Phys 331: Ch 4. Conservative Forces & Curvi-linear 1-D Systems 1

Transcription

1 Phys 33: Ch 4. Consevative oces & Cuvi-linea -D Systes Mon. / Tues. / Wed. /3 i. /5 Mon. /8 Wed. / Thus / Cuvilinea -D, Cental oce (.6) Hooke s Law, Siple Haonic (Coplex Sol ns) What (eseach) I Did Last Sue: AHoN 4p Review fo Exa Study Day Exa (Ch -4) HW4b (4.C-.) HW5a (5., 5.) Equipent o Inclined plane with cat, ass, sting, pully o Disc & ing to oll down plane o Ai puck with belt aound it and sting to deonstate v=w o Ppt and python of pola coodinates o Plotting Tutoial handout Coodinating fo Wednesday evening: I ll get goodies, Alan and I will bibe the eshen and Sophooes, we ll be in AHoN 6 at 4p. Note on coputational poble: I encouage you to fist wite the shot code that s in this Plotting tutoial; once that woks, you ll be eady to define you own functions to plot. Tie fo enegy consideations You eebe when we wee noinally taking a foce appoach, but we ephased the -D vesion of Newton s nd dv as a elationship between velocity and position (ug v v ) so we dx could say how speed depended on position without botheing with tie. Siilaly, now that we e noinally ug an enegy appoach, we still ephase it to pull out a tie. Hee s how that goes. If enegy is conseved, E T x, then: T x E x, which can be used to find the velocity as a function of position: x x E x /. (note: this ay look a little oe failia if I ultiply both sides by and ecognize the lefthand-side as being oentu p x E x. If you e taking quantu ight now, thanks to the elationship between p and wavelength, this coes in handy.) The velocity is two points: x dx dt, so dt dx. This can be integated to find the tie fo otion between x

2 t x dx x x x dx x x E x / o o The book goes though this; howeve, fo any situations this can be difficult to calculate because the integand goes to infinity as it appoaches the tuning point whee x. Even fo the siple pendulu, thee is no analytical solution (see Pob. 4.38). Enegy consevation is typically not a good way to get infoation about tie. That disclaie issues, hee is a tactable exaple. Exaple 3: (. of owles & Cassiday 5 th ed.) A paticle of ass is eleased fo est at x b and its potential enegy is x k x. (a) ind its velocity as a function of position. (b) How long does it take the paticle to each the oigin? (a) At x b, the kinetic enegy is T so the total enegy is E b k b. Since enegy is conseved: E k b T x. x x k x, so taking the negative oot because the potential attacts the paticle towad the oigin: An exaple of this is shown below. k x x. x b x (b) Since x dx dt, the tie uied to ove fo b to is: t dt t dx dx b k x x k b b x b se the integal (fo the font cove of the text): b x dx b x

3 y dy y y y y with the change of vaiables x by and dx b dy. The integal fo the tie becoes: t b k by b dy b by b 3 k y dy y b 3 k y y y t b 3 k b 3 k t b 3 8k Cuvilinea One-Diensional Systes: With foces and oentu, you wee dealing with vectos, so it was paticulaly ipotant to define a good, othogonal, coodinate syste in ode to keep tack of the coponents. But with enegy, you e dealing with scalas; you don t cae about the diection of the velocity, just its agnitude. So it becoes less ipotant, soeties unnecessay o even inconvenient to think stictly in tes of othogonal coodinates. Anothe diffeence between a foce appoach and an enegy appoach is that with enegy, potential enegy in paticula, we e geneally oe inteested in the change than in any specific value. What this adds up to is that we can be athe cavalie in descibing ou syste s enegy in tes of whateve vaiables ae convenient. Many systes can be descibe by one vaiable, o coodinate (a distance, angle, etc.) even if the otion is not along a line! An exaple of such a -D syste is a bead sliding along a cuved igid wie. If the distance along the cuve fo an oigin O. s O The speed of the bead is s and the kinetic enegy is: The foce along the wie (tangential) is: T. s 3

4 s tang. Taking it as a given that the bead stays on the sting, than any foces acting upon it, even if they depend upon, say the bead s elevation, should be able to be ephased in tes of the bead s location along the wie (ce, even elevation is a function of how fa the bead is along the wie.) So If all of the tangential foces ae consevative, a potential enegy can be defined as: s tang ( s) s s tang s Of couse, the total enegy of the syste depends on the kinetic and this potential: E T s. ds Phag the enegy-foce elationship the othe way, tang If d ds, the bead is at a point of uilibiu. If s is a iniu (axiu), the uilibiu is stable (unstable). s. Geneally, if a syste depends on only one coodinate s, then (as long as we e in an inetial efeence fae), thee s an uilibiu at d ds. (Counte exaple: can t do this fo a pendulu on an acceleating tain.) If two objects within you syste inteact in such a way that thei sepaation easued along this coodinate is constant, then thee s no net wok done by that inteaction, (block A exets foce fowad on block B while they ove fowad, we ll block B exets foce backwad on block A while they ove fowad no net wok.) This does not affect the total echanical enegy, so consevation of enegy can still be used if all othe foces ae consevative. 4

5 Do on CHALK boad so I can tuck it away and then etun to it. Exaple (I do ost of): Suppose the asses 6 kg and 4 kg ae initially at est. Ignoe fiction and assue that the ass of the pulley is sall. What will the speed of be when it hits the gound? = 35 We could use a nice catesian coodinate syste to descibe the otion of the two blocks, o aybe even two diffeent coodinate systes, one fo each block. Then again, we could siply call the distance they ove s (which is the sae fo both.) They ae tied togethe, so they ove the sae aount (until hits the floo). s h = s s = 35 The total echanical enegy of the syste is: (walk the though this pocess) Q: ist in vey geneal tes: E T T Q: next, in tes of speeds and heights: E v v gh gh Q: now in tes of the coon distance they ove, s and coon speed s : E s s gs gs Check signs: obseve that s potential becoes oe negative with s while s becoes oe positive. Okay, now that we ve expessed the enegy in tes of this vaiable, we ll use this expession to find the final speed. The initial condition of the syste is s and s, so E o. Consevation of echanical enegy gives: 5

6 s gs gs. Solving fo the speed and putting in the final condition s gives: s gs 9.8 /s 6 kg 4 kg 6 kg 4 kg /s Exaple / Execise: (Pob. 4.36) The ball (ass ) has a hole though it and slides on a fictionless vetical od. A light sting of length l passes ove a sall fictionless pulley and attaches to anothe ass M. The positions of the objects can be specified by the angle. (a) Wite an expession fo the potential enegy. (b) ind whethe o not the syste has an uilibiu position and fo what values of and M. Ae any uilibiu positions stable? b H h M (a) ind an expession fo the potential. ist, puely in tes of how fa the asses ae down fo the ceiling, h and H, what s the potential? gh MgH Now that we ve got that expession, ephase it in tes of the angle. b H h=b/tan( ) b/( ) M 6

7 The length of the sting fo the pulley to the ball is b. The heights ae h b tan and H l b (calling the total length of the sting l ). Since these distances ae easued below a efeence point, the PE is: M gh g tan b MgH tan Mgl Mg l b M cos Mgl. The last te is just a constant that will have no beaing on whethe a paticula angle is an uilibiu. (b) Okay, what about uilibia? The deivative of is: d d cos M cos It ll be convenient to put this all ove a coon denoinato, so athe than canceling off the s, I ll actually ultiply be anothe facto of on the last te d d M cos M cos cos Okay, at an uilibiu angle, d d M cos M cos M cos cos So it ust be that o M cos cos. M This only has solutions if M. When M, the answe is, which is not possible fo a finite length of sting. Theefoe, if M thee is an uilibiu point at: cos. M The uieent that M akes sense because in uilibiu the tension of the sting ust be ual to Mg, but upwad foce on is only a faction of the tension. Theefoe, ust be salle if they ae to both be in uilibiu. Take the second deivative of to check the stability: 7

8 d d 4 M M cos cos We aleady figued out that at the uilibiu point, the te in squae backets is zeo, so we e left with: d d 4 M M Since we know 9, o is positive. Theefoe, d d and the o uilibiu is stable. (we could go a step futhe and use that cos M to get an expession fo this second deivative, but we ve aleady figued out what we need to: that it s positive. Handling systes with ultiple coodinates (not necessaily Catesian, pola, etc.) will be easie ug Lagange s appoach (Ch. 7) Rigid Bodies: Befoe doing anothe one-coodinate exaple, I want to eind you about how to handle igid bodies with an enegy appoach. The end of the chapte discusses ulti-paticle systes. One coon situation is if they define a igid object. Then the total kinetic enegy can be ephased as the kinetic enegy of the cente of ass + the kinetic enegy of all the pats about the cente of ass: i vi. c i i c i i c I So, fo a igid body otating about an axis in a fixed diection (we genealize this in Ch. ): T T CM T ot MV I. Exaple : (elated to Ex. 4.9) If they stat fo est, which will ake it to the end of a ap faste, a cylinde (disk) o a thin ing? Do the asses o adii atte? h v Define the PE to be zeo at the botto of the ap, so initially o Mgh and finally f. The initial KE is T o and the final KE is 8

9 o an object that is olling without slipping, v R, T f T CM T ot Mv I. Pause and deonstate this with puck with sting. x = R while = whee R is the adius. The oent of inetia fo a cylinde is I cyl MR, so and fo a thin ing it is I ing v 3 T f. cyl Mv MR Mv 4 MR, so v T f. ing Mv MR Mv R R cylinde : ing : 3 4 T Mv Mv vc Mgh Mgh v 4gh 3 The cylinde will be going faste at any point along the ap, so it will each the botto fist. This esult does not depend on the ass o adius, just how the oent of inetia depends on the shape. o any ound object, it will be I shape facto MR. Question: How would a solid sphee copae to the othe two shapes? Answe: It would beat both because a lage faction of its ass is nea the axis of otation. (Its oent of inetia is I 5 MR, but you don t need to know that.) gh Single-coodinate pobles. Okay, ealie we d looked at this poble but with a assless pully; now let s say it has ass. Exaple : Suppose the asses 6 kg and 4 kg ae initially at est. Ignoe fiction, but assue the pulley is a cylinde of ass p kg and adius R.. Also, the ope does not slip ove the pulley. What will the speed of be when it hits the gound? 9

10 = 35 This syste is descibed by one paaete, s, the distance that the asses have oved. They ae tied togethe, so they ove the sae aount (until hits the floo). = s / R s s h = 35 The angula speed of the pulley is elated to the speed of the ope ( s ) by s R. The oent of inetia of the cylinde is I pr. The total echanical enegy of the syste is: E T T T s p I gs gs E s pr s R gs gs E s p gs gs The initial condition of the syste is s and s, so E o. Consevation of echanical enegy gives: s gs gs. p Solving fo the speed and putting in the final condition s gives: s gs p 9.8 /s 6 kg 4 kg 6 kg 35 kg 4 kg 3.7/s which is slight salle than what we found fo a assless pulley.

11 Cental oces: Speaking of othe coodinates, in this chapte the book intoduces (though doesn t yet do uch with) spheical pola. If a foce is always diected towad o away fo a fixed point ( foce cente ), it is natual to take that point as the oigin and to descibe the foce in spheical pola coodinates. These ae shown below. PPT. Note that the definitions of and ae usually evesed in ath textbooks! A cental foce can be witten as: ˆ. If the foce only depends on the distance fo the oigin and not on and, it is spheically syetic o otationally invaiant. A cental foce is consevative if and only if it is spheically syetic. In that case, the potential enegy only depends on : d, o o flipping the elationship aound: ˆ.

12 Now, even if we don t have a cental foce, it ay still be convenient to wok in spheical coodinates. In that oe geneal case, we have o Which begs the question of what is a diffeentially sall step in spheical coodinates? Rathe than saying d dxxˆ dyyˆ dzzˆ (baby steps in thee othogonal Catesian diections) We say, in tes of sall changes in,, and, is d dˆ d ˆ d ˆ. The iage below helps illustate that each te epesents a baby step in each of the thee othogonal diections. d d d d d d How do we expess the potential enegy foce elation in these coodinates? What s the opeato expessed in spheical coodinates?

13 As we ve aleady seen in this chapte, d d Then substituting in ou new expession, d dˆ poduct gives us d d d d d. d ˆ d ˆ, and pefoing the dot Now, we can tun this elationship aound, expess in tes of a deivative of just as we did when dealing with Catesian coponents. Hee we go: The little bit of wok done while the object just oves adially (d = d = ) is d, const d and so we d say that the patial deivative of with espect to d is Saying that elation the othe way aound, we see that the coponent of the foce is Siilaly, iagining oving the object by just swinging down a little futhe with, we see that O sweeping aound a little futhe with, we see that. Then expesg the foce in tes of these elations fo its coponents gives ˆ ˆ ˆ ˆ ˆ ˆ. Just as we did fo Catesian coodinates, we identify the opeation we e pefoing on as : ˆ Then we again have ˆ ˆ 3

14 Again, this ay see a little out of place in chapte 4, but we ll ake use of all this soon enough. Retun to this exaple if thee s tie Exaple 3: (Pob. 4.37) A assless (o vey light) wheel of adius R is ounts on a hoizontal axis. A ass M is attached to the i of the wheel and a ass is hung by a sting wapped aound the i. (a) Wite an expession fo the total PE as a function of the angle. Choose when. (b) ind any positions of uilibiu and discuss thei stability. (c) Suppose the syste stat at est at. o what values of the atio M will the syste oscillate? h R M H (a) As the wheel tuns though an angle, ass M ises by H R cos (this woks fo any angle!) and ass descends by h R (the aclength unwound). The total PE is: (b) The condition fo stability is: MgH gh MgR cos gr. d d MgR gr M. This only has solutions if M. If M, thee is one solution at. If M, thee ae two solutions, one with (M below the axis) and one with (M above the axis). The stability is deteined fo the second deivative: d d MgRcos. This is positive (negative) and the uilibiu is stable (unstable) fo ( ). o ual asses, the uilibiu at is a saddle point because d d 4

15 (phi) (c) o the given initial conditions, the total enegy of the syste is E. Plot the potential. The syste will oscillate if thee ae tuning points on both sides of the uilibiu. If M.75, this condition is et. /M=.7 /M=.75 /M= phi - phi - phi 5