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# Mechanics PHY 262

WFU

GPA 3.91

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This 66 page Class Notes was uploaded by Miss Lucienne Hamill on Wednesday October 28, 2015. The Class Notes belongs to PHY 262 at Wake Forest University taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/230735/phy-262-wake-forest-university in Physics 2 at Wake Forest University.

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Date Created: 10/28/15

Lecture 9 Wednesday Februaw 06 2008 1019 AM Please add the handout to your reading for Friday Last time we discussed friction and drag We will continue with a 39 39Cs We f39 39 with 1D motion includin linear drag L J Vet Jr 1 Wk Al 1 Kl i m V l ill1L 6 7 x T C Lecture9 Page 1 Lecture 8 Fndav Februarv 02 2007 11 17 AM Jumthj Mn SMMM ll 44 0 1 4 v03 L N R Lecture9 Page 2 Now let39s consider projectile motion How do we include friction into projectile motion Lecture9 Page 3 Lecture 9 Wednesdav Februarv 06 2008 10 29 AM Two ways to solve an equation like this 1 Perturbatively 2 Numerically Perturbative solution require that we be able to find a small parameter and to expand our equations for that parameter KT T W H m 91 First let s expand this in terms of a small parameter to 3rd order Then some algebra Next to last thing to do Check that its okay for k0 Replace TA2 by Tllt0quot2 Lecture9 Page 4 Wednesday January 15 2008 LCC her 1019AM Plank Mk 14 a 9mm See me MJrcr class 79 a I m T001 9 In h o H mm 7 COOKATn ul39e mny74ms mm vs min f mm 5 01 mm 3 400 W Wha 15 n vac rl Lecturel Page 1 Ln Irtr 0 mdhm mly oleAm vquw an Scalar w med 0 dab var limit nsI M Q rhlu qffu A w luau L W39 n 3 x x Y7quot 2 XL x 1 x1 N i a f was 7 retuer w Lecturel Page 2 4 AM 1 3k 4 C ml Wuhs quot r 7n 04 Ma 77 quot ag v 5 f 63 4 I y 093111 1030 x Y Nil VIN m1 er n 1 M 3 39 2x as 6 Jal r 1 zwdw Amt Lemuel Page 3 I RM 1 quot Ar hc SHIN 4 Vat h PM 0 r we 393 x quot ifkjyifpz an V an law Aim lc mnsnm mvt 4 saw I F 2 M IOJ39 44m 1 B 39a Q Ma a Wt m 7 Our Eightyh rt Ilka7m rsMg TWAIN m4 3 mm a sat 2741131 Lecturel Page 4 an m tha o gilt 3 l hpfsk O H Iq kj DWIm 01 Mn WI Wmltm 1554 N n mph inlfuj m I gum SW 7414 f M hoh art Sumh Wk l pqu rI39AVAV I r 44 f 1 1kg fquot IUIhghk 7 Cquot 1 39339 F 7 539 6 ch iFL M H Symbl Lizztni4 Z5mibl 1 L 39PW Chill yr ixIff tgn 01min 0 hpquot 7 ptrnu M n n1 5 MALI In CNS anHts H 1 art fq39l I4 Lecturel Page 5 flA IV 1 I1 Hf Cnt n TWD 1 4 647110 Al a iy mm A 0 A 39 4 Bf 74665 649 M E 39 gt i in SalIr a I WWM if 394 B m Lecturel Page 6 WA prJ af x C 1 4 x15 K c Lsan quot39k v WELZXI LBk AU CMMJ 74 X39n An nk zlka gm Lecturel Page 7 Lecture 4 Friday January 25 2008 1102 AM Last time we Practiced vector algebra Started a discussion of calculus and kinematics Today Continuation and discussion of curvilinear coordinates Reminder HW due next Friday at 4pm Turn it into Samrat Dutta39s Box in Olin 100 Working in groups is encouraged copying ie plagarism is not You must show your work No work No credit Your HW must be clearly organized and legible Lecture4 2 Page 1 Monday Januaw 22 2007 Le 303 PM Calful NJ h Vluhb enml VelaC b istkh k r chug f AWL pr y 1 0H cader Tm fiva39k n than an VIMj A 1 AM 1 JLI 739 IF So has new w Sgw t quot 9 34 015 mth an 3 33 40 wh Ts 07 W 139 KM AnH S m 33993 i SXJMM 5 9 wing away 1 139 V 4 I 30 1 3 1 39 f Ur Wm h L 0 2 7 r0391 12 H Yo Lecture4 2 Page 2 Monday January 22 2007 303 PM 1 Arm 5 10 mm v f ypguv Emlmq rez o 3950 hm 0 3 quot g 7 Palquot Ia39 m b in 6 wldd 0 arm I r xtvuse y F S he b CHM ME 50 We Jr U 5 3 5433 39Vquot re gtre skmalV NA a39im gisiiquot gar if 7 Srhm ri fLA Jx wlg 2 dls hhl V min In mm Q 391 W 75 ATs hnu 7 H1 m7 1 Al n c 4 1 tiffIn Lecture4 2 Page 3 Munday January 22 2mm 3 n3 PM c mm L A q Q2 C IS 2 5m arma 3 AL L 1 1 V K 1 139 3 05 L SphinxU muz quot1 u 4 0M X a 1 tap quotlam LT I I Cu mi 44 X 393 1 a 2 NM an xk39e ta r Iquot 39 l a L L a 39 a 39 t in 1 39 1 r r 3964 quotUh quot r r1 41 2 1 Lecture4 2 Page 4 Monday January 22 2007 303 PM VatHr AGNYIA 4 Q 3155 quotIndusb rlt wmL 9 r in Has 3lequ may symm a va 6 a o A 39 39 x r K 9 A A g l a A aquot quot A t A 1 W 340 Lecture4 2 Page 5 Lbs5 39 539quot Age inIsa 1 o U vsMl Wu U F3 Q J W Ca 1 Lix kw Lecture4 2 Page 6 Wednesday January 24 2007 851 AM Lech Lt ymml mic 9 MI OWN WAVE 3n 9 v76 lqr l esTuh uhA verb 7 t HM AMTV m c and ik Acrtwhyc 4 6 0 Ml eriwqivs A 39 A N A v b ZMWQ ltN QM Wm ign an mp hgy Lecture4 2 Page 7 LVIA 4 qu SprTlh HIM 415 9 L39AX 47 er y IJ II Lecture4 2 Page 8 Lecture 8 Friday February 02 2007 1117 AM MA fg imhm yr mm pry a Par wt 35 la ex 9 wxrk J vHw a39t rohrm vy Pattap m ICC Cur h 57Mpk Anof zrc id aim in ppnp 39 ll 7 I LACquot LLJ L Z icyr N H W E LectureS 2 Page 1 Last time we Went over Newton39s 3 laws Frames of reference Started on how to solve mechanics problems Reminders Two mathematics texts on reserve at ZSR for your edification LectureS 2 Page 2 LA m u l which 39fzdey m W W Imam n r M mung 6 4 75 WI km E k duh Aim agxl wl m put 1 d 39 ni i vk all but 40quot 29 rv 39 E A39a13tf vhH U a LectureS 2 Page 3 ll Aau Mud 15 Hm W ab k cr q Wt f rwq rt I A 5 hnlt XI 139 m Ln hm 0 inth m c Mr K Sm ha 50 WLUIP Hum v x 98 1 V M tltb LectureS 2 Page 4 I Md 61 m M gt NW 02 4 711 mquot M Lectures 2 Pages m n mu 1 1 1 44 m 3514 MM grim an W quot 1L 4p m I P5 28quot M44 MM m mATh 0 what J all quotWkn vk l UmU Hm km 0 V0 q End Jul m WHAT Ht rLT nl an at Mung I 9a 0 WM 5 A 59 Ht quot1quot 01 67 la quot JihadEm 75 HR ml vm w F1 5N 145N IHF1L0 01 393 7 r5 a so In 96 I 5M 2 kn6LM5 J 339 0c h ZM5 Mm a39 J l LectureS 2 Page 6 Vk kw 10 5 mt puma erch hilt Htk uL wll Jl slhl fr quot t 9 W4 4 E to Inga Znal IVflu Ml39kth MIL k 339 quot ruB Hn 2 M F V 019 ail V 12144q MimiMW u D 4 b m kv n2 nn 391 ranp hgtrr1 m ms r c 54 11va ms Fh1mhnn LectureS 2 Page 7 Mk than InL T w m M 79quot 4v Ania 39mlfhl39wzfnw W1 gt W a K kl EA 1 my xmvy 46 J0 l4 VIA 3 39 39 V 7 V01quotC l 4 my VOSo I t 0 3X0 LectureS 2 Page 8 J pul M 3 M T KMV Sam M iv n Fzmv MJ ICMV 0 AV E 4 77 K V l MUGe5 quotI c lc 7 ltv l L clCc V i t K t V V k V0 1 L c 39 km03 e K V LectureS 2 page 9 Lecture 3 Wednesday January 23 2008 1020 AM 1 We went over Vec roducts Last time Iq 39d J 2 Worked through some interesting products Showed that the cross product does not commute Worked roug 3 Defined Unit Ve tors and Ethan M 1 SD 4 Ipromised you that you will get to work through an identity that We usedlasttime l quotk 39 Sn SJ Sm S 0 Lecture3 Page 1 V 52 4 82 z mo 7l 139 f91 f 0amp4qu BM 0 3750 L 6 bunk IMAM Q J in I 9quot 11 5 6 5 11quot quot394 9 39 34r u x LectureS Page 2 Slalom olefinHm b 5 W M l byway Mth 4 Mat hams ms Lecture3 Page 3 6 Lecture3 Page 4 1 JAY L 7 15 L 01 3 1 2 I199 Mondav Januarv 22 2007 3 03 PM Ax A B nk a x 0 no rwa J I m Vgg I LClA rgAM an quotn t 1 9 9y D 2539 El53 is Cu hSIan Busch A uh nun an 4 ialrkna 51yko kinda an R mlrt us 3J4 44 mnnk m cm 41 n m He wluu V L a 5 m is iest hm h AlF PerQA f oarbvudt S 5 a k WP bum Hru fMquott 39397 74k PhyTGu q fquot M Lecture3 Page 5 Monday January 22 2007 303 PM n Calal an Cinench 427 WW4 Isthhk r ch 11quot JfSplUMA I V 17 430 I 3 a 5 Myh eh WHY TS 0 T 3 04 7 7 ML fquot r r EH SEECH T SDlt4J 5 9 My 1 V e 0 39 o C1 f A 5 WF M IH 6 m 7 H9912 5th 15 mm hn H 5 U Lecture3 Page 6 Lecture 11 MandaV February 11 2mm 11 us AM Lasttime we discussed conservation laws and energy Today nish up consetvative elds and energy ghuhm a by Laid K5 irks PAIL LmsnyAiw zxdsyssbwe FAx lgt Rem m 41111 Haw Leanne 17mm Page 1 Lecture 11 man my sz m 2cm 5 5va 52 ng a Tvrgw 9 F90 A 3i 3 F 3 14 am 5 JO 7 1 M HUN urliwn M an valml LAM WE 31 mm M We 4 V 33 O 17 a f ifquot LAL I IIJH 4110 yr 91 Wle Leanne 175x131 Page 2 anqzrm h r7 may 5475 WM 3 Tm ilmv r mg n m imil qu 77 k 3W0 1L pilnKA1Vf4D 0 507mm whim quot 7 W w m y W rm quotJ Athsrr f U wim b 3713mmth inwm u Mr M Lecmlljmalhgei Dxaw a paxabola on the bond and discuss motion W R NA in HQ p ukhm 44M 13 J m 5 Leauenjmx PagaA Classification of points Expand Ux in a series for simplicity let39s say around x0 If the first term is zero the we have an equilibrium point Pick out where they are on the graph Stable vs Unstable Lecturel lifinal Page 5 Turning points Places where particle turn ie Change direction of motion Where EUx so they depend on the energy Where are the turning points on the graph Lecturel lifinal Page 6 17 Swap Wk 3 gt197 Hm MAL m HHIIVL H wam r w 4r W4 M WW m 739 114 r LEE if 1 A w Hm MWM 1 yiri Mg am 439 my m M A M K M h A 1 H mm m m 9W mmwm W Qul h vt w 0M4 t1le MUM Lecture Lena Page 7 SMWK a AWL v5 WV u 74 F 39kx VLX H1 6 mm W Q MM M W Mk N l 0mm H MW 6 Vkm E JEH WMWM Lecture 133ml Page a Lecture 6 Wednsday January 30 2008 959 A Last time we discussed Vector differential operators Div grad curl and laplacian Then We started a practice problem Today more Vector calculus Integration more on physical pictures of Vector differential operators and then back to problems given time How do We integrate Vectors 4 mum WWW l V It 6 m Cali 3511 ALL My 74 uni ll h kil in 7H Lecture Page 1 Wednesday January 30 2008 10 04 AM Mm tomltu l uh lf u MM 3 1th Iur 4 mr AimsMd SMAJQ S E 1 85 F i 5 L4 45 Q1 gc E 15 a 3L rm THHL Law Law W dinkall tsplliuly 1a WV garAlli 0 I 44 N1 L 439 a h Mimi quotIn a 395405 m 391 Lecture Page 2 in 391 M Yttran S marl plWKL F 1 lx39fA V111 3 95 A1 6 Mn VL A n 39 Lu ANIMAL HA splvrxf ynhb a hm SV L AV 4 5 l Lecture Page 3 SEAL SCVLXJTM NW W Slim fYMplej 5 VJUUS pm 71 iww WW Lecture Page 4 A B C D Find the components of the acceleration vector in spherical coordinates Do it in parts Find how to express units vectors in spherical coordinates in terms of cartesian coordinates a I wrote this on the board for you last time Express the postion vector in spherical coordinates trivial Find the time derivatives for the unit vectors in spherical coordinates in terms of each other Find the velocity vector in spherical coordinates Lecture Page 5 Wednesday January 30 2008 1048 AM Lecture Page 6 Thursday February 3907 2008 655 PM Lectu re 1 0 T C V t2 5 k g 4w g k LlcT L4 ph l L gmquot 6 1 WV T 37 T 14 elm n y o quotleTL ltlzj39f1 4 k WT I r L vs 1 C K 0 quot VLVT 39 t VTZ JVLQT L L 992 1 l P PQA39W 044 quot CLT 37 OZVIKVTvT kk39r T L V 2 Lecture 1 O Page 1 Last time we looked at drag and perturbation theory Today we will nish perturbation theory and discuss conservation laws LecturelO Page 2 7 QekvM LOWTLE a 4 HCV Vl L I 39139 i 739ka 3 quotIr 39Vrrkl 7f fecr rlcfvt 7 so 1 Liv my a 1 x 73 HO 939 L Lab 41 CM 511 Hm rr T M 4 1 IJI MHzk T7 M TUUd TM 33 v11 31 WI 441 6quot L quot I 1 D kw gm1 U L9 77 Di l l WI 3 1 39 L if viii J 339 i it Wit1n Y Lch 11M LecturelOPage 3 11stqu Hva mem M f a ul mm mm Ammu H W A zmjma m M m L MM 10 a umLinxm slam Q vanllmlb Mr 1 AM mm In 9 msprydwn 11quot MW fmnmMvn I n W 5 Invarnmla MA rrf rm in h msnhhm H 7qu ma nah11 O mwrimu MA Hm mum eJ 1 rmizwnrm 3sz Tkm lws MU 19y LAW Lecture Page A Mm 50mm 9 m wnm v1 lVW MMV39H W 7 fat W I m Am 19 L n M SIM MAM w MDWm 3 MW M M MM o 77 is m M 1530 34311 g 3Q 1 MmHM If MJMIN MAW e m mm er A w W WW munm m C4 Maul ZLW 3395 quot 3 50 L pgww 1 I t quot I 5 Y icyL 1 5 solg po 5quot 7 134 Amn uk m w Mm m mm M M 739 MM HRH V5 4 LmW M 73W mm Mal w my LE mElUPagES What does it mean to be a conservau39ve eld QM Tws K 11 110 0 I5 Hm PM MyrnJIw mm ix A7 A q FAxk Fkiz xy by Hm Abld W1 A 1 r m Lecturel Page a Lemxem pug 1 ElU7 7 ENE0 41w 0 1F F 1 5 w 7 h sphrtul Maw 1 WI V 81 Q a L A c v N l quot 1 Mr O W a wquot LEM ML MAMA w to M La 4 W Lecture 2 Wednesday January 16 2008 331 PM Last time we discussed coordinate transforms vectors scalars and vector multiplication Major results Equation for the effect of a coordinate transformations on axes m X 39 gtl J39 3 SumN q l Dh M Definition of a Scalar and a Vector Scalar is under 139 t t M mm 3 MM m V MA I A Vector is a set of components t at transform as axes do 4 W Definition of Scalar and Vector Products with some conventions and notations 9 1 I v 3 39 LectureZ Page 1 Recall that we defined direction cosines 5 I5 I 9 H MIL gtJ J 1 K in Van39iYP i39i39lu W X JB Ma rix ix J a mn5 M4344 omfria N Then wt am whit X 2 NJ at I aminxtthn 3ltv5x TWO Mpk Qti mg Has MW W LectureZ Page 2 Whn rMW39 lmhe 1 V0 04195 A0 vtrbr am 501 I 539 B Kg M an parnah Mime an M VLMJI Hvr timSnn4Tm n 15mlart P am 2 UK 06 02 b 1 a A KW A K F LectureZ Page 3 T3 vegan W3 w u not h 4 qu an Ca 31 K W4 1 6 8 W4 hlw m hm ns39tt 11 5qu 01 1 MM Ase cm h Hl valvesquot g LectureZ Page 4 TQM for iaii Cm vllpmc 115 4 so Ax 114 2 Am 95 j LectureZ Page 5 Ufcquot 1 IMATWU Haw 10 w Admva a V544 in rm 394 03994 W47 0 than unil waftrs wwrn m m WW 1 m4 439 wk if kqu39 2gtK M Kfm LectureZ Page 6 Sink MA Mob m 0 aptLl Q 54 Atlumpoil 4 WA in 9915 V7quot Ava 71 W A i 1 3 4 16 Q Jk 9 RM Um cx tags I A C M LectureZ Page 7 Lecture 7 Friday February 01 2008 1106 AM By 4pm you will place your first 2 weeks HW in Samratt Dutta s mailbox in Olin 100 rm m l W 7 5M ms Z Last time we Went over integration of vectors including 2 vector V thms Looked at simulations of particles moving under different vector elds Looked at the divergence and curl of the different elds to get a physical picture of div d curl We are one with our mathematical introduction If you need more details or refreshing thoughout the course there are 2 texts on reserve at ZSR D Advanced Calculus Today we will discuss amp Newton 5 ads 9 Vlfl WV Galileo For Monday the reading assignment is long We will spend at least 2 lectures on Section 24 and Appendix E is so you can familarize yourself with the integrals tabulated Lecture7 Page 1 Newton s 3 laws First an litl in mrlivh 8 In r le vznl af u lll In 19 ah ly l mal 34 Second Note Newton39s second law tells us what inertial I d mass 5 Lecture7 Page 2 Hidden assumptions 1 Inertial framW a 96 M 0 2 Point particles Where the force depends only on the distance between them central forces V Examples inertial frames Noninertial 4 IN Mugl Zml tl uh lhy 0145 flail Mull 39n l l ml m owl Ml q i 3 M Central forces Noncentral i39 n j Cl4 5 quotl th with AV d 4M W MW m Lecture7 Page 3

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