PHY2053 Summer 2012 Exam 2 Solutions N F o f k
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1 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 s cnsered gy gy U K U gy gy K g( y y ( /s (9.8 /s (60 0 /s. Mechancal energy s nt cnsered snce the sled sts ng. nc K U ( K K ( U U (0 0 (0 gy gy ( kg kg(9.8 /s ( 8800 J
2 Frctn des the dssate wrk nc k r csθ k nc r cs J ( ( 00 N. he rce nratn ges the rce cnstant r the srng F k k F 80 N N/ Mechancal energy s cnsered as the ball ets the gun k U K k 0 0 U k K k ( N/ 0.08 kg 0 /s. he wrk dne by the engne ncreases the car s knetc energy. K K K (000 kg(0 /s (000 kg(0 /s 7. 0 J he wer utut by the engne rate the wrk dne by the engne t. 0 0 s 7 J he rce s und r the ulse-entu there. Bere the cllsn 6 /s
3 Ater the cllsn /s he ulse-entu there s r r Fat Snce ths s a ectr equatn, we ust take cnents. Snce the tn s nly alng the -as, nly the -cnent s needed. he change n entu s F, a t (. kg( /s (. kg( 6 /s kg /s he aerage rce s F t kg /s 0.0 s a, 60 N 7. Lnear entu s cnsered snce the elsn s an nternal rce. Bere the elsn 0 /s ( kg(0 /s 600 kg /s Ater the elsn
4 0 /s Snce lnear entu s cnsered, (600 kg /s ( kg(60 kg /s 0 kg 0 /s 8. Use the equatns r a ne densnal elastc cllsn dered n lecture /s /s kg kg ( kg kg kg ( /s ( /s. /s kg kg kg kg 9. Mentu s cnsered n the cllsn. Bere the cllsn /s 00 kg 0 /s 00 kg
5 Fr the -cnent (00 kg( /s.8 0 kg /s Fr the y-cnent y (00 kg(0 /s. 0 kg /s Ater the cllsn θ 700 kg Fr the -cnent csθ And the y-cnent y snθ Usng the cnseratn lnear entu.8 0 kg /s csθ And. 0 y y kg /s snθ here are tw equatns
6 kg /s csθ kg /s snθ sle r drectly, square the equatns and add the tgether. ( csθ ( snθ (.8 0 kg /s 8 ( (cs θ sn θ.9 0 kg kg /s (. 0 /s kg /s kg /s (700 kg 8.7 /s 0. he bject lks lke he dentn the center ass s c he subscrt reers t the rd and the subscrt reers t the addtnal ass. Slng r : c ( c ( c c c ( ( ( c c c Measurng the lcatns r the let end the rd, the lcatn the rd s, the lcatn the added ass s 0, and the lcatn the center ass s c 0.7. S ( ( c ( 0.7 ( kg (0.7 0 c.0 kg. he bject cnssts tw arts. he rtatnal nerta can be decsed nt I I dsk I ass
7 he rtatnal nerta the dsk s I dsk R dsk ( kg( kg he rtatnal nerta the etra ass s I dsk r ( kg( kg Fnally, I I dsk Iass 0. kg 0.6 kg 0.90 kg. 960 N he rces n the bea are θ he cndtn r equlbru s τ 0 akng trques abut the let end (the hnge τ τ τ 0 0
8 he trque due t the weght s clckwse. Its alue s τ r ( 960 N( 80 N nd the trque due t the tensn, we need the angle θ he trque s cunterclckwse, τ he tensn can be und θ tan 7 r (( sn 7 (. τ τ 80 N (.. he rtatnal nerta the h s N. 600 N I MR ( 00 kg( 00 kg Its angular acceleratn ω (0 0 rad/s α 0.0 rad/s t 00 s e dn t care abut the drectn the acceleratn. Dr the nus sgn. he trque s τ Iα τ Iα (00 kg (0.0 rad/s 0 N. he astest bject reaches the btt rst. Use energy t nd the astest. ake stn at the t the ra and stn at the btt the ra. K U K 0 gy ( U Iω 0 he rtatnal nerta r the shaes can be suarzed (lke ur tet des by
9 I βr Fr the shere β /, the cylnder β /, and the rng β. Als use ω /R n the energy relatn: gy gy gy ( β ( β βr he largest β wll be the slwest. he rder wll be shere, cylnder, and rng.. Angular entu wll be cnsered. L L I ω I ω R 6. N. he te r ne rtatn ( s related t the angular seed (ω Substtutng 7. At the deth I ω I ω π I I π I I ω π π ω I (.8 s I 0.90 s gd (.0 0 a (000 kg/ (9.8 /s (. 0 Duble that nuber and nd the deth a
10 gd d g (. 0 a.0 0 a (000 kg/ (9.8 /s 8. Fr the densty and the ass the lue s und V kg V 000 kg/.0 0 Use Archedes rncle t nd the buyant rce F B gv (000 kg/ (9.8 /s ( N 9. Call the stn at the btt the e and the stn at the t the e. Alyng Bernull s equatn he e s daeter des nt change s gy gy By the cntnuty equatn A A A A Snce the end the e s esed t the atshere at. Heghts are easured r the lwest nt s y 0. Makng these substtutns
11 a 0.60 (6 (9.8 /s (000 kg/ a gy gy gy gy at at 0. seulle s law s 0 a/ (0.0 s a 0 (8(.0 /s 0 (.0 8 / / 8 π π η η π r t V L r L t V
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