CHAPTER If two balls swing in initial momentum is 2 mv and balls 4 and 5 will swing out.

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1 HPTER 4 4. Slutn: Mentu s cnsered. Therefre 5 th ball es ut wth sae elcty as st ball n pact. 4. If tw balls swng n ntal entu s and balls 4 and 5 wll swng ut. 4.3 Slutn: L n L L n 4.4 Slutn: M b M. kg kg 4 kg b 5 /s. /s b. b b b b b M 4 3 /s (elcty f bullet between & ) nseratn f entu fr entre eent. b b M 4 4. ( M ) b 4.3 /s

2 4.5 st pact between & ( ) ( ) 36. ft/s 6.67 ft/s fps 5ph 5 3 e e g g e e nd Ipact between & ( ) ( ) ft/s ft/s " " " " e e e There wll be anther pact f & ( ) " " e Slng: ( ) ( ) [ ] ( ) ( ) [ ] " " e e e e ( ) ( ) [ ] ( ) ( ) [ ] ft/s ft/s N addtnal pacts.

3 4.6 Fr equatns deelped n 4.5 nd pact: 3 rd pact: ft/s 5.5 3( ) ft/s 5.5 " " 4.4 ft/s ( ) ft/s ( ) ft/s 4.7 Slutn: e Usng equatns deelped n Prb ft/s nd pact " " " ft/s 8 Ths culd hae been btaned by equatng ntal entu t fnal entu. 4.8 Slutn: f ( ) 3.76 ft/s f 4kg 5 6 L L x y z x y z 6 3

4 5 x Slng L L yelds: 6 y z The fllwng MTL fle s entered and saed: r[3;-;-];[5;4;-];r[3;5;];[5;-5;4];r[8;;];[-4;;-]; ;;4; rp[7;-;-];p[-;-;];rp[;-;-];p[-;.5;-]; L*** p(/)*(l-(*p*p)) H*crss(r,)*crss(r,)*crss(r,); HH-(*crss(rp,p)*crss(rp,p)) rpcrss(p,h)/(*dt(p,p)) Runnng ths MTL fle yelds: L p H rp

5 4.9 Slutn: r c r r r r c H c ( r r ) ( r r c ) ( r r c ) 54 H c c c ddng the fllwng lnes t the MTL fle fr 4.8: rc(*r*r*r)/() c(***)/() Hc*crss((r-rc),)*crss((r-rc),)*crss((r-rc),) Runnng agan yelds: rc c Hc

6 4. Ths s the slutn drectly as t wuld be typed n Mathcad (usng apprprate equal sgns) 3 r 5 4 r 35 5 r E E 5 7 r 5 5 r D 45 5 r c 5 ( r r r r D r E ) 4 r c 9 c D E 3 c D 9 H r r r r D D r E E H H c ( r r c ) ( r r c ) ( r r c ) K ( r D r c ) D ( r E r c ) E H c Next the MTL cde fr akng the sae cputatn s gen The fllwng MTL fle s entered and saed: r[3;5;-];[-5;5;-];r[-;-5;-5];[;-5;-]; 6

7 r[4;35;];[-8;;6];rd[;-45;5];d[3;-9;]; re[-5;5;4];e[-9;-5;-7];rc(/5)*(rrrrdre); cde; Hcrss(r,)crss(r,)crss(r,)crss(rD,D)crss(rE,E) Hccrss((r-rc),)crss((r-rc),)crss((r-rc),)crss((rDrc),D)crss((rE-rc),E) Runnng ths MTL fle yelds H Hc Slutn: In Mathcad type: In MTL type: The fllwng cands cpute r: EDU>[3;-;3];H[-;3;3]; EDU>r(/4)*crss(,H)/dt(,) r H r H 4.75 r

8 4. Usng the drect ectr slutn, a Mathcad cputatn yelds: In MTL, the cputatn s: EDU>r[-.5;;-.5];H[-;6;6]; EDU>r-(/)*crss(r,H)/dt(r,r) r r H r H r r nseratn f entu crate wth car, () 5 cs3,( ).3 /s The cuplng requres e, thus, ( ),( ), r.7 /s (nus t left), The elcty f the rtatng ass at any te s f f z r r cs r snj k M M r r sn r csj x y Mentu f the syste n the -drectn s cnsered. 8

9 M ( M ) M at, M nstant r sn nstant nstant ( M ) M ( M ) sn 4.5 Slutn: H r 3 ˆ ˆj 4kˆ Let r x ˆ yˆj zkˆ [ kˆ ] General H ( 4y z) ˆ ( 3z 4x) ˆj ( 3y x) Fr H t hae nly î cpnent and z 4 3 x y 3 H x 8 3 x 8 3 x x Therefre there s n r that wll prduce ths cndtn. 4.6 Slutn: ( ) ( cs 45 ), f 4, cs /s f f f 9

10 4.7 Intal entu f syste Mentu durng walkng x M / t M / M L ( ) H M M / t M / M / M / Mtn f bat: t 5 ft 4.8 In Mathcad type: 3 r 3 4 r F F 6 r 3 3 F L 8 L 7 H r ( ) r ( ) r ( ) H 33

11 r c r r r.667 r c.5 H c ( r r c ) ( ) ( r r c ) ( )K ( r r c ) ( ) H c The MTL cde fr cputng the slutn s Type the fllwng MTL fle t cpute the desred alues 3;;;r[;3;];r[4;;];r[;;]; [3;-;];[5;;];[-3;-;]; F[;;];F[-3;-6;];F[3;-;]; L***; Hcrss(r,*)crss(r,*)crss(r,*); rc(*r*r*r)/() Hc*crss((r-rc),)*crss((r-rc),)*crss((r-rc),) Runnng ths MTL fle yelds; rc Hc

12 4.9 In Mathcad enter: a c F F F a c dh c ( r r c ) F ( r r c ) F ( r r c ) F dh c The sae slutn n MTL s: dd the fllwng tw lnes f cde t the MTL fle f prble 4.8 ac(fff)/() dhccrss((r-rc),f)crss((r-rc),f)crss((r-rc),f) Then runnng ths yelds: ac dhc

13 4. Slutn: y N 3 E - x, / g ph ˆ 93.3 ft/s ˆ 3 / g ( sn 3ˆ cs 3ˆj ) 5 ph ( sn 3ˆ cs 3ˆj )ft/s ( ) w ˆ.57 6ˆ j 3 3 W w 87ˆ.86ˆ j ft/s && x x& 87 x 87t && y y&.8 y.8t && z 3. z& 3.t z 6.t, te t ht grund. t 35.5 s x,5 ft y ft Wreckage wll be,5 ft East and 99.4 ft Suth. 4. The center f ass fllws the fllwng path && x && z g x& 5 z& gt t x 5t z g t 3 t.473s.365 x grund The crdnate f c.. s (.365,,) 3

14 Therefre x 5 3. y rdnates f the -kg partcle are (.393,-.5,) 4. t axu alttude, the entu s L 6 cs8 ˆ 4.ˆ (there s n ertcal entu). fter the explsn: 5 L Therefre: Fr Prble 4.5 Intal energy 3 3. ( 36.7) 6,744 ft lb V 3x V 3y V 3z 4. The fnal energy 3 3. ( 4.6) 5 3. ( 6.6) Energy lss 9,49 ft lb 4.4 Fr Prble 4.7 Intal energy 6,744 ft.lb 8 3. Fnal energy ( 3.76 ) 3,5 ft.lb Energy lss 39,4ft.lb ( ) 43,54 ft lb 4

15 4.5 Fr prble 4.8, as entered n Mathcad: 4kg The MTL cde fr ths s: E ( ) E ( ) E 8.5 E 43 E E 85.5 J The fllwng MTL fle akes the desred calculatns: ;;4; [5;4;-];[5;-5;4];[-4;;-]; [5;-6;-];[-;-;];[-;.5;-]; E(/)*(*dt(,)*dt(,)*dt(,)) E(/)*(*dt(,)*dt(,)*dt(,)) dee-e runnng ths fle yelds E 8.5 E 43 de 85.5 where the energy s gen n Jules. 5

16 4.6 The fgure belw llustrates the dependence. V / Lnear entu n the x and y drectn yelds V sn cs / cs cs / sn Energy yelds V ( ) / The abe yelds three equatns fr, /, (ddng each by the ass): V sn / cs () cs sn () / V / (3) Fr () / tan. Substtutn nt the reanng equatn yelds: V / V V cs / / cs tan V / cs / 6

17 ut V / sec / / V sec / tan sec V / tan sec sec V tan 3 tan sec V V sec 3 tan / V 3 cs tan, whch expresses as a factr f and V sn 3 cs sn, whch expresses as a functn f 4.7 Slutn: V / Lnear entu cnseratn n the x and y drectn s V 3 3 sn V cs / / cs sn () () Energy cnseratn s V ( / ) (3) The three equatns ay be wrtten: 7

18 8 V sn cs 3 / ( ) cs sn 3 / ( ) / 3 V (3 ) Fr ( ) tan 3 / (4) Fr (4) and ( ) V cs / (5) Substtutng (4) and (5) nt (3 ) / / / / / / / / / tan sec sec sec tan 3 sec sec tan 3 cs tan 3 V V V V V V ne slutn wuld be / but ths leads t n cllsn as and V. Therefre 3 tan tan tan 3 sec 3 tan tan sec tan 3 3 tan sec tan 3 sec tan 3 4 sec / V V V V V V

19 4.8 Slutn: The centers fr an equlateral trangle. nseratn f lnear entu n ectr fr yelds r fr cpnents sn 3 cs 3 sn 3 cs3 ( ) () Energy cnseratn yelds: (3) Fr () Fr () cs3 Therefre 4 ( 4 cs 3 ) cs3 cs 3. cs cs3 cs Usng Mathcad drectly (frst the ntal guess): 3 β 6 deg β deg 3 9

20 Gen cs sn ( β ) 8 3 cs( β3 ) ( β ) sn( β ) Fnd (, 8, 3 ) where the elctes are gen n /s. In MTL t s st cnenent t set up a gernng MTL fle that cntans all the predefned cnstants and then calls the teratn rutne newt. t sle the set f equatns gen n fle set4_9.. The gernng MTL fle culd be f the fllwng fr, here t was als desgned t utput the results f the teratn neatly n the screen: % Excercse 4.9 % nte: "newt." and "set4pt9." ust be n acte drectry % gen elctes and angles 3; % [/s] beta 6; % [deg] beta3 45; % [deg] % nte: cnert t radans beta beta*p/8; beta3 beta3*p/8; % unknwns:, 8, 3 % ntal guesses IG [; ; ]; % nuber f teratns t ; sln newt(set4pt9,t,ig,[],[],,beta,beta3); fprntf(\nfr %3.f /s, the slutn s:\n,); fprntf(\n %6.3f /s\n,sln()); fprntf(8 %6.3f /s\n,sln()); fprntf(3 %6.3f /s\n,sln(3)); functn qset4pt9(p,,beta,beta3) % p(); 8 p(); 3 p(3); q zers(3,); % set f nnlnear equatns q() *cs(beta) 8 3*cs(beta3) - ; q() *sn(beta) - 3*sn(beta3); q(3) ^ 8^ 3^ - ^; % EDU>Exc4_9 Fr 3. /s, the slutn s:

21 .877 /s /s 3.99 /s EDU> 4.3 Wrkng drectly n Mathcad: Gen cs sn β β 6 deg 45 deg ( β ) 8 3 cs( β3 ) ( β ) sn( β ) Fnd (, 8, 3 ) gan, newt. s requred t sle the nnlnear set f equatns. % Excercse 4.3 % nte: "newt." and "set4pt3." ust be n acte drectry % gen elctes and angles 5; % [/s] beta 6; % [deg] beta3 45; % [deg] % nte: cnert t radans beta beta*p/8; beta3 beta3*p/8; % unknwns:, 8, 3 % ntal guesses IG [; ; ]; % nuber f teratns t ; sln newt(set4pt3,t,ig,[],[],,beta,beta3); fprntf(\nfr %3.f /s, the slutn s:\n,); fprntf(\n %6.3f /s\n,sln()); fprntf(8 %6.3f /s\n,sln()); fprntf(3 %6.3f /s\n,sln(3)); functn qset4pt3(p,,beta,beta3) % p(); 8 p();

22 3 p(3); q zers(3,); % set f nnlnear equatns q() *cs(beta) 8 3*cs(beta3) - ; q() *sn(beta) - 3*sn(beta3); q(3) ^ 8^ 3^ - ^; % EDU>Exc4_3 Fr 5. /s, the slutn s: 3.9 /s 8.76 /s /s 4.3 Fr Mathcad: Gen sn 4 β 3 deg β 3 deg β 3 deg ( β ) cs( β ) sn( β ) ( β ) sn( β ) cs( β ) cs Fnd (, ), The ther slutn crrespnds wth 4 β 3 deg β 3 deg β 3 deg.73 3

23 Gen sn ( β ) cs( β ) sn( β ) ( β ) sn( β ) cs( β ) cs Fnd ( ) Ths slutn s nt pssble because t wuld late cnseratn f entu durng the frst cllsn. The slutn n MTL s based n agan usng newt. t sle the nnlnear set f equatns. % Excercse 4.3 % nte: "newt." and "set4pt3." ust be n acte drectry % gen elctes and angles 4; % [/s] beta 3; % [deg] beta 3; % [deg] beta 3; % [deg] % nte: cnert t radans beta beta*p/8; beta beta*p/8; beta beta*p/8; % unknwns:,, % ntal guesses IG [.5; ; ]; % nuber f teratns t ; sln newt(set4pt3,t,ig,[],[],,beta,beta,beta); fprntf(\nfr %3.f /s, the slutn s:\n,); fprntf(\n %6.3f /s\n,sln()); fprntf( %6.3f /s\n,sln()); fprntf( %6.3f /s\n,sln(3)); functn qset4pt3(p,,beta,beta,beta) % p(); p(); p(3); q zers(3,); % set f nnlnear equatns q() *sn(beta) *cs(beta) *sn(beta) - ; q() -*cs(beta) - *sn(beta) *cs(beta); q(3) ^ ^ ^ - ^; % EDU> Fr 4. /s, the slutn s: 3.33 /s.579 /s 3

24 3.666 /s 4

25 4.3 The angular entu abut the ntal pstn s zer. Therefre the fnal angular entu abut the pnt s zer. r r r / / / ( ) ( ˆ sn 3 ˆj ) ( ˆ cs3 ˆj ).73 cs3 3 sn 3 3ˆ.73ˆj xˆ yˆj 4.33 Wrkng n Mathcad type: Gen ( ) (.73x 3y) kˆ r.73x 3y sn 7 β 3 β 3 β 3 ( β ) cs( β ) sn( β ) ( β ) sn( β ) cs( β ) cs Fnd (,, ) The MTL cde s dependent n the ersn f MTL aalable t yur students. Ths s all dscussed n pages 98, 99 and f the MTL Suppleent, whch dscusses the slutn f a syste f nnlnear algebrac equatns. Use the cde gen there wth q()a*sn(3*p/8)b*cs(3*p/8)c*sn(3*p/8)- q() -a*cs(3*p/8)b*sn(3*p/8)c*cs(3*p/8) q(3)a^b^^-^ and adjust the rest f the cde accrdngly. Use ectr f ntal guesses t be x[;;] 5

26 4.34 Slutn: g l a a g M d dt H g g [ ] ( l a) cs ga cs a ( l a) ( l a) cs ( l al a ) d dt l g ( l a) al a cs d dt α Slutn s dered fr the free-bdy-dagra: M d dt H (l-a) a g(l a)cs( ) gacs() d [ dt a (l a) ] g(l a)cs() (l al a )α d dt g(l a) l al a cs() 4.35 Slutn: M g ( l a) cs c ( l al _ a ) α d dt H ( l al a ) α [ g( l a) cs c] 6

27 4.36 The Mathcad slutn s.8 t. t α t (, ) L L ( ) a.3 al a (, ) α t g 9.8 c.4 t [ g( L a) cs c] The dapng ceffcent c was chsen by tral and errr untl the apprprate tn was btaned. Wrkng n MTL use an MTL fle and the enu fr dfferent alues f untl the desred slutn s btaned. T sle the nnlnear dfferental equatn we use de45.. The syste n frst rder fr s gen n sys4_36.. % Excercse 4.36 % clear wrkspace clear all % defne te nteral t ; % ntal te [s] tf 8; % fnal te [s] % defne ntal cndtns x ; % [rad] ; % [rad/s] % defne syste prpertes g 9.8; % [/s^] ; % [kg] L ; % [] a.3; % [] c.4; % [N--s] % defne useful arables l *a - L; l L^ - *a*l *a^; % sle DE [t,x]de45(sys4_36,[t tf],[x ],[],l,l,,g,c); % plt dsplaceent n degrees! fgure(); clf; plt(t,x(:,)*8/p,b-); ttle([(l-a),nustr(l-a), a,nustr(a),,...,nustr(), kg c,nustr(c), N..s]) xlabel(te (s)) ylabel(dsplaceent (deg)) 7

28 grd functn xp sys4_36(t,x,flag,l,l,,g,c) % DE n frst rder fr xp [x(); -c//l*x() - g*l//l*cs(x())]; (L-a).7 a.3 kg c.4n..s 8 Dsplaceent (deg) Te (s) 8

29 4.37 Slutn: d M H dt T d dt ( 3a ) (.5) ( 5) α 3.75α T 3 Desred α α6 α.667 rad/s Therefre the requred trque s T 6.5 N 4.38 Slutn: L L ( ) Energy s cnsered L L 3 ( 4) ( 3 ) 3 /s () 3 ()() 3 3 ()( 3) gl( cs ) 7.9 9

30 Slutn: l g dt d H M ( ) α sn sn cs l l l l l g g d d g dt d g 4.4 Wrkng n Mathcad: ( ) ( ) ( ) t t L g t t t g L α α, cs,

31 T sle the nnlnear dfferental equatn we use de45.. The syste n frst rder fr s gen n sys4_4.. % Excercse 4.4 % clear wrkspace clear all % defne te nteral t ; % ntal te [s] tf 8; % fnal te [s] % defne ntal cndtns x p; % [rad] ; % [rad/s] % defne syste prpertes g 9.8; % [/s^] 4; % [kg] L ; % [] % sle DE [t,x]de45(sys4_4,[t tf],[x ],[],,g,l); % plt dsplaceent n degrees! fgure(); clf; plt(t,x(:,)*8/p,b-); ttle([l,nustr(l), xlabel(te (s)) ylabel(dsplaceent (deg)) grd,nustr(), kg]) functn xp sys4_4(t,x,flag,,g,l) % DE n frst rder fr xp [x(); g/l*cs(x())]; 8 L 4 kg 6 4 Dsplaceent (deg) Te (s) 3

32 3 4.4 Slutn: N L g dt d g L N H M sn α µ cs l l Nr g k 4.4 In Mathcad the slutn prceeds as fllws: Select 4 L r.3 µ k.9 ( ) ( ) ( ) ( ) ( ) µ α r N Lg L g L N t t t k, cs, sn,.. ( ) t t α,

33 5 deg t Ntce that the scllatn dap ut but at a slw rate. Whle the dapng ceffcent s relately large, t s a lng pendulu and dapng s sall. The MTL cde fllws. T sle the nnlnear dfferental equatn we use de45.. The syste n frst rder fr s gen n sys4_4.. % Excercse 4.4 % clear wrkspace clear all % defne te nteral t ; % ntal te [s] tf 5.93; % fnal te [s] % defne ntal cndtns x p; % [rad] ; % [rad/s] % defne syste prpertes g 9.8; % [/s^] 4; % [kg] L ; % [] r.3; % [] uk.9; % sle DE [t,x]de45(sys4_4,[t tf],[x ],[],,g,l,r,uk); % plt dsplaceent n degrees! fgure(); clf; plt(t,x(:,)*8/p,b-); % ttle([(l-a),nustr(l-a), a,nustr(a),,... %,nustr(), kg c,nustr(c), N..s]) xlabel(te (s)) ylabel(dsplaceent (deg)) grd functn xp sys4_4(t,x,flag,,g,l,r,uk) % nral frce N *L*x()^ *g*sn(x()); % DE n frst rder fr xp [x(); -(uk*n*r)/(*l^)*sgn(x())g/l*cs(x())]; 33

34 4.43 Slutn: ˆ csˆ d ρv dt F ρv net. ( cs ) 4.44 Slutn: F (.6) ( cs3 ) 6.8 N 4.45 Slutn: T d a d f dt dt ( )6 9 N T 5. d a dt 4.46 Slutn: d d T ( 4 ) 3 dt dt D.5g cs T D g sn Ma a x M d 3 g dt x ( sn.5 cs ) 4.47 Slutn: F ( 4.) [ ( sn3 )] 94 lb 4.48 The geetry f the cneyr belt β β.8 34

35 Ipulse-entu free-bdy dagra x t W t y x y t y t && x && y x& y& gt gt x t y te x.5 t.s g y&.96/s (.37ˆj ) 3..98ˆ.5ˆ.96ˆj Lnear Ipulse entu x t ( ) x (.84) 56.8 N t kg/s Ipulse entu n ertcal drectn. t y y y y y y t N W t [.3 (.96) ] ( 9.8) ( 3.75) Ment pulse angular entu: Snce the structure s at rest, ents can be taken abut any pnt: hse Slutn: [ ] y t( 3.5) W t 3.(.37)3.5 3.(.98)..5(.) y [ ( ) ] 58 N y 8 N 35

36 z.3 /s ˆ ( j sn 45ˆ k) cs 45ˆ x y Nˆ k t F t gˆ j t.3ˆ.77ˆ j.77ˆ k ( ) Nˆ k F 96.k ˆ.5ˆ 3.54ˆ j 3.54ˆ k F.5ˆ 3.54ˆ j N 99.5ˆ k 36

37 4.5 Slutn: F ρ ( ) π 6 F. 55 N 4 W g 96 N F W a a.795/s up 4.5 Slutn: 4.5 Slutn: π 6 F N a 5.46 /s M t 4 ( r ) r.5ˆ.3ˆ j.3ˆ k 6.577ˆ.577ˆ j.577 k ˆ ( ) Mg.748 N 4.53 Slutn: ( )( )( ˆ 3 4 cs3 sn 3 ˆj ) 78ˆ ˆ N F j y The frce n the bat s equal and ppste. x 4.54 Slutn: The upper part f the chan s at rest. x ass ρ unt length Ipulse hange n lnear entu 37

38 ρgx t ρ ρgx ρx ( x x)( ) ρx ρx ρ x ρ x ρx d dt ρ dx dt ρx 3 hgher rder Dfferental equatn 4.55 Slutn: d x dt g x x Fρgx If Integratng: F a ρgx ρla ρl d x g x dt L d g x dx L g L x gx L when x L, ( x x ) gl 4.56 Slutn: 3 4 /s && x x& 4 cs3 x 4cs 3 t && y g y& gt 4sn 3 gt y 4sn 3 t.5 t.386s when y 38

39 x & F F g b /s y& t F g /s (.5ˆ 3.464ˆ 5.787ˆj ) 78.56ˆ 3.45ˆN j 4.57 Slutn: F F w frean ( csˆ snj) ρv ˆ F w 39

PHY2053 Summer 2012 Exam 2 Solutions N F o f k

PHY2053 Summer 2012 Exam 2 Solutions N F o f k HY0 Suer 0 Ea Slutns. he ree-bdy dagra r the blck s N F 7 k F g Usng Newtn s secnd law r the -cnents F a F F cs7 k 0 k F F cs7 (0 N ( Ncs7 N he wrk dne by knetc rctn k r csθ ( N(6 cs80 0 N. Mechancal energy