Funny & Jokes

Phys 331: Ch 3. Angular momentum for Systems of Particles 1. p 3. So, the total angular momentum of the cloud, relative to the star, is simply PDF

Phy : Ch. ngua oentu fo Syte of Patice Mon. 9/4 Tue. 9/5 Wed. 9/6 Thu. 9/7 i., 9/8 Mon. / Tue. / Wed. / i. /5.5 ngua Moentu fo utipe patice 4.-., 4.9 Wok & Enegy, oce a a Gadient, Patice nteaction Science
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Phy : Ch. ngua oentu fo Syte of Patice Mon. 9/4 Tue. 9/5 Wed. 9/6 Thu. 9/7 i., 9/8 Mon. / Tue. / Wed. / i. /5.5 ngua Moentu fo utipe patice 4.-., 4.9 Wok & Enegy, oce a a Gadient, Patice nteaction Science Pote Seion: Hedco7~9p Cu of Conevative oce, Vaying Potentia, -D yte Cuviinea -D, Centa oce 5.-. (.6) Hooke Law, Sipe Haonic (Copex So n) What (eeach) Did Lat Sue: HoN 4p eview fo Exa HWb (.D-.G), Poject Topic HW4a (4., 4.B) HW4b (4.C-.) ngua Moentu fo Syte of Patice: 9.4 ngua oentu of utipatice yte Now that we ve got a ea hande on angua oentu of a inge object, we e going to banch out and conide yte of patice. When deaing with the otion of yte of patice in the pat, we ve found it convenient to epaate out ou deciption of the cente of a otion and the intena otion of the patice; thi tie wi be no exception. Conide ou od favoite the intetea ga coud. Whie each individua patice i ipping thi way and that, et ay that a a whoe it obiting oething, pehap a ta; heck, pehap ou coud i a poto-panet itef in the poce of coaping to fo a new panet. So, et ook at the tota angua oentu of the coud about the ta. o a tat, each individua patice ha it own inea oentu (eative to the ta otion). Since we e inteeted in the angua oentu about the ta, we need to conide each patice poition eative to the ta. p p p So, the tota angua oentu of the coud, eative to the ta, i ipy L L L p p p o hee, the book bee-ine it fo the toque-angue oentu eation fo a utipatice yte. dl dl dl (ngua Moentu Pincipe with) Muti-patice yte.ppt o Change in angua Moentu fo each patice. Thi i the patice poition vecto coed into the u of foce on the ext patice. ext ext ext ext.ext.ext.ext Hee, we ve epoyed the Pincipe of ecipocity fo foce. Now adding the thee equation to each othe give the ate of change in tota angua oentu of the yte of patice. Now, fo centa foce, uch a we ve thu fa encounteed, the toque due to inteaction between the patice u to, jut a the foce theeve do. Thi eave jut the extena foce to woy about. d L o L L ext ext ext d L tot ext ext ext Jut ike fo a point object. net. ext Siiay, in the event that thee i no net toque, the we have Conevation of ngua Moentu fo the yte of patice. d Ltot Note: the net toque i not ipy the net foce coed by oe epeentative poition vecto; it the u of individua toque which ae the net foce on each individua pat coed by the poition (eative to the axi) Unifo otation & oent of inetia coon pecia cae of angua otion fo a yte of patice i when they have the ae angua peed. That ake it convenient to ephae angua oentu in te of that haed w athe than individua v, and facto out the w. What you e eft with i the object oent of inetia. Suppoe pin an object, ay a ba; we ca the axi of otation the -axi. So we decibe the otion in poa coodinate. Let focu on jut one oe of the ba, on patice. The poition of a patice in the object i ˆ ˆ. f the object otate with an angua peed but aintain contant agnitude of and (going aound and aound, not in o out, up o down), then it veocity i ipy v ˆ ˆ. The angua oentu of the patice i: p v ˆ ˆ ˆ. The co poduct of the unit vecto ae ˆ ˆ ˆ and ˆ ˆ ˆ (ee the diaga beow), o: ˆ ˆ. ˆ ˆ ˆ x y f the axi we e conideing i an axi of yety, then the inwad coponent of the angua oentu fo thi oe cance with that fo anothe; howeve, we ti have the -coponent to conide. The coponent of the angua oentu fo a patice i, o the coponent of the tota angua oentu, uing ove a patice that ake up the object i: L N N We can wite L if we define the oent of inetia,, by: N (ubcipt becaue we coud have conideed oatation about a diffeent axi) whee i the ditance of a fo the axi. The othe piece often cance fo an object (eebe ˆ i not in the ae diection fo a point). We wi be concened with the whoe anwe in Ch.. o, then we have L,. What i oent of inetia conceptuay? o t pay an anaogou oe to a fo inea otion. o o L p v L p v net. net. netia i the idea that an object going to keep on doing what it doing. The bigge the inetia, the hade it i to change it tate of otion. o inea otion, a epeent inetia, but fo angua otion, the oent of inetia doe that job. The bigge the oent of inetia, the bigge the toque i equied to change the angua veocity. You can ee that in the equation, eating toque and angua veocity. o ook at the N definition of oent of inetia,. the oe a and the futhe it i fo the axi of otation, the hade it i to change it otation. Befoe we ue oent of inetia in decibing otion, et think about deteining the oent of inetia. Exape : ind the oent of inetia of a thin od of a and ength about it cente. Chooe the x axi to be aong the od with the oigin at the cente. Divide the od into a piece of ength dx (a hown beow). The a of each piece i dx. dx x The oent of inetia i: cente dx x dx x x dx cente x Execie: What i the oent of inetia about the end of the od? o thi execie, ove the oigin to the end of the od. end dx x x dx 4 end 4 4 Notice: Thi i an exape of the paae axi theoe.. about. axi about. Thi i age than the eut fo Exape becaue oe of the a i fathe fo the axi. Exape : (Pob..) ind the oent of inetia of a phee of a M and adiu an axi though it cente. So, we e going to need to integate ove the whoe phee, that ean we tat out by defining diffeentiay a oe of a, and then uing ove the; howeve, if we e ceve about the geoety of thoe oe, we can teaine the poce. cente M Vo dvo Q: on what hape ie a point that contibute the ae to the oent of inetia that i, that have the ae? Divide the phee into thin hoow cyinde (thickne d ) becaue the each cyinde wi have the ae ditance fo the axi (ee beow). d h The height of each ing i h. The voue of a ing i h d, o it a i: 4 d M M 4 d 5 The oent of inetia i: M 5 M d d M d. Make the change of vaiabe q and dq d to get: M q q dq. dd and ubtact q in the integand: M q q dq M q q dq M 5 M M q q M Notice that pheica poa coodinate whee not ued. Now fo oe ipe ue of oent of inetia in decibing otion. Exape : Suppoe fou 6-kg chiden ae tanding at the edge of a ey-go-ound that i pinning at p. ue the ey-go-ound i a unifo -kg dik of adiu. What wi happen if the chiden each wak unti they ae fo the cente? ue that thee i no fiction. i=p =? ad/ f=? chid.i = chid.f =/ The oent of inetia of the ey-go-ound i M. The oent of inetia fo each chid (teating the a patice) i c, whee i the ditance fo the cente. Conevation of angua oentu (the diection if fixed, o jut conide the agnitude) give: 6 L initia L fina 4 co o 4 cf f Cance out on both ide to get: M 4 o M 4 f. M 4 o M 4 9 f. Sove fo the fina angua peed: f M 4 kg 4 6 kg M 4 9 o kg 4 6 kg 9 p 7.8 p. The ey-go-ound wi pin aot fou tie a fat! ngua Moentu about the CM: The eut that L ext wa deived fo an inetia efeence fae (unacceeated). Howeve, the ae eut hod if L and ext ae eaued about the cente of a, even if the CM i acceeated! o exape, the otion of a pojectie can be decibed by the otion of it CM and otation about the CM. nothe way to put that i, you can beak the pobe up into the otion of the cente of a and the otion about the cente of a even if the CM i acceeating. Exape 4: Suppoe two a phee of a ae attached by a ight ting of ength b. Suppoe oeone hod one of the phee and pin the othe aound at a peed v o in a pane that pae though vetica. f the phee ae eeaed when the ting ake an ange with epect to vetica, decibe the ubequent otion. CM v o The diaga above how the yte at the intant of eeae. The CM wi tave on a paaboic path a if it wee a patice of a with initia veocity: vo V vo. 7 n othe wod, the initia veocity of the CM i v o at an ange above hoionta. t the intant of eeae, the ange of the ting eative to vetica at the CM i eated to the initia peed by: o: v b, o v o b. Thee i no net toque about the CM (the weight give oppoite toque), o the angua oentu about the CM and ae contant. What foow i oe additiona dicuion of angua oentu taken fo y Phy note. So thi i an extended efehe. ngua oentu about cente of a Thi uch i faiia, but now et note that each poition vecto coud be e-expeed a the u of two vecto, one fo the ta to the coud cente of a, the othe fo the cente of a to the dut patice. o exape: p CM CM p p So, we can ewite ou expeion fo the coud angua oentu about the ta and epaate out the contibution due to obiting the ta and the contibution due to otating about it own cente of a. p p p p p p p p p p p p p p p ptot L L L L Ltanation Lotation Tanationa /obita. The fit te now ook jut ike the angua oentu fo a point object, ocated at the cente of a and with the net inea oentu of the coud. Thi te i what we d ca the tanationa o obita angua oentu, fo it decibe the whoe body up and tanating o obiting about the ta. otationa. The econd te decibe how the coud otate about it cente of a, thu it caed the otationa angua oentu. 8 o Condenation of coud to panet: Conevation of angua oentu. Now ay biion upon biion of yea have paed and the dut coud ha indeed coaped into a oid panet. Q: How houd the otationa peed of the newy foed, a, dene panet copae with that of the age, diffue coud it ued to be? : Much age. Q: Why? o : Jut a with the pinning figue kate, negigibe extena toque, o the angua oentu i coneved. deceae, inceae to aintain L ot =. o igid body pecia cae. Now that we ve got a faiy igid body, the panet, we can take ou expeion fo the otationa angua oentu a tep futhe. Each of the patice of the panet now ha the ae angua veocity about the cente of a, we ca that the otationa angua veocity ince it coepond to the panet otating on it axi, ot. Lot p p p Lot ot ot Lot ot So, Lpanet ta p. p p. p. ot Now, we ve been taking about dut coud, panet, and ta, but thoe pecific wee jut fo the ake of concetene- the eation we ve eaoned out appy oe boady. Say you pin a baton, then the angua oentu of that baton about you head can be deteined uing the above eation. Deo: Spin baton obit head and otate about cente of axi. o Deo: Spinning Baton otate baton aound e o that it away point at the Q: What kind() of angua oentu doe it have? o : Jut obita: L tan. baton head b. h pb. o otate it o that it keep ynched up with e Q: What kind() of angua oentu doe it have? o : obita and otationa: Lb h b. h pb. b ot o, cick once Both poitive otate and twi with oppoite diection of pin. Q: Doe it have oe o e angua oentu now? o : Le. What i the diection of the obita angua oentu? What i the diection of the otationa angua oentu? Lb h b. h pb. b ot One poitive, one negative 9 o, cick a few tie to get angua oentu vecto pointing in oppoite diection. o Vecto: eebe, angua oentu, ike inea oentu, i a vecto quantity, and it add vecto-wie. So, fo exape, you coud have two puck pinning with equa and oppoite angua oentu, then the cobined yte woud have angua oentu. That iia to two cat unning at each othe with equa and oppoite inea oentu. dl tot dl dl dl dl n te of Cente of Ma Tanation and otation. o Lat tie, we found that it wa convenient to ephae the tota angua oentu of a yte in te of it otation, o pin, and it tanation, o obit. So we can beak the eft-hand ide up that way. Siiay, we can beak the ight-hand-ide up in te of toque about the cente of a and toque about the efeence point,. ext ext ext dlot ext ext ext dlot ext ext ext ext ext ext dlot net. ext ext ext ext dlot Tie fo HW quetion Next two cae: Wedneday Wok, Kinetic & Potentia Enegie iday Conevative oce & -D Syte
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