0 mm PROEM..4 m 45 mm (a) Determine the torque that causes a maximum shearing stress of 45 MPa in the hollow cylindrical steel shaft shown. Determine the maximum shearing stress caused by the same torque in a solid cylindrical shaft of the same cross-sectional area. SOUION (a) Given shaft: J c 4 c 4 Solid shaft of same area: J (45 0 ) 5.89 0 mm 5.89 0 m c J J c 4 4 4 4 (5.89 0 )(45 0 ) 5.89 0 N m 45 0 (45 0 ).54 0 mm c c c or c.54 mm 4 c J c, J c ()(5.89 0 ) 87. 0 Pa (0.054) 5.7 kn m 87. MPa PROPRIERY MERI. Copyright 05 McGraw-Hill Education. his is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. his document may not be copied, scanned, duplicated, forwarded, distributed, or posted
54 mm =. kn m 4 mm C = 0.8 kn m 4 mm D = 0.4 kn m 40 mm E C D PROEM. Under normal operating conditions, the electric motor exerts a torque of.4 kn m on shaft. Knowing that each shaft is solid, determine the maximum shearing stress in (a) shaft, shaft C, (c) shaft CD. SOUION (a) (c) Shaft : Shaft C: Shaft CD:.4 0 N m, c d 0.07 m c (.4 0 ) 77.5 0 Pa 77. MPa J c (0.07) C.4 kn m. kn m. kn m, c d 0.0 m C c ()(. 0 ).788 0 Pa.8 MPa J c (0.0) CD 0.4 0 N m c d 0.0 m CD c ()(0.4 0 ) 0.99 0 Pa 0.9 MPa J c (0.0) PROPRIERY MERI. Copyright 05 McGraw-Hill Education. his is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. his document may not be copied, scanned, duplicated, forwarded, distributed, or posted 7
C 40 mm PROEM. 00 mm 000 N m D torque of magnitude 000 N m is applied at D as shown. Knowing that the diameter of shaft is 5 mm and that the diameter of shaft CD is 4 mm, determine the maximum shearing stress in (a) shaft, shaft CD. SOUION CD 000 N m r 00 CD (000) 500 N m r 40 C (a) Shaft : c d 0.08 m ()(500) c 7.50 0 J c (0.08) 7.5MPa Shaft CD: c d =0.00m ()(000) c 8.7 0 J c (0.00) 8.7MPa PROPRIERY MERI. Copyright 05 McGraw-Hill Education. his is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. his document may not be copied, scanned, duplicated, forwarded, distributed, or posted 8
00 N m 00 N m C 44 mm D 48 mm. m 0.9 m PROEM.5 he electric motor exerts a 500-N m torque on the aluminum shaft CD when it is rotating at a constant speed. Knowing that G 7 GPa and that the torques exerted on pulleys and C are as shown, determine the angle of twist between (a) and C, and D. 40 mm m SOUION (a) ngle of twist between and C. J C 00 N m,. m C 9 c d 0.0 m, G 7 0 Pa 4 9 c 7.97 0 m C (00)(.) 4.57 0 rad GJ (7 0 )(7.97 0 ) C / 9 9 ngle of twist between and D. 9 CD 500 N m, CD 0.9 m, c d 0.04 m, G 7 0 Pa 4 4 9 4 JCD c (0.04) 5.5 0 m (500)(0.9) CD /.980 0 rad 9 9 (7 0 )(5.5 0 ) C /.84 D / C / CD / 4.57 0.980 0 5.7 0 rad D /. PROPRIERY MERI. Copyright 05 McGraw-Hill Education. his is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. his document may not be copied, scanned, duplicated, forwarded, distributed, or posted 95
PROEM.8 00 N m mm 800 N m 0 mm 75 mm C D 50 mm he aluminum rod ( G 7GPa) is bonded to the brass rod D ( G 9GPa). Knowing that portion CD of the brass rod is hollow and has an inner diameter of 40 mm, determine the angle of twist at. 400 mm SOUION Rod : Part C: Part CD: 9 G 7 0 Pa, 0.400 m 800 N m c d 0.08 m 4 4 9 J c (0.08) 4.89 0 m (800)(0.400) / 7.875 0 rad 9 9 GJ (7 0 )(4.89 0 ) 9 G 9 0 Pa 0.75 m, c d 0.00 m 800 00 400 N m, J c (0.00).74 0 m (400)(0.75) C / 8.7 0 rad 9 GJ (9 0 )(.74 0 ) 4 4 4 c d c d 0.00 m 0.00 m, 0.50 m J c c (400)(0.50) CD / 5.08 0 rad 9 GJ (9 0 )(.00 0 ) 4 4 4 4 4 (0.00 0.00 ).00 0 m ngle of twist at. / / / C C D 05.080 0 rad.0 PROPRIERY MERI. Copyright 05 McGraw-Hill Education. his is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. his document may not be copied, scanned, duplicated, forwarded, distributed, or posted 98
PROEM.8 he design specifications of a.-m-long solid transmission shaft require that the angle of twist of the shaft not exceed 4 when a torque of 750 N m is applied. Determine the required diameter of the shaft, knowing that the shaft is made of a steel with an allowable shearing stress of 90 MPa and a modulus of rigidity of 77. GPa. SOUION 4 750 N m, 4 9.8 0 rad,. m, J c ased on angle of twist. 4 GJ Gc c ased on shearing stress. J c 9 90 MPa 90 0 Pa G 77. GPa 77. 0 Pa c c ()(750)(.) G (77. 0 )(9.8 0 ) 4 4 9 ()(750) (90 0 ) 7.44 0 m 8.0 0 m Use larger value. c 8.0 0 m 8.0 mm d c.mm PROPRIERY MERI. Copyright 05 McGraw-Hill Education. his is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. his document may not be copied, scanned, duplicated, forwarded, distributed, or posted 44
PROEM.5 7 mm 54 mm Steel core luminum jacket.5 m torque of magnitude 4kN mis applied at end of the composite shaft shown. Knowing that the modulus of rigidity is 77. GPa for the steel and 7 GPa for the aluminum, determine (a) the maximum shearing stress in the steel core, the maximum shearing stress in the aluminum jacket, (c) the angle of twist at. SOUION Steel core: GJ c d J c orque carried by steel core. G J / 4 4 9 0.07 m (0.07) 84.79 0 9 9 (77. 0 )(84.79 0 ) 4.44 0 N m luminum jacket: GJ c d 0.07 m, c d 0.0 m J c c 4 4 4 4 4 (0.0 0.07 ).8055 0 m 9 (7 0 )(.8055 0 ) 48.70 0 N m orque carried by aluminum jacket. G J / otal torque: ( GJ GJ) / (a) 40 5.50 rad/m G J G J 4.440 48.70 0 Maximum shearing stress in steel core. 9 G Gc (77. 0 )(0.07)(5.5 0 ) 7.7 0 Pa 7.7 MPa Maximum shearing stress in aluminum jacket. 9 G Gc (7 0 )(0.0)(5.5 0 ) 4.4 0 Pa 4.4MPa (c) ngle of twist. (.5)(5.5 0 ) 88.8 0 rad 5.0 PROPRIERY MERI. Copyright 05 McGraw-Hill Education. his is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. his document may not be copied, scanned, duplicated, forwarded, distributed, or posted 4
PROEM.58 5 m ' s the hollow steel shaft shown rotates at 80 rpm, a stroboscopic measurement indicates that the angle of twist of the shaft is. Knowing that G 77. GPa, determine (a) the power being transmitted, the maximum shearing stress in the shaft. 0 mm 5 mm SOUION c d 0 mm c d.5 mm 4 4 4 4 J c c [(0) (.5) ] 4 4.4 0 mm.4 0 m 0.05 rad GJ 9 GJ (77. 0 )(.4 0 )(0.05) 997. N m 5 ngular speed: f 80 rpm rev/sec Hz (a) Power being transmitted. P f ()(997.) 8.800 W P 8.80 kw Maximum shearing stress. m c J (997.)(0 0 ).4 0 4. 0 Pa 4. MPa m PROPRIERY MERI. Copyright 05 McGraw-Hill Education. his is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. his document may not be copied, scanned, duplicated, forwarded, distributed, or posted 4