth ts moton. The total work done on an object, by all orces, equals the change n knetc energy o the object. Isaac Newton ded n 177 at the age o 85. The remander o the 18 th century was a perod o tremendous scentc development, or whch Newton s nluence played a sgncant role. Part o ths development was the creaton o new deas and correspondng new dentons that would make t easer or scentsts to descrbe and understand the physcal world. The most mportant o the new dentons rom ths perod s energy. The concept o energy s by ar the most mportant that we wll use ths semester, and even ths entre year. It s used throughout scence, and we wll use t n nearly every chapter or the remander o the year. Energy s nature s accountng system, and t serves the same uncton or studyng nature that the concept o money serves or studyng economcs (.e. human behavor.) Energy s our way o quantyng every nteracton between any two objects n nature. To develop the denton o energy, we have to start wth basc dentons o moton rom Chapter. An object has some ntal velocty and ntal poston, and later has some nal poston and velocty (ater t has accelerated or some tme.) We wrote two equatons: v v at and + + v t + 1 at We also know, rom Chapter 4, that there are orces causng the acceleraton o the object. Usng Newton s Second Law: F ma Page 1 o 8

2 Physcs A Chapters 6 - Work & Energy Fall 017 And now we can do a bt o algebra. When we worked wth the two moton equatons n Chapter, t was mportant that tme was present n both o our equatons. Tme was an essental element or the study o Moton n One Dmenson. But now we take advantage o the act that tme appears n both equatons... to elmnate t algebracally. v v t substtute ths epresson nto the second equaton a + v v v a + 1 v v a a Ths looks lke qute a mess. To clean t up a lttle, we ll subtract rom both sdes and multply everythng by a. ( ) v ( v v ) + ( v v ) a Net, we ll get rd o the parentheses on the rght sde: ( ) v v v + v v v v a + Smply the rght sde and replace a wth F/m, usng Newton s Second Law: F ( ) v v m In the nal steps, we multply both sdes by m, dvde everythng by, and replace - wth d, the dsplacement o the object: Fd 1 1 mv mv Ths epresson s the Work-Energy Theorem. Page o 8

3 Physcs A Chapters 6 - Work & Energy Fall 017 Work The let sde o the work-energy theorem represents the work done on the object. We dene the work done as the product o the net orce (.e. all the orces combned) appled to the object and the dsplacement the object moves through. Note that orce and dsplacement are both vector quanttes, so t s reasonable to ask whether or not work s a vector quantty. It s not. Work s a scalar quantty, but t s ound by takng the product o two vector quanttes. In other words, drecton o the orce s mportant and drecton o the dsplacement s mportant, but work tsel has no drecton. Ths leads us to a very useul and smple result: only the relatve drecton o the orce and dsplacement,.e. ther drectons relatve to each other, matter when calculatng the work done by a orce. Another observaton: when addng vector quanttes (.e. addng dsplacements, veloctes, orces, etc) the result s always the same vector quantty and drecton matters. There are two derent ways to multply vector quanttes: one results n a scalar quantty, and we wll use ths method n Chapter 6 to determne work; the other results n a vector quantty, and we wll use t n Chapter 10. The work done on an object by a orce s dened as: W F d We say ths F dot d, where the dot ndcates the method by whch we multply two vector quanttes to get a scalar as a result. Ths process s sometmes reerred to as the dot product. SI Unt o Work: Newton-meter whch s dened as a Joule and s also a kg-m /s In the orgnal epresson or the work-energy theorem, the F represented the net orce or the combnaton o all orces actng on the object. It wll be useul n Chapter 6 to consder the work done on the object by ndvdual orces and then combne these ndvdual work to get the total work. The real advantage to ths process s that work s a scalar quantty, so we can determne the work done by ndvdual orces, the total work s the smple sum (.e. no drecton necessary) o the ndvduals. O course the drecton o the net orce must be n the drecton o acceleraton. But ndvdual orces can be wth the drecton o moton, aganst the drecton o moton, perpendcular to the drecton o moton, or anywhere n between. We use ths act, and the epresson o work rom the work-energy theorem, to urther rene our denton o work: Work done by any ndvdual orce s the product o the dsplacement o the object and the component o the orce n the same drecton as the dsplacement. Page 3 o 8

4 Physcs A Chapters 6 - Work & Energy Fall 017 Ths means a partcular orce s n the same drecton as the dsplacement o the object, the work done by that orce s smply the product o the orce and dsplacement. But a orce acts perpendcular to the dsplacement o the object, that orce dd not cause the object to change ts speed. I an object accelerates horzontally, then vertcal orces dd not cause the acceleraton. As the object moves horzontally, the vertcal orces dd zero work. Another way to look at ths s that the vertcal orces must be balanced n ths knd o stuaton, so together they contrbute zero to the net orce that s acceleratng the object. I a orce s at an angle to the dsplacement, so that the orce has a component parallel to and a component perpendcular to the dsplacement, then the parallel component o the orce does work (causng the object to accelerate) whle the perpendcular component does no work. Ths means that the work done by the orce s W ( F cosθ ) d Fd cosθ where θ s the angle between the drecton o the orce and the drecton o the dsplacement. We now have a denton o the dot product : t s the product o the magntudes o the two vectors and the cosne o the angle between ther drectons. The denton provdes us wth a ew smple results: Force s n same drecton as d Force s perpendcular to d Force s n opposte drecton to d θ 0 cos θ 1 θ 90 O cos θ 0 θ 180 O cosθ -1 Work Fd Work 0 Work -Fd Page 4 o 8

5 Physcs A Chapters 6 - Work & Energy Fall 017 Note that work can be postve, the orce s n the same drecton as the dsplacement, or t can be negatve, the orce s opposte the drecton o the dsplacement. (O course t can also be zero, the orce s nether wth or aganst the dsplacement.) Ths brngs us to another concluson regardng work: I the orce s n the same drecton as dsplacement, the orce wll push the object aster and the work done wll be postve. I the orce s opposte the drecton o dsplacement, the orce wll push the object slower (.e. aganst ts drecton o moton) and work done wll be negatve. Knetc Energy The rght sde o the work-energy theorem ncludes only three thngs: the mass o the object, ts nal velocty and ts ntal velocty. We make a denton, nspred by the result on the rght sde o the workenergy theorem: Knetc energy o a movng object: 1 K mv Usng ths denton, the rght sde o the work energy theorem can be wrtten as the derence between the nal knetc energy o the object and the ntal knetc energy o the object. In other words, the rght sde o the work-energy theorem s the derence (ncrease or decrease) o the knetc energy o the object. The work-energy theorem can now be stated as: the total work done on an object s equal to the change n the knetc energy o the object. Note that: I the total work s postve, the orce pushes the object aster, and knetc energy ncreases (.e. change n knetc energy s postve.) I the total work s negatve, the orce pushes the object slower, and knetc energy decreases (.e. change n knetc energy s negatve.) Knetc energy s a scalar quantty, and the SI unt o knetc energy s the Joule. Note that snce the velocty o the object s squared when calculatng knetc energy, the result wll always be postve. Snce knetc energy s always postve, there s no method by whch to ndcate ts drecton. So there s no drecton assocated wth knetc energy. Page 5 o 8

6 Physcs A Chapters 6 - Work & Energy Fall 017 Sprngs Robert Hooke lved around the same tme and place as Isaac Newton. Hooke made varous contrbutons to the advancement o scence, ncludng Hooke s Law or the behavor o an deal sprng. Hooke determned that an deal sprng eerts a orce that s proportonal to the dstance the sprng s stretched or compressed. The orce the sprng eerts s also n the opposte drecton o the stretch or compresson. So we can wrte Hooke s Law as: Hooke s Law or an Ideal Sprng: F k where F represents the orce the sprng eerts, s the dstance the sprng s stretched or compressed, and k s the sprng constant, a measure o the strength o the sprng. The hats on F and are remnders that these are vector quanttes,.e. we have to consder ther drecton. The mnus sgn ndcates that the drecton o the orce s opposte the drecton o. We dene a relaed sprng as one that eerts no orce, so by denton 0 or a relaed sprng. A sprng can apply a orce to an eternal object, and when that object moves the sprng wll have done work on the object. The work done by the sprng s the product o the orce appled by the sprng and the dstance the object moves. But there s one mnor complcaton: whle the orces o gravty, normal, or knetc rcton reman constant whle they act on an object, the orce the sprng eerts changes as the sprng s compressed or stretched. Ths means we cannot smply determne the work done by the sprng by usng the sngle orce eerted by the sprng and the total dstance the object moves. Instead, we have to consder the work as t s done n tny steps... nntely tny steps. Ths dea, consderng physcal processes as the sum o an nnte number o nntely tny steps, s the bass o Newton s ntegral calculus. And ntegral s the mathematcal tool that we use to add that nnte number o nntely tny steps. In ths case: Tny bt o work done by the sprng: dw Fd kd Add all o the tny bts o work: W Fd kd Or, ater crunchng the ntegral: W k 1 k 1 k Page 6 o 8

7 Physcs A Chapters 6 - Work & Energy Fall 017 Notce that: I the sprng s ntally relaed and then s compressed or stretched by contact wth an eternal object, wll be zero and the work done by the sprng wll be negatve. Ths means the sprng wll cause the object to slow down. I the sprng s ntally compressed or stretched, but then pushes (or pulls) the object, t wll do postve work on the object. Ths s because wll be somethng and wll be zero. Work Done By Specal Forces We can now consder the work done by specal orces,.e. orces that have a specal denton o magntude and drecton. Usng ther dentons we can create epresson or the work done by these orces, whch wll make solvng work-energy problems much smpler. Gravty: the orce o gravty s mg and downward. We can dene the vertcal dsplacement o an object as ts change n heght, or h. Note that h s the nal heght mnus the ntal heght o the object. I we consder that h s up when postve, then the angle between the dsplacement and orce s 180 O. So: Work by gravty: W g -mg h Normal orce: an object sldes along a surace, ts dsplacement s parallel to the surace and the normal orce s perpendcular to the surace. The angle between the orce and dsplacement s 90 O, so: Work by normal orce: W n 0 Knetc rcton: when an object sldes on a surace, knetc rcton always acts opposte the drecton the object sldes. Knetc rcton always slows the objects speed, dong negatve work. The angle between the orce and dsplacement s 180 O. I we use d or the dsplacement o the object and the orce o knetc rcton s µ k n, then: Work by knetc rcton: W -µ k nd Sprngs: the epresson or work done by a sprng was derved above. Work by a sprng: W s ½k - ½k Page 7 o 8

8 Physcs A Chapters 6 - Work & Energy Fall 017 The Elevator Ecepton! For nearly all stuatons, we can clam that work done by the normal orce or statc rcton s zero. The eceptons to ths generalzaton occur when the surace tsel moves. The normal orce can do work the surace moves vertcally (e.g. or an object n an elevator.) The statc rcton orce can do work the object moves parallel to the surace wthout sldng on the surace. Ths can happen or a object n the bed o a truck that does not slde but moves as the truck moves. Statc rcton s also the orce by whch tres pushng aganst the road can move a vehcle orward. Power Power s dened as the rate at whch energy s produced or used. In other words, power s energy per tme. The SI unt o power s the Joule per second, whch s dened as a Watt. There are no specal rules or consderatons when usng the concept o power. It s smply related to energy by a actor o tme. For ths reason, problems that menton power are energy problems and you can determne the relevant amount o energy by usng the denton that energy s the product o power and tme. Work-Energy Problems The rules or work-energy problems are arly smple and streamlned (.e. non-techncal) when compared to the rules or problems rom earler chapters. Most o your eort to solve work-energy problems wll be concentrated n organzng your normaton, as the algebra (one equaton, one unknown...) s about as smple as t can be. The rules : 1. Draw pctures (multple pctures!) In Chapter you drew one pcture and dented ponts o nterest n the moton o the object. Your pctures n work-energy problems wll play a smlar role. Draw a pcture or the ntal stuaton o the object and the nal stuaton. There could be more than one nal stuaton, so you should be prepared to draw more than two pctures, necessary.. Label all relevant normaton n each pcture, ncludng v (speed o the object), h (heght o the object), and (stretch or compresson o a sprng.) 3. Make a lst o orces that act on the object. 4. Wrte the work-energy theorem. Replace the let sde wth a lst o W, one or each orce on your lst. 5. Smply your equaton and solve. You should have just one unknown wth one equaton. Page 8 o 8

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