Phys 331: Ch 4. Conservative Forces & 1-D Systems 1. 1 W b

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1 Phys 33: Ch 4. Conservaive orces & -D Sysems Thurs. 9/3 ri., / Mon. /4 Wed. /6 Thurs. /7 ri. / Curl of Conservaive orce, Varying Poenial, -D sysems Curvilinear -D, Cenral orce 4.9 Energy of Paricle Ineracion (.6) Hooke s Law, Simple Harmonic (Comple Sol ns) HW3, Projec Topic HW4 Announcemen: SPS Come Oserving rip wih CSUSB Pah-Independence of Work: The condiion ha he line inegral of he force is pah independen is equivalen o he condiion ha he line inegral for any closed loop is zero as is illusraed elow. In he diagram elow, if W a W, hen W closed loop ecause W W. W a W W a W Then W d r closedloop Think aou wha i would ake for his o no e he case. Say, on he upper ranch he force poins up and a lile o he righ, and on he lower ranch i poins down and a lile o he lef. Then posiive work is done headed along he upper ranch from o and posiive work is done headed along he lower ranch from o clearly hose wo posiive works aren going o add o. Visualize ha force as a funcion of posiion, i migh look somehing like W W a

2 Tha force curls around from poining one way o poining anoher clearly he depends on he y componen of he posiion. Clearly, if a force curls around like his, hen he work is pah dependen, and he work done in raveling a closed pah isn necessarily. Now we ll make ha idea mahemaically concree. Crossing he del operaor ino a force would mean muliplying say d/dy y, i.e., seeing how much he componen of he force depends on he y-componen of he locaion; looking a he illusraion aove, i s ha kind of dependence ha characerizes curl. So, we define curl. of. f de ˆ ˆ y ˆ z y z y y z y y z ˆ z z y ˆ y y z ˆ (if one mus, and someimes, indeed one mus, one can ranslae his ino polar or spherical coordinaes; see he ack inside cover) As you migh hen epec, if his doesn evaluae o, if he force has curl, hen he work is pah dependen. Wha you migh no guess is ha his operaion direcly relaes o he closed-pah work inegral in a really simple way: from vecor calculus, Soke s Theorem is closed loop dr area enclosed ˆ n da, where ˆ n is normal o he area. Those of you who ve already had E&M have seen his proven, hose of you who haven ye, have somehing o look forward o. I s a fun derivaion, u we ve go oher fish o fry oday. Taking i as a given, hen if and only if he curl of a force vanishes:, is he work pah-independen. If he force also depends on only he posiion, i is conservaive and a poenial energy can e defined. So, le s play wih his. Eercise: Which of he following is/are a conservaive force? a a yˆ ˆ y z ˆ z yˆ ˆ y z 3ˆ z

3 The curls of he forces are: a de de ˆ ˆ y ˆ z y z z ˆ y z ˆ ˆ y ˆ z y z z ˆ y z 3 The second force is conservaive and he firs one is no. 4.5 Time Dependen Poenial Energy The only poin I wan o make here is ha he auhor unfolds his sory in an unnecessarily myserious way and par of ha has o do wih his poor choice of associaing poenial energy wih a single ojec raher han he wo ineracing ojecs and his compleely overlooking he sep of defining his sysem. The poenial energy is shared y he charges on he conducing sphere and he disan charge and also wihin ha sysem are all he ineracions of he charges on he sphere wih each oher. Looking a ha sysem, he slow reconfiguring of he charges ha were iniially on he sphere changes he poenial energy erms for heir ineracions wih he disan charge, u if nohing eernal is ineracing wih he sysem, hen he oal energy in he sysem is consan. Alernaively, if you look a he disan charge as he sysem, hen is ineracion wih he charges on he sphere is an eernal one, and hus no validly represened as a poenial energy of he charge. The eernal ojec does work on he charge, and is kineic energy (he only kind i classically can have all o iself) changes. ˆ z 4.6 Linear -D Sysems: Of course, everyhing s simpler in -D. If a paricle is consrained o move along a line (call i he ais), work and energy are simple. The work done y a force is an ordinary inegral: W d. As long as a force only depends on he posiion (and no velociy v or ime ), i is conservaive! If we choose U o a he reference poin o, hen he associaed poenial energy is: U d. o Relaive o ha a he reference poin. (Noe: like momen of ineria, angular momenum, and posiion, poenial energy values are relaive.) (assuming ha he oher pary o he ineracion implied y ha force isn moving oo) The force in -D is relaed o he poenial y: 3

4 Graphically, you can hink of a plo of he poenial energy vs. posiion as a roller coaser rack. There will always e a force on an ojec in he downhill direcion (see diagram elow). U() du d. ( ) ( ) A paricle is in equilirium if he ne force on i is zero. when du d, which corresponds o a minimum or maimum of U vs.. An equilirium is sale if a small displacemen from i resuls in a force ack oward he equilirium. This occurs where U is a minimum and d U d. An equilirium is unsale if a small displacemen from i resuls in a force away from he equilirium. This occurs where U is a maimum and d U d. (A saddle poin where du d and d U d is unsale ecause a small displacemen in one direcion will resul in a force ha pushes he paricle furher away.) or a sysem ineracing y only conservaive forces, he oal energy: is conserved. E T U Poins where U E are known as urning poins. A hese poins, T and v. Ineracing paricle will no separae eyond his disance, o locaions where U is larger, ecause conservaion of energy would require ha he kineic energy e negaive. Regions where U E are classically allowed and hose where U E are classically foridden Energy diagrams or he ime eing, his relaion is mos useful in ha i helps us o inerpre energy diagrams. Rollercoaser Track. o An energy diagram is like a roller coaser rack. By jus looking a a rollercoaser rack, you can ell how he cars will move where hey ll speed up and where hey ll slow down, and where, if i weren for a chain pulling hem, hey would sop and roll ackwards. I s he same wih a energy diagram. Say you saw he following srech of roller coaser rack. 4

5 Energy o Q: If you imagine placing he car high on he lef slope, wha would i do, wha direcion would i go, where would i e speeding up, where would i e slowing down? A: Speed-up o he righ unil i his he oom, hen slow down on i s way furher righ unil i his he lile peak, hen gradually speed up as i goes righ. o Q: Wha is he lowes saring poin on he lef ha would allow he car o sill clear he hump on he righ? A: Jus even wih he hump. o Poenial Energy Curve. If, insead, his curve ploed he graviaional poenial energy of he rollercoaser (which varies proporional o elevaion afer all), hen wha would he orce Poenial Energy relaion say? On he far righ, where he slope of he poenial energy is srong and negaive, wha should he force e like? The force is large and posiive (o he righ). Where he slope is? The force is : an equilirium poin. Where he slope is posiive? The force is negaive (o he lef). Moral: Your inuiion aou his rollercoaser rack holds perfecly for a plo of poenial energy vs. posiion. Nuclear Poenial. The poenial energy of a nucleus and a proon plos like his. If he proon says near in, i is araced y he srong force, u if i ges far enough ou, he srong force apers off, u i sill feels he elecric repulsion of all ha oher proons. Since K + U = -mc + E and mc is generally consan and E is consan in an isolaed sysem, we have ha K + U is consan. K= r r K= K+U = Cons r K(r ) K(r ) 4-poenial-energy-well.py Choose differen K+U lines, see how he kineic energy varies as he paricle moves across he poenial conour. Bound Saes = we say a paricle is in a ound sae if i can escape he poenial well. This is he case for he proons and neurons in nuclei. One would require 5

6 addiional energy o surmoun he arrier and escape. Similarly, any all you can hrow is graviaionally ound o he Earh i will only ge a few meers ino he air efore is kineic energy comes o, and i falls ack down again. In his way, he idea of energy helps us o see wha a sysem can and canno do. Eample : (Pro. 4.9) The force eered y an ideal spring wih is lef end fied is k, where he spring is unsreched a. The parameer k is called he spring consan. (a) If we choose U a he equilirium, wha is he corresponding poenial energy? () Suppose a spring is hung verically from he ceiling wih a mass m aached o he oher end and consrained o move verically. If y is he displacemen downward from he equilirium posiion, find he oal poenial energy. (a) If ( ), he spring is sreched (compressed) eers a force o he lef (righ). The poenial energy is: U spr o d k d k () Le o e he equilirium posiion and y e he disance from equilirium as shown elow. unsreched o y = o + y In equilirium, he spring force and weigh mus alance so k o mg or o mg k. The oal poenial energy is (for graviy, i decreases as he mass moves downward): U k mg k o y mg o y U ky k o mg y k o mg o U ky consan 6

7 The consan has no physical consequence, so he ehavior is he same as for a mass aached o a spring ha moves horizonally wihou fricion. You could define away he consan. Eample : A -kg paricle moves in one dimension under a force: csin a, where a = m -, = N/m, and c = N. The argumen of he sine is in radians. (a) ind he poenial energy wih he reference poin a he origin so ha U. Skech he poenial and show he classically allowed and foridden regions if he oal energy is E = -.5 J. () Idenify he hree poins of equilirium and deermine if each is sale or unsale. (a) The poenial is found y inegraing he force (wih a minus sign!): U U d c sin a d c a cos a c a cos a c, a. The consans a,, and c are all one and each erm is in Joules when is in meers. The graph elow shows U vs.. or large, he dominaes he oscillaing erm. The dashed line is E = -.5 J and he allowed (A) and foridden () regions are laeled. The paricle is only allowed o e where E U ecause E T U and T mus e posiive A - A (m) () The poins of equilirium are where du d. This gives he ranscendenal equaion: c sin a The soluion corresponds o an unsale equilirium ecause d U d. The oher wo soluions can e found approimaely y making successive guesses o ge.896, which are sale ecause d U d. 7

8 If energy is conserved, E T U, hen: T m E U, which can e used o find he velociy as a funcion of posiion: E U. m The velociy is d d, so d d. This can e inegraed o find he ime for moion eween wo poins: o d In pracice, his can e difficul o calculae ecause he inegrand goes o infiniy as i approaches he urning poin where Ý. Even for he simple pendulum, here is no analyical soluion (see Pro. 4.38). Energy conservaion is ypically no a good way o ge informaion aou ime. m Eample 3: (. of owles & Cassiday 5 h ed.) A paricle of mass m is released from res a and is poenial energy is U k. (a) ind is velociy as a funcion of posiion. () How long does i ake he paricle o reach he origin? o (a) A, he kineic energy is T so he oal energy is E U k. Since energy is conserved: E k T U E d U m k, so aking he negaive roo ecause he poenial aracs he paricle oward he origin: An eample of his is shown elow. k. m. 8

9 () Since d d, he ime required o move from o is: d d m d m k k Use he inegral (from he fron cover of he e): y dy y sin y y y d wih he change of variales y and d dy. The inegral for he ime ecomes: m k y dy y k y dy y k sin y y y k sin k 8k Ne wo classes: Monday Curvilinear -D Sysems & Cenral orces Wednesday Muliparicle Sysems 9