CHAPTER 2. Problem 2.1. Given: m k = k 1. Determine the weight of the table sec (b)

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1 CHPTER Problem. Give: m T π 0. 5 sec (a) T m 50 g π. Deermie he weigh of he able sec (b) Taig he raio of Eq. (b) o Eq. (a) ad sqarig he resl gives or T T mg m 50 g m lbs m g 0.5. Deermie he laeral siffess of he able. Sbsie for m i Eq. (a) ad solve for : 40 6π m 6π 6.4lbs i Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

2 Problem.. Deermie he aral freqecy lb i. m lbsec i m Deermie iiial deflecio. 98. rads sec Saic deflecio de o weigh of he iro scrap ( 0) 00 i Deermie free vibraio. ( ) ( 0) cos cos ( 9. 8) 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

3 Problem.3. Se p eqaio of moio. mg/ mü mg m mg. Solve eqaio of moio. ( ) cos Bsi mg 0, ( 0) 0 ad ( 0) 0 mg, B 0 mg ( ) ( cos ) 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 3 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

4 Problem.4 v 0 m m 0 0 m lb sec i lb sec i m 0 00 lb i. Coservaio of momem implies mv ( m m) ( ) ( 0 mv 0 0 ). m m 857 f sec 34.9 i. sec 0 fer he impac he sysem properies ad iiial codiios are Mass m m lb sec i. Siffess 00 lb i. Naral freqecy: m m rads sec Iiial codiios: ( 0) 0, ( 0) 34. 9i. sec The reslig moio is () ( 0) si si ( ) i. 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 4 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

5 Problem.5 h m f S m m m m g Wih measred from he saic eqilibrim posiio of m ad, he eqaio of moio afer impac is ( m m ) m g (a) The geeral solio is m ( ) cos Bsi g (b) m m The iiial codiios are (c) m ( 0) 0 gh (d) m m The iiial velociy i Eq. (d) was deermied by coservaio of momem drig impac: m ( m m) ( ) where 0 g h Impose iiial codiios o deermie ad B: mg ( 0) 0 ( 0) B B m m m gh Sbsiig Eqs. (e) ad (f) i Eq. (b) gives (e) (f) m h m ( ) g g ( cos ) si m m 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 5 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

6 Problem.6. Deermie deformaio ad velociy a impac. mg 0 ( 0) 0. i. 50 ( 0) gh ( 386)( 36) i./sec. Deermie he aral freqecy. g ( 50)( 386) rad/sec w 0 3. Compe he maximm deformaio. ( 0) () ( 0 )cos si o ( 0. )cos 368. si ( ) [()] ( ) 38. i. 4. Compe he maximm acceleraio. ( ) ( 38. ) o o 7334 i./sec 8.98g 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 6 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

7 Problem.7 Give: 00 m 6. lb sec f 3. f Hz Deermie EI: 3EI 3EI EI 3 3 lb f L 3 9 EI f π m π EI ( 4π) lb f 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 7 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

8 Problem.8 Eqaio of moio: m c 0 (a) Dividig Eq. (a) hrogh by m gives 0 (b) where. Eqaio (b) hs reads 0 (c) ssme a solio of he form (). Sbsiig his solio io Eq. (c) yields ( s s ) e 0 s Becase e s is ever zero, he qaiy wihi pareheses ms be zero: e s or s s 0 s ( ) ± 4 (doble roo) The geeral solio has he followig form: () e e where he cosas ad are o be deermied from he iiial codiios: ( 0 ) ad ( 0 ). Evalae Eq. (d) a 0 : (d) 0) (e) ( Differeiaig Eq. (d) wih respec o gives ( ) e ( ) e (f) Evalae Eq. (f) a 0 : ( 0) ( 0) ( 0) ( 0) ( 0) (g) Sbsiig Eqs. (e) ad (g) for ad i Eq. (d) gives { [ (0 ] } e ( ) ) (h) 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 8 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

9 9 Problem.9 Eqaio of moio: m c 0 (a) Dividig Eq. (a) hrogh by m gives 0 (b) where >. ssme a solio of he form e s (). Sbsiig his solio io Eq. (b) yields 0 ) ( s e s s Becase e s is ever zero, he qaiy wihi pareheses ms be zero: s s 0 or s ± ± 4 ) ( The geeral solio has he followig form: exp exp ) ( (c) where he cosas ad are o be deermied from he iiial codiios: ( ) 0 ad ( ) 0. Evalae Eq. (c) a 0 : 0) ( (d) Differeiaig Eq. (c) wih respec o gives exp exp ) ( (e) Evalae Eq. (e) a 0 : ] [ or or (f) Sbsiig Eq. (f) i Eq. (d) gives (g) The solio, Eq. (c), ow reads: ( ) D D e e e ) ( where D D D 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

10 Problem.0 Eqaio of moio: 0 (a) ssme a solio of he form The geeral solio is () e e Deermied from he iiial codiios 0 ad ( 0 ): 0 ( 0) (j) (i) () e s Sbsiig his solio io Eq. (a) yields: F s s H I K es 0 Becase e s is ever zero s s 0 (b) The roos of his characerisic eqaio deped o. (a) Uderdamped Sysems, < The wo roos of Eq. (b) are F H I K s, ± i (c) Hece he geeral solio is () e s e s which afer sbsiig i Eq. (c) becomes where e j (d) i D i D () e e e D Rewrie Eq. (d) i erms of rigoomeric fcios: () e ( cos D Bsi D ) (f) Deermie ad B from iiial codiios 0 ad ( ) 0 : B ( 0) 0 D Sbsiig ad B io Eq. (f) gives () ( 0) e F H si (b) Criically Damped Sysems, I K (e) (g) The roos of he characerisic eqaio [Eq. (b)] are: s s (h) Sbsiig i Eq. (i) gives () ( 0 ) e () (c) Overdamped Sysems, > The roos of he characerisic eqaio [Eq. (b)] are: F H I K s, ± (l) The geeral solio is: s s () e e (m) which afer sbsiig Eq. (l) becomes F H I K F H () e e I K Deermied from he iiial codiios ( 0) 0 ad ( 0 ): ( 0) Sbsiig i Eq. () gives ( 0) e () (d) Respose Plos F HG e e Plo Eq. (g) wih 0.; Eq. (), which is for ; ad Eq. (p) wih. () () / ) I K J () (o) (p) 0 /T Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 0 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

11 Problem. l j F HG I F l KJ j H G I π 0. K J j 0% j 0 % l ( 0) π π 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

12 Problem. i exp i (a) 00. : (b) 005. : (c) 05. : π i i i i i i Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

13 Problem.3 Give: w 0.03 ips (empy); m ip-sec /i. (8.) 6.4 ips/i. c ip-sec/i. m (a) T π π sec. c (b) m ( 6. 4)( ) 94. % 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 3 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

14 Problem.4 (a) The siffess coefficie is lb/i. The dampig coefficie is c ccr m c lb-sec/i. (b) Wih passegers he weigh is w 3640 lb. The dampig raio is c 59. m (c) The aral vibraio freqecy for case (b) is D ( ) / rads/sec 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 4 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

15 Problem.5. Deermie ad. π j l l % π 0. j Therefore he assmpio of small dampig implici i he above eqaio is valid. 3 T D sec; T T 05. sec ; π rads sec.. Deermie siffess coefficie. m ( 4. 89) lbs i. 3. Deermie dampig coefficie. c c cr m ( 0. ) ( 4. 89) lb sec i. c cr ( ) lb sec i. D 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 5 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

16 Problem.6 50 (a) 08. m 3. 5 lbs i. w lb sec i. g rads sec m (b) ssmig small dampig, l F HG F HG j I KJ I j π 0 l l ( ) ( ) KJ π This vale of may be oo large for small dampig assmpio; herefore we se he exac eqaio: or, l F HG j l ( 8) I KJ jπ ( ) π ( ) (c). 69 rads sec D Dampig decreases he aral freqecy. 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 6 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

17 Problem.7 Readig vales direcly from Fig...4b: Pea Time, i (sec) Pea, i (g) T D 0.35 sec g l % π (30) 0.50g 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 7 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

18 Problem.8. Deermie bclig load. L θ w cr wcr ( L θ) θ w cr L. Draw free-body diagram ad se p eqilibrim eqaio. f I L θ w f S O where MO 0 fi L fs wlθ (a) w fi L θ fs θ (b) g Sbsiig Eq. (b) i Eq. (a) gives w L θ ( wl ) θ 0 (c) g 3. Compe aral freqecy. or wl wl ( w g) L ( w g) L w (d) w cr 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 8 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

19 Problem.9 For moio of he bildig from lef o righ, he goverig eqaio is m F (a) for which he solio is ( ) cos B si (b) F Wih iiial velociy of ( 0 ) ad iiial displaceme ( 0) 0, he solio of Eq. (b) is () ( 0) si F (cos ) (c) ( ) ( 0) cos si (d) he exreme righ, ( ) a ( 0) F F 0 ; hece from Eq. (d) Sbsiig 4 π, F 05. i. ad ( 0 ) 0 i. sec i Eq. (e) gives or a 0 4π si ; cos (e) Sbsiig i Eq. (c) gives he displaceme o he righ: π (. ). (. ). i. fer half a cycle of moio he amplide decreases by F i. Maximm displaceme o he rer swig is i. 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 9 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ

20 Problem.0 Give: F 0. w, T 05. sec F 0. w 0. mg 0. g 0. g F ( π T) 0. g i. ( 8π) The redcio i displaceme amplide per cycle is 4 F i. The displaceme amplide afer 6 cycles is.0 6 (0.44) i. Moio sops a he ed of he half cycle for which he displaceme amplide is less ha F. Displaceme amplide a he ed of he 7h cycle is i.; a he ed of he 8h cycle i is i.; which is less ha F. Therefore, he moio sops afer 8 cycles. 0 Pearso Edcaio, Ic., Upper Saddle River, NJ. ll righs reserved. This pblicaio is proeced by Copyrigh ad wrie permissio shold be obaied from he pblisher prior o ay prohibied reprodcio, sorage i a rerieval sysem, or rasmissio i ay form or by ay meas, elecroic, mechaical, phoocopyig, 0 recordig, or liewise. For iformaio regardig permissio(s), wrie o: Righs ad Permissios Deparme, Pearso Edcaio, Ic., Upper Saddle River, NJ