Chapter 4 or an Energ 4 or an Energ You have one qute a bt of problem solvng usng energ concepts. ac n chapter we efne energ as a transferable phscal quantt that an obect can be sa to have an we sa that f one transfers energ to a materal partcle that s ntall at rest, that partcle acqures a spee whch s an ncator of how much energ was transferre. e sa that an obect can have energ because t s movng (netc energ, or ue to ts poston relatve to some other obect (potental energ. e sa that energ has unts of oules. You have ealt wth translatonal netc energ K = mv, rotatonal netc energ K = Iw, sprng potental energ U =, near-earth s-surface gravtatonal potental energ U = mg, an the unversal gravtatonal potental energ U = G m m r corresponng to the Unversal Law of Gravtaton. The prncple of the conservaton of energ s, n the opnon of ths author, the central most mportant concept n phscs. Inee, at least one ctonar efnes phscs as the stu of energ. It s mportant because t s conserve an the prncple of conservaton of energ allows us to use smple accountng proceures to mae prectons about the outcomes of phscal processes that have et to occur an to unerstan processes that have alrea occurre. ccorng to the prncple of conservaton of energ, an change n the total amount of energ of a sstem can be accounte for n terms of energ transferre from the mmeate surrounngs to the sstem or to the mmeate surrounngs from the sstem. Phscsts recogne two categores of energ transfer processes. One s calle wor an the other s calle heat flow. In ths chapter we focus our attenton on wor. Conceptuall, postve wor s what ou are ong on an obect when ou push or pull on t n the same recton n whch the obect s movng. You o negatve wor on an obect when ou push or pull on t n the recton opposte the recton n whch the obect s gong. The mnemonc for rememberng the efnton of wor that helps ou remember how to calculate t s or s Force tmes Dstance. The mnemonc oes not tell the whole stor. It s goo for the case of a constant force actng on an obect that moves on a straght lne path when the force s n the same eact recton as the recton of moton. more general, but stll not completel general, how-to-calculate-t efnton of wor apples to the case of a constant force actng on an obect that moves along a straght lne path (when the force s not necessarl recte along the path. In such a case, the wor one on the obect, when t travels a certan stance along the path, s: the along-the-path component of the force F tmes the length of the path segment r. = F r (4- Even ths case stll nees some atonal clarfcaton: If the force component vector along the path s n the same recton as the obect s splacement vector, then F s postve, so the wor s postve; but f the force component vector along the path s n the opposte recton to that of s mentone before, the potental energ s actuall the energ of the sstem of the obects an ther fels as a whole, but t s common to assgn t to part of the sstem for booeepng purposes as I o n ths boo. 6
Chapter 4 or an Energ the obect s splacement vector, then F s negatve, so the wor s negatve. Thus, f ou are pushng or pullng on an obect n a recton that woul ten to mae t spee up, ou are ong postve wor on the obect. ut f ou are pushng or pullng on an obect n a recton that woul ten to slow t own, ou are ong negatve wor on the obect. In the most general case n whch the component of the force along the path s contnuall changng because the force s contnuall changng (such as n the case of an obect on the en of a sprng or because the path s not straght, our how-to-calculate-t efnton of the wor becomes: For each nfntesmal path segment mang up the path n queston, we tae the prouct of the along-the-path force component an the nfntesmal length of the path segment. The wor s the sum of all such proucts. Such a sum woul have an nfnte number of terms. e refer to such a sum as an ntegral. The Relaton etween or an Moton Let s go bac to the smplest case, the case n whch a force F s the onl force actng on a partcle of mass m whch moves a stance r (whle the force s actng on t n a straght lne n the eact same recton as the force. The plan here s to nvestgate the connecton between the wor on the partcle an the moton of the partcle. e ll start wth Newton s n Law. Free o Dagram m F a a = F m Solvng for F, we arrve at: a = F m F = ma On the left, we have the magntue of the force. If we multpl that b the stance r, we get the wor one b the force on the partcle as t moves the stance r along the path, n the same recton as the force. If we multpl the left se of the equaton b r then we have to multpl the rght b the same thng to mantan the equalt. F r = ma r 63
Chapter 4 or an Energ On the left we have the wor, so: = ma r On the rght we have two quanttes use to charactere the moton of a partcle so we have certanl met our goal of relatng wor to moton, but we can untangle thngs on the rght a bt f we recogne that, snce we have a constant force, we must have a constant acceleraton. Ths means the constant acceleraton equatons appl, n partcular, the one that (n terms of r rather than reas: v = v a r Solvng ths for a r gves o a r = v v o Substtutng ths nto our epresson for above (the one that reas = ma r we obtan whch can be wrtten as = m = v v o mv mvo Of course we recogne the m v o as the netc energ of the partcle before the wor s one on the partcle an the m v as the netc energ of the partcle after the wor s one on t. To be consstent wth the notaton we use n our earl scusson of the conservaton of mechancal energ we change to the notaton n whch the prme smbol ( sgnfes after an no super- or subscrpt at all (rather than the subscrpt o represents before. Usng ths notaton an the efnton of netc energ, our epresson for becomes: = K K Snce the after netc energ mnus the before netc energ s ust the change n netc energ K, we can wrte the epresson for as: = K (4- Ths s nee a smple relaton between wor an moton. The cause, wor on a partcle, on the left, s eactl equal to the effect, a change n the netc energ of the partcle. Ths result s so mportant that we gve t a name, t s the or-energ Relaton. It also goes b the name: The or-energ Prncple. It wors for etene rg boes as well. In the case of a rg bo that rotates, t s the splacement of the pont of applcaton of the force, along the path of sa pont of applcaton, that s use (as the r n calculatng the wor one on the obect. 64
Chapter 4 or an Energ In the epresson = K, the wor s the net wor (the total wor one b all the forces actng on the partcle or rg bo. The net wor can be calculate b fnng the wor one b each force an ang the results, or b fnng the net force an usng t n the efnton of the wor. Calculatng the or as the Force-long-the-Path Tmes the Length of the Path Conser a bloc on a flat frctonless nclne that maes an angle wth the vertcal. The bloc travels from a pont near the top of the nclne to a pont, a stance n the own-thenclne recton from. Fn the wor one, b the gravtatonal force, on the bloc. F g = mg e ve rawn a setch of the stuaton (not a free bo agram. e note that the force for whch we are suppose to calculate the wor s not along the path. So, we efne a coornate sstem wth one as n the own-the-nclne recton an the other perpencular to that as F g = mg an brea the gravtatonal force vector up nto ts components wth respect to that coornate sstem. 65
Chapter 4 or an Energ F g = mg F g F g F F g = F cos = mg cos g = F sn = mg sn g g Now we reraw the setch wth the gravtatonal force replace b ts components: F g F g F g, beng perpencular to the path oes no wor on the bloc as the bloc moves from to. The wor one b the gravtatonal force s gven b = F = F g = mg(cos = mg cos hle ths metho for calculatng the wor one b a force s perfectl val, there s an easer wa. It nvolves another prouct operator for vectors (beses the cross prouct, calle the ot prouct. To use t, we nee to recogne that the length of the path, combne wth the recton of moton, s none other than the splacement vector (for the pont of applcaton of the force. Then we ust nee to fn the ot prouct of the force vector an the splacement vector. 66
Chapter 4 or an Energ The Dot Prouct of Two Vectors The ot prouct of the vectors an s wrtten an s epresse as: = cos (4-3 where, ust as n the case of the cross prouct, s the angle between the two vectors after the have been place tal to tal. The ot prouct can be nterprete as ether (the component of along, tmes, the magntue of or (the component of along, tmes, the magntue of, both of whch evaluate to one an the same value. Ths maes the ot prouct perfect for calculatng the wor. Snce F r = F r an F r s, we have = F r (4-4 means of the ot prouct, we can solve the eample n the last secton much more qucl than we before. Fn the wor one on the bloc b the gravtatonal force when the obect moves from pont to Pont. F g = mg 67
Chapter 4 or an Energ e efne the splacement vector to have a magntue equal to the stance from pont to pont wth a recton the same as the recton of moton (the own-the-ramp recton. Usng our efnton of wor as the ot prouct of the force an the splacement, equaton 4-4: r = F wth the gravtatonal force vector Fg beng the force, an beng the splacement, the wor can be wrtten as: F. = g Usng the efnton of the ot prouct we fn that: = F cos. g Replacng the magntue of the gravtatonal force wth mg we arrve at our fnal answer: = mgcos. Ths s the same answer that we got pror to our scusson of the ot prouct. In cases n whch the force an the splacement vectors are gven n,, notaton, fnng the wor s straghtforwar. The Dot Prouct n Unt Vector Notaton The smple ot prouct relatons among the unt vectors maes t eas to evaluate the ot prouct of two vectors epresse n unt vector notaton. From what amounts to our efnton of the ot prouct, equaton 4-3: = cos we note that a vector otte nto tself s smpl the square of the magntue of the vector. Ths s true because the angle between a vector an tself s 0 an cos 0 s. o = cos0 = Snce the unt vectors all have magntue, an unt vector otte nto tself els ( whch s ust. =, =, an = Now the angle between an two fferent Cartesan coornate as unt vectors s 90 an the cos 90 s 0. Thus, the ot prouct of an Cartesan coornate as unt vector nto an other Cartesan coornate as unt vector s ero. 68
Chapter 4 or an Energ 69 So, f = an = then s ust ( ( = ( ( ( = = = = The en result s that the ot prouct of two vectors s smpl the sum of: the prouct of the two vectors components, the prouct of ther components, an the prouct of ther components. Energ Transfer or vs. Center of Mass Pseuo-or I ntrouce the topc of wor b statng that t represents one categor of energ transfer to a sstem. s such, wor s energ transfer wor. There s a quantt that s calculate n much the same wa as wor, wth one subtle fference. I m gong to call the quantt center of mass pseuo-wor an I m gong to use a couple of specfc processes nvolvng a frctonless horontal surface, a sprng, an a bloc (an n one case, another bloc to stngush energ transfer wor from center of mass pseuo-wor. Suppose we attach the sprng to the wall so that the sprng stcs out horontall an then push the bloc towar the wall n such a manner as to compress the sprng. Then we release the bloc from rest an start our observatons at the frst nstant subsequent to release. Let our sstem be the bloc. The sprng pushes the bloc awa from the wall. The sprng transfers energ to the bloc whle the sprng s n contact wth the bloc. The wor one can be calculate as the ntegral of vector force ot vector nfntesmal splacement whch I'll loosel state as the ntegral of force tmes stance. The stance n ths case s the splacement (the nfnte set of nfntesmal splacements of the pont of applcaton of the force. Ths n of wor s energ transfer wor. It s the amount of energ transferre to the bloc b the sprng.
Chapter 4 or an Energ Now let's sconnect the sprng from the wall an attach the sprng to the bloc so that the sprng stcs out horontall from the bloc an agan push the bloc up aganst the wall, compressng the sprng, an release the bloc from rest. Let our sstem be the bloc plus sprng. The bloc goes slng off as before, ths tme wth the sprng attache. The wall oes no energ-transfer wor on the sstem because the part of the wall that s eertng the force on the sstem s not movng there s no splacement. However, we fn somethng useful f we calculate the ntegral of the vector force (eerte b the wall ot vector nfntesmal splacement of the center of mass of the sstem loosel state, force tmes stance move b center of mass. I'm callng that "somethng useful" the center of mass pseuo-wor eperence b the sstem. It's useful because our Newton s Law ervaton shows that quantt to be equal to the change n the center of mass netc energ of the sstem. In ths case the sstem gane some center of mass netc energ even though no energ was transferre to t. How that happen? Energ that was alrea part of the sstem, energ store n the sprng, was converte to center of mass netc energ. So what s the subtle fference? In both cases we are, loosel speang, calculatng force tmes stance. ut n the case of energ transfer wor, the stance s the stance move b that element of the agent of the force that s n contact wth the vctm at eactl that pont where the force s beng apple, whereas, n the case of center of mass wor, the stance n force tmes stance s the stance move b the center of mass. For a partcle, there s no fference. For a trul rg bo unergong purel translatonal moton (no rotaton there s no fference. ut beware, a trul rg bo s an eale obect n whch no bt of the bo can move relatve to an other bt of the bo. Even for such a bo, f there s rotaton, there wll be a fference between the energ transfer wor an the center of mass pseuo-wor one on the obect. Conser for nstance a bloc at rest on a horontal frctonless surface. You appl an off-center horontal force to the bloc for a short stance b pressng on the bloc wth our fnger. The wor ou o s the ntegrate force tmes stance over whch ou move the tp of our fnger. It wll be greater than the ntegrate force tmes the stance over whch the center of mass moves. Some of the wor ou o goes nto ncreasng the center of mass netc energ of the rg bo an some of t goes nto ncreasng the rotatonal netc energ of the rg bo. In ths case the energ transfer wor s greater than the center of mass pseuo-wor. Conclung Remars t ths pont ou have two was of calculatng the wor one on an obect. If ou are gven nformaton about the force an the path ou wll use the force tmes stance efnton of wor. ut f ou are gven nformaton on the effect of the wor (the change n netc energ then ou wll etermne the value of the change n netc energ an substtute that nto the wor energ relaton, equaton 4-: = K to etermne the wor (or center of mass pseuo-wor as applcable. There s et another metho for calculatng the wor. Le the frst metho, t s goo for cases n whch ou have nformaton on the force an the path. It onl wors for certan ns of forces, but when t oes wor, to use t, the onl thng ou nee to now about the path s the postons of the enponts. Ths thr metho for calculatng the wor nvolves the potental energ, the man topc of our net chapter. 70