COSMOS Chapter 8 Solution 116

COSMOS: Complete Online Solutions Manual Organization System Chapter 8, Solution 116. (a) For minimum mC with blocks at rest, impending slip of A is down/left. Note: φs = tan −1 µ s = tan −1 0.30 = 16.7° < 30°, so mC min > 0 FBD A: ( ) WA = ( 6 kg ) 9.81 m/s 2 = 58.86 N ΣFy = 0: N A − WA cos 30° = 0, N A = WA cos 30° Impending slip: FA = µ s N A = 0.30WA cos 30° ΣFx = 0: TA + FA − WA sin 30° = 0, TA = WA ( sin 30° − 0.30 cos 30° ) = 14.1377 N FBD Drum: If blocks don’t move, belt slips on drum, so 0.2 0.87266 ) 14.1377 N = TA = TC e µk β = TC e ( = 1.19069 TC so TC = 11.8735 N FBD C: ΣFy′ = 0: NC − WC cos 20° = 0, NC = WC cos 20° Impending slip: FC = µ s NC = 0.30WC cos 20° ΣFx′ = 0: 0.30WC cos 20° + WC sin 20° − 11.8735 N = 0 WC = 19.0302 N, mC = WC = 1.93988 kg 9.81 m/s 2 mC = 1.940 kg
continued Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P. Beer, E. Russell Johnston, Jr., Elliot R. Eisenberg, William E. Clausen, David Mazurek, Phillip J. Cornwell © 2007 The McGraw-Hill Companies. COSMOS: Complete Online Solutions Manual Organization System (b) For motion of A to impend up/right FBD A: As in part (a) N A = WA cos 30°, FA = 0.30WA cos 30° ΣFx = 0: TA − WA ( sin 30° + 0.30cos 30° ) = 0 TA = 44.722 N Also, as in part (a) TA = TC e µk β = 1.19069 TC , so TC = 44.722 N 1.19069 TC = 37.560 N FBD C: As in part (a) FC = 0.30WC cos 20° ΣFx′ = 0: WC ( sin 20° − 0.30cos 20° ) − 37.560 N = 0 WC = 624.83 N, mC = WC = 63.69 kg 9.81 m/s 2 mC = 63.7 kg
(c) For uniform motion of A up and B down, and minimum mC , there will be impending slip of the rope on the drum. FBD A is same as in (b) but FA = µk N A = 0.20WA cos30° and ΣFx = 0: TA − WA ( sin 30° + 0.20cos 30° ) = 0, TA = 39.625 N Drum analysis, with impending slip, TA = TC e µs β 39.625 N = TC e 0.30( 0.87266 ) = 1.29926 TC or TC = 30.498 N FBD C is same as in (b), but FC = µ k NC = 0.20WC cos 20° and ΣFx′ = 0: WC ( sin 20° − 0.20 cos 20° ) − TC = 0 WC = 30.498 N = 197.933 N, 0.154082 mC = 197.934 N 9.81 m/s 2 mC = 20.2 kg
Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P. Beer, E. Russell Johnston, Jr., Elliot R. Eisenberg, William E. Clausen, David Mazurek, Phillip J. Cornwell © 2007 The McGraw-Hill Companies.