ME 236 Engineering Mechanics I Test #4 Solution
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1 ME 36 Enineein Mechnics I est #4 Slutin Dte: id, M 14, 4 ie: 8:-1: inutes Instuctins: vein hptes 1-13 f the tetbk, clsed-bk test, clcults llwed. 1 (4% blck ves utwd ln the slt in the pltf with speed f & ( 4t /s, whee t is tie in secnds. he pltf hs dius f 3 nd is ttin t cnstnt te f 6 d/s. Ine the size f the blck. If the blck stts f est t the cente, deteine 1.1 the nitude f its velcit when t 1 secnd; nent 1. the nitude f the cceletin when t 1 secnd; c Pth 1.3 the tie when the blck leves the pltf.. 1 φ c Slutin 1.1 Velcit when t 1 s l displceent: t t dt & 4tdt t, thus In this cse, we hve tw velcit cpnents: when t 1 s. v & v & nd, thus v v & ( & v (4 1 ( (/s 1. cceletin when t 1 s hus && & (/s && & & ( (/s (/s Plese nte tht in this cse & t nd n & becuse the vin pth f blck is spil cuve nd thus its nl nd tnentil diectins e nt sipl the dil nd tnentil diectins f the cicle f dius. 1.3 ie when the blck leves the pltf When the blck leves the slt f the pltf, 3, nel hus, we hve t & dt t t 4tdt t 3 / 1.5 (s 3
2 he sses f the tw blcks e nd. th the sttic nd kinetic fictin cefficients between blck nd the und e equl t. ssue n fictin n the pulle sufce nd tet the blcks s pticles, i.e., ine ll the ents..1 (% Deteine the cceletin f blck nd the tensin in the cble.. % Shw tht, t keep the sste f vin, the iniu is? Slutin:.1 ee-bd di f blcks nd e shwn bve. Equtins f tins f the tw blcks: in Y : blck in Y : blck in X : blck Eliintin f the fist tw equtins, we hve the cble kinetics, we knw becuse s s ll the tie. hus, we slve the bve tw equtins f the cceletin f blck nd the cble tensin s fllws (,. When the sste is nt vin, ust hld. heefe, f which we find the iniu s
3 nthe ethd f slvin pble. When the sste is t ipendin tin, the sste ust still be in equilibiu. heefe, we hve the fllwin equtins f the entie sste (tetin bth nd s whle: Σ Σ ( Substitutin the nl fce f the secnd equtin int the fist equtin, we btin 3 If the bll hs ss f 3 k nd speed v 4 /s t the instnt it is t its lwest pint (when. elect the size f the bll. 3.1 (% Deteine the tensin in the cd t the instnt when. 3. % Deteine the nle t witch the bll swins t the instnt it entil stps. n Slutin t ee-bd di is shwn bve, f which we hve the fllwin equtins f tin v n n cs n t t sin t ( 3.1 pute the tensin when the bll is t its lwest pint, i.e.,. eq., we hve v cs cs( ( 3. Deteine the nle when the velcit f bll beces ze, i.e., v. eq.(, we find the tnentil cceletin
4 t sin Since, we hve knwn the initil nd finl velcities s well s the epessin f tnentil cceletin, we cn slve the pble b intetin the fllwin ful, which leds t ds t vdv Slvin it, we btin the nle (cs cs 1 cs sin ( d 1 ( vdv 37.38
5 E: his pble is equied f this test. Hweve, if u slve it, the esultin cedits will be dded t the ne f u pevius tests which hs the lwest de. 4 (3% cn, hvin ss f.5 k, ves ln ved hizntl slt shwn in the fiue. he slt is in the f f spil, which is defined b the equtin.1 etes, whee is in dins. If the is ttin t cnstnt te & 4 d/s in the hizntl plne, deteine the fce it eets n the cn t the instnt when π dins. elect the fictin nd the size f the cn. Slutin: nent When π, the nle between the nl nd dil diectins cn be cputed f c Pth.1 d / d.1 1 tn φ.1 π hus, 1 tn 1 φ 17.7 π Equtins f tin: φ c ; ; Kinetics when π : cs sin & 4 (d/s; &&.1.1π.314 (; &.1 &.4 (/s; && & (/s && & & (/s && Substitutin the bve kinetics esults int the equtins f tin, we cn btin,.64 (,.8 ( he netive vlue f c ens the fce is in the ppsite diectin s ssued.
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