Chapter 4 Circular and Curvilinear Motions

Transcription

1 Chp 4 Cicul n Cuilin Moions H w consi picls moing no long sigh lin h cuilin moion. W fis su h cicul moion, spcil cs of cuilin moion. Anoh mpl w h l sui li is h pojcil..1 Cicul Moion Unifom Cicul Moion Risi I is moion wih cclion icion of h loci chngs; Th cclion is lws ppnicul o h ph of h moion. Th cclion lws poins ow h cn of h cicl of moion (i s no n moion of consn cclion!). This cclion is cll h cnipl cclion. 1

2 Th mgniu of h cnipl cclion is gin b c / Nwon s scon lw ics h F m m Th Pio, T, is h im qui fo on compl oluion. c T π /

3 Empl 1. Conicl Pnulum Fin n pssion fo n h pio T. Answ: Equilibium in icl icion: T cos mg 0 Unifom cicul moion in hoizonl pln: m T sin m Lsin Combin h bo wo qns. n h pio τ Lg sin n π πl sin Lcos π Lg sin n g 3

4 Empl. Bnk Row A c of mss m ls consn sp oun bn of ius on o bnk n ngl. Th cofficin of ficion bwn h c s s n h o sufc is nλ, wh λ <. Show h: () If h c ls wih no nnc o slip g n (b) If h c is bou o slip ouws, g n ( λ) (c) If h c is bou o slip inws g n ( λ) 4

5 Answ: () In his cs h is no ficionl foc cing. Th quion m of moion R sin, n R cos mg 0 So g n (b) Th ficionl foc cs own h slop n ks is limiing lu, i.. F µr Th quions of moion / α F R mg m R sin µ R cos, n R cos µ Rsin mg 0 R mg (sin µ cos ); R m cos µ sin Hnc n g(sin µ cos ) cos µ sin g(n µ ) g(n n λ) 1 µ n 1 n n λ g n ( λ). 5

6 (c) Th ficionl foc cs up h slop n ks is limiing lu µr. Th quions of moion / R F R sin µ R cos m R cos µ Rsin mg 0, n α mg Thn g(sin µ cos ) cos µ sin g(n n λ) 1 n n λ g n ( λ). 6

7 Empl 3. Th Roo ω Roo is qui ofn foun in musmn pk. I is hollow f clinicl oom h cn b s N s o o bou h cnl Along im icl is. A pson ns R h oo n sn gins h wll. Th oo gull incss is oing sp up o ps on n h floo blow h pson is opn ownw. Th pson os no fll own bu mins oing wih h oo. Dmin h minimum sp which h boom floo cn b opn n h mn is sf, gin h sic ficionl cofficin µ. Th oo s wll mg 7

8 Answ: Th mn os wih h oo n h cnipl foc which cs on him is poi b h wll s noml foc, N m R. As h mn os no fll own, h ficionl foc in h upw icion blnc wih his wigh,.g. mg µn. Hnc, mg m µ s ( R ) n so gr µ s No: I os no pn on h mss of h mn. 8

9 Nonunifom Cicul Moion (moing ing sp in cicul ph) In iion o il componn of cclion, h is ngnil componn of cclion of mgniu / ; Th ol cclion is hn Th mus b n foc on picl h is inclin o : F F F n F, sponsibl fo cnipl F n ngnil cclions, spcil. 9

10 Empl 4. Kp ou on h bll A smll bll of mss m is ch o h n of co of lngh R n s ino moion in icl cicl bou fi poin O. Dmin h ngnil cclion of h bll n h nsion in h co n insn whn h sp of h bll is n h co mks n ngl wih h icl. Answ: Th ngnil foc on h bll is F mg sin m g sin Th il foc on h bll is T mg cos Rg T mg cos m R F 10

11 A op n boom, hfo T op mg Rg 1, T boom mg 1 Rg Th minimum sp whn h bll is h op, such h h nsion on co is non-nishing T op mg Rg 1 0 op gr 11

12 Angul isplcmn, loci n cclion In scibing cicul moion, on m us h s of quniis o spcif h s of moion, h Angul isplcmn, ngul loci ω, n ngul cclion, α. s, ω lim, 0 ω α O s No: s, ω, n α. 1

13 Kinmic Equion of Cicul Moion consn ngul cclion: Cicul Moions Lin kinmics ω ω α 0 u 1 1 ω0 α s u ω ω α 0 u s 13

14 . Pln Pol Cooins Fo picl moing on pln, ins of cngul (, ) cooins, i is ofn connin o scib h moion b using h pln pol cooins (, ). Dfin wo Uni Vcos in h pln pol cooins (, ), which ppnicul o ch oh (simil o cos ( i, ) j in (, ) cooins): cos i sin j sin i cos j So posiion P(, ) cn lso b spcifi s P(, ) in pln pol cooins, o in co fom, wh ( ). ê P ê 14

15 15 obiousl, Fuhmo Th loci co is n sin cos ( ) / n 1 j i j i sin cos cos sin ( ) ( ) ω is h il componn of loci long, n is h ngul componn long. ê ê

16 16 Th cclion is ( ) ) ( ) ( ) ( ) ( n gin, n h il n ngul componns of h cclion.

17 Cicul moion in pln pol cooins R ( ), R R Rω, ( R cnipl R ) Rω ( R Rα ngnil R ) 17

18 Empl 5. Bug wlking on oing whl A lbi s ou o wlk consn sp u long spok of oing whl wih consn ngul loci ω. Th pln of h whl is hoizonl n is oion is is fi icl. Assum h lbi ss off 0, 0 0. Fin h mgniu of h loci n cclion of h bug n oh im. Epss is locus in ms of n. Answ: Fom Fom ω, on hs u ( ω ) u ( uω) ( ) ( ), on gs 4 4 ω 4u ω ω u 4u ω ( No : 0; u, ω) 18

19 19 Finll, ( ) ( ) j i u j i u sin cos sin cos ω ω u u ω ω sin cos so, u u / ( ) ( ) sin cos Thn, u u ω ω

20 .3 Cuilin Moion Gnl moion is no lin, no cicul, bu long cu ph. Th cclion chngs fom poin o poin. A n insn, such co quni cn b sol ino wo componns bs on n oigin h cn of cicl (s figu) cosponing o h insn: il (noml) componn long h ius, n ngnil componn ppnicul o h cicl. n 0

21 1 Th ph is ih spcifi b n quion ( ) f n Rius of Cuu ( ) is: k ρ In Csin cooins s k

22 O h ph cn b pmiz s in which cs, h Rius of Cuu is: ) ( ) ( ( ) k ρ 1 3

23 3 Fo h l cs, h loci n cclion is sigh fowl i: n ) ( ) ( ) ( ) (, Mgniu Dicion n, n

24 4 Fo h fom cs, on is ofn gin n/o. Th ol loci n cclion is hn ibl: wh n, Mgniu Dicion n, n

25 Empl 6. A c on s ck gos ino un scib b , 0, wh n msu in ms n in scons. Fin h cclion of h c 3.0 scons. Answ: Hoizonl cclion: s , so / 0.6 n / 1. A 3.0, 3.6 Vicl cclion: 0, so / 0 4 n / 4 A 3.0, 4 Now (3.6 ( 4) ) 1/ 5.38 m/s n n -1 ( / ) n -1 (-4/3.6) 31 o fom h posii -is. 5

26 Empl 7. A picl mos long h ph scib b 4 (cm). Th hoizonl loci is consn 3.0 cm/s, fin h mgniu n icion of h loci whn h picl is h poin (-1, -1). Answ: I is gin h / 3 cm/s. Now w n o fin (knowing h 4, n -1) / (/) 4(/ ) 0 ( 1)(3)4(3) 6. So w h 6 cm/s. Th mgniu of h loci is hn: [( ) ( ) ] 1/ 6.7 cm/s. Th icion of h loci is gin b: n -1 ( / ) n -1 (6/3 ) 63.4 o 6

27 Empl 8. A j pln ls long icl pbolic ph fin b 0.4. A poin A, h j hs sp of 00 m/s, n is incsing h of 0.8 m/s. Fin h ol cclion of h j poin A. Answ: 7