INTRODUCTION BIOECHANICS HUAN OTION ANALYSIS, THIRD EDITION SOLUTIONS ODD-NUBERED PROBLES b D. Gdn E. Rben, PhD, CSB Schl Humn Knec, Une Ow Cpgh 013 (eed 11 Decembe 013) INTRODUCTION (p. 1) Cnen c e ken m Tble 3 n pge 8. () (d) (e) () (g) (h) () (j) (k) (l) 350 0.4536 81 1557 N 6.50 / 0.4536 14.33 lb. 168.5.54 48 cm (10 100) /.54 394 nche mle 609 km 1hu 1000 m 70.0 33 m/ hu 1mle 3600 1km mle 609 km 80.0 18.7 m/ hu 1mle 8.35 1.54 55 cm 440 0.9144 40 m (800 / 0.9144) 3 60 ee 50.0 609 80.5 km 0 / 609 154 mle 00 81 4 newn lb 0.4536 kg 50 114 kg 1 1lb W mg 114 81 115 N Thu, he 100 N pen wegh me hn he 50 lb. pen. 45. 45 1 540 n. 540.54 1376 cm 1716 m Thu, 175 m lnge hn 45 ee.
UNDAENTAL CONCEPTS (p. 6) cθ n θ θ n 1 () θ n 50 1 350 430 cm (350 / 50) 54.5 θ n.00 1 000 (000 /.00).4 kn 6.6 θ n 10.00 1 0.0.4 m/ ( 0.0 / 10.00) 116.6 (d) θ n 1000 1 000 (000 /1000) 40 N 64 (e) θ n 0 1 ( 30) 40 m/ (350 / 50) 54.5 () θ n 00 1 100 181 kn (100 / 00) 108.4 () 0 c 0 0n 0.7 m/ 10.57 m/ 10.00 c 30.0 10.00n 30.0 8.66 N 00 N 100.0 c 40 100.0n 40 70.7 m/ 70.7 m/ (d) 00c(π / ) 0.0 m 00n(π / ) 00 m (e) 0 c160.0 0n160.0 5 N 8.55 N () 0c(π / 4) 0n(π / 4) 168 m/ 168 m/ Sum de lw gee hn lengh hpenue. () ( 30.0 0.0, 50.0 40.0) (10.00,10.00) cm ( 400 500, 700 400) ( 100.0, 300) N R (750, 1000) N 150 N 51 deg 1.75 5 93 km
3 SPATIAL (3D) COORDINATE SYSTES (p. 30) P c α P / P cβ P / P c γ P / P c θ n P P α c 1 ( / ) P β c γ 1 () P cα cβ 350 / 6 0.56 c γ 50 350 50/ 6 450/ 6 450 0.40 0.73 6.5 P 1 3 74 cα /74 0.535 cβ 1/ 74 0.67 c γ 3/ 74 0.80 P 10 ( 0) 34 cα 10/ 40.7 0.46 cβ 0/ 40.7 0.49 c γ 34 / 40.7 0.836 40.7 (d) P 1000 000 5000 cα 1000/5477 0.186 cβ 000 / 5477 0.365 c γ 5000/5477 0.913 5477 (e) P 5 cα ( 35) 5/ 68 55 0.358 cβ 35/ 68 0.501 c γ 55/ 68 0.788 68 () P cα 5/ 5 0.1961 cβ 15/ 5 0.588 c γ 5 15 0 / 5 0 0.785 5 50 350 430 cm 1.4 m () θ n 1 450 cm (350 / 50) 54.5 θ n 1 00 m (1/ ) 6.6 θ n 10 1 44 34.0 m ( 0).4 m ( 0 / 10) 64 180 (d) θ n 1000 1 5000 mm 000 40 mm (000 /1000) 64
4 (e) 5 θ n 1 50 cm ( 35) 40 cm ( 35/ 5) 54.5 () 5 θ n 108.4 1 15 0.0 cm 181cm (15/ 5) 76 180.0 0c30 0.5 cm 100c30 0.376 m () 0n 30 40 cm 14.34 cm 100n 30 0.500 m 034 m 65c 40 47 cm 80c10.0 4.5 cm 65n 40 0.0 cm 47 cm (d) 80n10.0 50 cm 76 cm
5 RESOLUTION ORCES n COPONENTS (p. 37) n θ cθ wegh 1 Gm m mg W G 495 N 81.5 50 mg W 669 N 81 68. mg W 68. kg.045 lb 1kg 150 lb. 67 kg 81 65 g W m 6) (.03, 10) 5) ( 56 0, 5) ( (4.53 ) ( 00) 15, ( 5) 1 5,10 10 (0 ) ( 68) (34, 5) 56 3 5, 4.53 (3 ) ( 40.4) (35, 56) 1 30 4.53, 10 (5 ) ( R d R c R b R (100,18.00) 30)] (1 5), [(10 ) ( 00) 15, ( 10) 1 5 0, 10 5 ( ) ( (8.75,10.50) 1) (30 4 1 10), (.53 4 1 ) (.78) (87, ) 56 ( 1 6, (4.53) 1 ) ( R h R g R R e 0.00 N 60.0 kg 98.1 N (81)60.0 6 1 pce pce mn m 1 6 3 11 71 m/ ) 10 (396 10 6.419 10 6.673 m m m G g
OENT ORCE (p. 46 7) 6 d n θ d 500 303 N 650 () () R Σ 50.0 0 10 50.0 0.00 N R Σ d 50.0(0.0) 0(0.05) 10(0.15) 50.0(0.5) 0 5 18.75 50 00 N.m Σ A 0 10(dC) 50. 0(0. 0) 0(0.05) 50.0(0.5) 0 10(dC) 0. 0 0(0.05) 50.0(0.5) dc (5 50)/10 0.1100 The ce C huld be med 100 cm he gh A. Σ 00(.50) 50(.0) 50. 0 Theee, Ch wll g up. Jll mu me 0.0 cm wd Ch he w mmen e equl. Σ 0 00(.50) 50(.0) 150.0( d ) 0 50.0 d 0.333 150.0 Theee, In mu 33 cm m ulcum n Ch de. () d 300(0.5) 70 N.m A O A C cθ(0.0) 00(c 0 )(0.0) 36 N.m Ne, ecl cmpnen C cncel ecl cmpnen D nd hnl cmpnen D h n mmen bu A. d 56.5(0.650) 36.7 N.m
LAWS STATICS (p. 6 4) 7 Σ Σ Σ 0 0 A 0 ( Σ 0 Σ Σ( ) 0 ) k () ( 0.35 80.0 0.0 50.0) 18.00k N.m ( 0.35 0.0 0.0 30.0) 100k N.m (d) (e) () [ 0.10 (80.0 0.0) 0.30 (50.0 30.0)] 16.00k N.m ( 0.10 0.0 0.30 0.0) 00 N.m ( 50 50.0.50 80) 75 N.m ( 0.10 50.0 0.30 80) 0.5k N.m ( cθ, n θ) (56.3c 34.3, 56.3n 34.3 ) (46.51, 373) (0.1366 373 0.05 46.51) 0k N.m () (d) ( 4.00 30 8.00 0) 60.0 N.cm 0.600 N.m ( 4.00 30 8.00 0) 340 N.cm 40 N.m ( 4.00 30 ( 8.00) 0) 340 N.cm 40 N.m ( 4.00 30 ( 8.00) 0) 60.0 N.cm 0.600 N.m L mg C A.00 300 019 m 109 cm 60.0(81) Σ Σ Σ cg g g knee knee knee ( 0 mg 0 knee knee ) ( g g ) 0
8 1 1 () Σ cble 0 : cble mg 0 80.0(81) 39 N pe cble jckng up c, png wh ble pene, helng hwng d, kckng, jumpng Σ Σ 0 : 1 00c30 0 : 1 1 1 0 17 N W 0 mg 00n 30 400 300 N Σ knee Σ 0 : knee 50c30 0 : knee knee ld knee ld ld 0 17 N ld W 0 mg 50n 30 40.0 165 N
SPATIAL (3D) LAWS STATICS (p. 67) 9 Σ 0 Σ Σ 0 A 0 Σ Σ( ) 0 j k Σ Σ A 0 0 Σ Σ ( ) ( ) j ( ) k Z 0 A 0 () (d) (e) () 0.0 90.0 40.0 80.0 1400 N.cm 14.00 N.m (30 90.0 40.0 50.0) 1150 N.cm 150 N.m 30 80.0 0.0 50.0 1800 N.cm 18.00 N.m 0.0 40 40.0 0.0 100.0 N.cm 000 N.m (30 40 40.0 30.0) 780 N.cm 8 N.m 30 0.0 0.0 30.0 1300 N.cm 100 N.m C D (50.0 30.0, 80.0 0.0, 90.0 40) (0.0,100.0,130) 30.0 130 0.0 100.0 4050 N.cm 40.5 N.m ( 10.00 130) 0.0 0.0) 1350 N.cm 150 N.m 10.00 100.0 30.0 0.0 1600 N.cm 16.00 N.m 5b 5( 10.00, 30.0, 0.0) ( 50.0,150.0, 0.0) 150.0 90.0 0.0 80.0 13500 N.cm 130 N.m ( 50.0 90.0 0.0 50.0) 4500 N.cm 40 N.m 50.0 80.0 150.0 50.0 11500 N.cm 110 N.m b (30 10.00, 0.0 30.0, 40.0 0.0) (0, 50.0, 40.0) 50.0 90.0 40.0 80.0 1300 N.cm 100 N.m (0 90.0 40.0 50.0) 50 N.cm.50 N.m 0 80.0 50.0 50.0 500 N.cm 00 N.m 10D 10( 30.0, 0.0, 40) ( 300, 00, 450) 30.0 450 0.0 00 13500 N.cm 130 N.m ( 10.00 450) 0.0 300) 4500 N.cm 40 N.m 10.00 00 30.0 300 7000 N.cm 70.0 N.m
Σ 0 Σ Σ Σ 0 1 1 150.0 N 0 60 N 0 1 1 1 1 1 600k N 150.0 N 60 N 350 N 600 N 50 N DRY RICTION (pp. 78 9) 10 c knec µ µ c knec nml nml Σ Σ n nml c knec n nml 0 : mg 50 81 490.5 µ 0 : µ µ c nml knec nml mg 0 nml mg 0 mg 35 81 344 Snce bjec n mn : knec Σ 0 : knec nml he ngle ule : nml 0.95 490.5 466 N 0.90 490.5 441 N be cnucng he ngle we ee h : 0.90 344 188.8 N n nege decn Snce hee e nl hee ce nd he mu dd e c u cn ppl Σ cn n nml nml c µ Aume Σ cn nml W 0 W n15 50 81 0.588 10 N up he nclne 0 : equlbum equlbum mg c1 n45 0 30 81 c1 00 n45 c 0 : nml mg n1 ppled 0.80 146.4 11 ppled 30 81 n1 00 c45 nce h bulue lue mlle hn c45 0 146.4 80. c cn 80. N dwn he nclne.
Σ 0 : n nml c 50 81 500 n13 600 µ c nml nml mg ppled n13 0 0.80 603 48.4 N 11 1 Aume Σ equlbum 0 : equlbum ppled 500 c13 48 nce blue lue gee hn Σ n nml c 0 : µ 50 c10 c nml nml Wc10 46. 0 c13 0 c 0.500 46. 110 N he bd mng (cn knec ). 1 1 1 T ge he b mng he cn mu equl Σ µ µ µ µ 0 : ppled c knec nml c knec nml c knec µ µ c c c knec Wc10 50n 10 nml nml 10.00 81 98.1 N c nml knec nml ppled 0 c 110 441 166.5 N 0.950(150.0 81) 1398 N 0.900(150.0 81) 134 N 0 0.55 98.1 0.0 0.04 98.1 30.0 81 94.3 N c nml knec nml 5 94.3 15 94.3 0.765 0.731. 1 N. B denn he cecen c cn nl ccu he nn he lppng ccu when he mmum cn l ccu.
LINEAR KINEATICS (pp. 94 5) 1 () 0 40 m: ½ ( ½( ) ( ) 9 0 9 (40 0) (40) 81 015 m/ 80 ) 40 70 m: 0 m/ 70 100 m: 100 70 9(4) ½(4) (30 36) 0.750 m/ 16 040 40 70 70 100 l ½ 9 0 8.89 015 70 40 33 9 4.00 8.89 33 4 16. 81 0 0 ½ 5 80 (d) 3 100 9 4 6.00 m/ 4 () ± 536 m/ ± 66 m 5 (0.005)(5 0) 536 50 11 0.005 ½ 0 5 10 35 0.5 33 m () ½( 0.3)(33) 0.0(333) 1 ½(0.005)10 0.0 33 0.300 (333) ½ 894 1736 cche : ( cche : ( 8.07 m 150 km/h 6 467 m/ 0 467 ) (0.300 0) 0 467 894 0.01440 14.40 m nd 0 467 ) (0.500 0) 0 467 1736 0.040 4.0 m The deence n he ccelen 1158 m/. The deence n he me 60 m.
3 00 546 m / 546 /500 8.36 m/ Bll peed mu be gee hn 8.36 m/. 13 1 () 100 m 1mn 4 mn 60 0.417 m/ 1 dwn e l l 66.7 cuen km 4.00 h 100.0 85 m 1h 60 mn / 84.8 /(4.00 60.0) 187 m/ 0 ( ( ± ) ) (.00)(6.00 0) 4.0 ± 4.90 1000 m 4.00 mn 66.7 m 1km The nl elc mu be 4.90 m/.
PROJECTILE OTION (pp. 101 ) 14 g g g( ½( ) ) () km 1000 m 1h 90 0 m/ h 1km 3600 cθ 5c5 4.90 m/ n θ 5n 5.179 m/ 10 0 0.40 4.9 g( ) g 0 (.179) (81) (d) () 10 0 8.6 m/ 0.350 ± ± 181 m/ 0 0.4 m 4. cm g( ) 0 (81)( 0 0) 34 ± 181 m/ g 181 0.0 81 0 (.0) 4.04 m () 1 0 0 g g 6 g g( (81)(0 10) ± 6.00 835 m (81) ) ± 14 m/ elec he nege elc 14 6.16 81 0.500(.16) 08 m θ m m me 30 8.66 00 70 8.83 019 45 07 07.55 10.19 44 60 00 8.66 8 8.83 765 10 0.319 81 13 0 g(0 5) 81 ± g p g 94 m/ 1 0.406 33 m/ 13 0 0.319 81 0 (81)( 0.795) 0.406 8 p
15 ANGULAR KINEATICS (p. 106) ω ω θ ω θ α θ ω ½α ω θ ½( ω ω α(θ θ ) ) () ω ω 5 α 500 / 3π d/ 4 d/ θ θ ω ½α 0 () ½(5) 1 θ 0 0 (0.35)36 7 elun ω ω α 6 0.750(5) 75 d/ 3 6π ω d/ 0.89 0.89 00 elun ω ω α 60 ( 0.00)00 00 d/ nl nl 1 ω nl ω α nl 0 0.0 133 500
RELATIONSHIP beween LINEAR nd ANGULAR EASURES (pp. 111 ) 16 () ω α ω ω 0.75(15) 15 m/ α 0.75(150) 115 ω 03 m/ 0.75(15) 168.75 115 () deg ω 0.900 573 00 m/ ω ω 10 0 α 5 α 0.9(10) 00 ω 90.4 m/ ω 0.9(10) 0.75(10.00) 50 m/ ω 0.75(10) 70 m/ 00 168.75 π d 180 deg 6.67 d/ 90.0 75 90.0 α 0.75(.00) 500 m/ 5 70 m/ 1 () ee ω.30(10) 0 m/ cg ω 30(10) 100 m/ 10 0 ω 00 d/ ee cg π π(.30) 14.45 m π π(3) 8.16 m 14.45 8.16 6.9 m (d) () ee ω 30 m/ l.30(10.00) 00.70 700 m ω 70(8.75) 14.875 m/ ω;, ω 5 70 ω 16.375 7 0.1714 m 8.75 ω 11 m/ 0.681 m/ 0.1714(8.75) π d 1mn ω 65 mn 1 60 6.81 d/ m ω 0.330(6.81).5 m/ puhwheel ω 0.100(6.81)
17 LAW ACCELERATION (p. 117) m m m Σ Σ Σ 80 N 350) 60.0( 350 m/ 60 81 0) (30 9 0 ) ( m N 47 40(81) 40().00 1 0 Σ mg m mg m le le 8.49 m 886) ( 10 0 0 Thu, 886 m/ 70 41 N 0.6 686.7 686.7 70(81) Σ knec knec nml knec nml knec m mg µ.70 111 0 3 111 m/ 900 1000 / 900 1000 Σ m 00 N 66.67) 0.180( 66.67 m/ 0.3 0 0 m 1 8.41 m/ 0 168. /.05 m/ 0 41/ 168. N 196. 8 3 0 5 40 N 5 58 35 Σ Σ mg m m
OENT ORCE (p. 1) 18 R [ ] ( ) Σ d I α α 35/ 150 d/ I d 150(0.35) 5 N.m α / I.10 d/ ω ω.35 4.45 α 0.40 00 Iα 0( 0.4) 144 N.m d 400 0.35 140.0 N.m d 500 0.40 00 N.m I / α 00 / 0 10.00 kg.m
OENT INERTIA (p. 19) 19 K I I cg k cg cg mk I cg cg /L m () k I cg I I m cg cg m 000 kg.m 0.5 8 90 00(0.50) 0.50 m 0.5 8(0.5) k KL 0.60(0.95) 0.57 m I I I cg hp b mk I cg 00(0.57) m I m cg 18 168. 184.0 kg.m 90 kg.m 6.90 kg.m 180 80.0(450) k 180 80.0 0.1975 0.444 m d 300(0.15) 40 N.m α (ω I / k I ω ) / 0 /1 0.0 d/ I / m 45/ 0 0.410 kg.m k pml pml I cg I.5 kg/m.5/ 90.0 m pml / m 0.1581 m 0.1489 05(0.195) 0.453/ 05 0.41 m.41cm
REACTION ORCES (p. 136) 0 R cenpe l mw m / Σ Σ n g g m / m g 7 / 58 mg m 4. m/ (1345 58 81) / 58 119 m/ θ / θ 100 / π 383 θ n 1 1 11 g 383 81 (0.3875) deg mg m 50 g 81 6371m 6.37 km 1 Σ g m m 686 N m g 50(.00)5 mg mg 60(0.75 81) m / mω 170 N m 75 (0 81) 14 70 N
LINEAR IPULSE nd OENTU (p. 150) 1 mmenum p m mpule d d m m 0 00 m 60.0 60( 10.909) 655 N 100 Impule m m 50(.50) 0 135 N. m g( ) 0 (81)( 350 0) 6.487 m 0 100 70(81) 3786 lndng lndng m m 00 / 70.0 m m 3786 / 70.0 6.476 147 m/ m lndng m d 0 00.86 m/ d W 37 m/ 0 ( 17) 87 m/ 0.400 60.0(87) 77 N 0.005(400) 40.0 N 0.050 1 m 0 4.0(1) / 4 6.00 m/
ANGULAR IPULSE nd OENTU (p. 155) ngul mmenum L Iω ngul mpule d Iω Iω d L Iω 00(5) 6 kg.m / π d L Iω 65.5 1 65(14.137) 79 kg.m ( d) 50(350)0.500 168.8 N.m. ( d) 68.0(0.30)5 108.8 N.m. k Kl 0.36(35) 0.406 m I mk 150(0.406) 8641 kg.m L Iω 8641(.55) 4.75 kg.m / /
CONSERVATION OENTU (p. 163) 3 p L p L m cnn Iω cnn () () L Iω (0) 40.0 kg.m I I I p p lnd ω p L L p L L / 30.0 333 kg.m L / 6.6 504 kg.m lnd L Iω 8.5 0.430ω ω 8.5 / 0.43 119 d/ / 40.0 kg.m / 119 0 Iα 0.43 16.50 N.m 0.5 ω L / I 8.95/16.6 0.5504 θ θ ω θ 0 p 14 ω 0.5504 I L / ω 36 / 5 10.95 kg.m
WORK-ENERGY THEORE (p. 174) 4 wk W E E E eneg E mg ½m ½Iω E mg ½m 70(81)150 ½(70)00 787 5040 5830 jule 83 kj E ½m ½(60.0).00 10.0 J E ½Iw ½(.00)3416 π 1mn ½(.00) 300 mn 1 60 987 J n 0 6.00n 0 (60.0 ) 11 m W E E mg mg 50(81)11 0 490.5(11) 60.4 kj W E E 0 E 0 ½m ½(0)5 313 J 1 () 0 3 0.750 4 0 3 0 6.00 m ( 0.75) 50 m 0.0( 0.75) 100 N W 100(6.00) 90.0 J
WORK ORCE OENT ORCE (p. 180 1) 5 W W W ce ce mmen c φ θ ( nφ) θ () W 358 0.517 181 J W 3 35 6.9 18 J W ce cφ 30 (0) c 30 45 N 196 J 697 J mg 0(81) 45 0.800 196. N W W nml knec µ 0.8 µ 0. nml knec µ knec 196. 10.00 45 0. 10.00 491J W ( Lg) 0(81)10(6.00) 196.(60.0) 1177 J 177 kj 1 1 () W ( Lg) 50(81)4000 137 340 J 133 kj W mg 60(81)0.350 06 J W l 441J W mmen mg 300(81)0.1500 θ ( d)θ (90.0 0.600)(1 d) 54.0 J n W l / W mmen 441 8 nd1/6 8.17 ccle 54 1 () W E E 0 ½ I ω ½ (0.450) 0.0 90. 0 J cn nmlµ knec 50(0.800) 10.00 N d 10.00 0.300 α I I 0.450 6.667 d/ ω ω 0 0.0 00 α 6.667
POWER (p. 185) 6 P W / E / P P P ce ce mmen cφ ω 1 70.0 6.00 840 mee W L P W 10.00 81 98.1 jule L 687 W P L / 98.1(840) /( 60) ( W g) / (5 81) 4000 /(6 60) 38 W P 0 An mec cncn de n mechncl wk. deg ω 00 π d 360 deg 49 P ω 50.0 39 174.5 W π d ω 3 18.85 d/ 1 P ω 50 18.85 1037 W P 5 0.555 14.9 W W P 14.9 50 437 J
CONSERVATION ECHANICAL ENERGY (p. 190) 7 W G ½m g E (81)3 67 m/ E mg 70 81 00 10 J mg cnn E ½m ½m E ½m 0 ½(60)4.15 557 6376.5 6936 J g mg mg ½ m W mg ½ m 6936 60 mg 60(81)10.00 14 14.61 m/ 50 30.5 54 m (81) 16 ½ m W 60.0(81)0.45 66 J ke ½ m ke p mg ½ m 66 J 66 60.0 585 J 585 0.100 ke 8.868.98 m/ mg (100 0.100)(81)(0.35) 585 J 1036.9 m/