2/4/2012. τ = Reasoning Strategy 1. Select the object to which the equations for equilibrium are to be applied. Ch 9. Rotational Dynamics

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1 /4/ Ch 9. Rtatna Dynamcs In pue tansatna mtn, a pnts n an bject tae n paae paths. ces an Tques Net ce acceeatn. What causes an bject t hae an angua acceeatn? TORQUE 9. The ctn ces an Tques n Rg Objects The mst genea mtn s a cmbnatn tansatn an tatn. The amunt tque epens n whee an n what ectn the ce s appe, as we as the catn the axs tatn. DEINITION O TORQUE agntue Tque (agntue the ce) x (Lee am) τ Dectn: The tque s pste when the ce tens t puce a cuntecckwse tatn abut the axs. SI Unt Tque: newtn x mete (N m) ctn ces an Tques Exampe The ches Tenn The tenn exets a ce magntue 79 N. Detemne the tque (magntue an ectn) ths ce abut the anke jnt. τ cs m Rg Objects n Equbum EQUILIBRIU O RIGID BODY g by s n equbum t has ze tansatna acceeatn an ze angua acceeatn. In equbum, the sum the extenay appe ces s ze, an the sum the extenay appe tques s ze. a x a y Reasnng Stategy. Seect the bject t whch the equatns equbum ae t be appe.. Daw a ee-by agam that shws a the extena ces actng n the bject. 3. Chse a cnenent set x, y axes an ese a ces nt cmpnents that e ang these axes. τ ( 7 N)( 3. m) cs55 5 N m x α τ y 4. ppy the equatns that specy the baance ces at equbum. (Set the net ce n the x an y ectns equa t ze.) 5. Seect a cnenent axs tatn. Set the sum the tques abut ths axs equa t ze.. Se the equatns the ese unknwn quanttes. 3 4

2 /4/ Exampe 3 Dng Ba wman whse weght s 53 N s pse at the ght en a ng ba wth ength 3.9 m. The ba has neggbe weght an s suppte by a ucum.4 m away m the et en. n the ces that the bt an the ucum exet n the ba. τ W W W W ( 53 N)( 3.9 m) y.4 m 48 N + W + 48 N 53 N 95 N 5 Exampe 5 Bybung The am s hznta an weghs 3. N. The et musce can suppy 84 N ce. What s the weght the heaest umbbe he can h? τ W a a W + (.5 m) sn3. Wa a + W ( 3. N)(.8 m) + ( 84 N)(.5 m). m sn3. 8. N Cente Gaty The cente gaty a g by s the pnt at whch ts weght can be cnsee t act when the tque ue t the weght s beng cacuate. Exampe Cente Gaty an m The hznta am s cmpse thee pats: the uppe am (7 N), the we am ( N), an the han (4. N). n the cente gaty the am eate t the shue jnt. W x + W x + L W + W + L ( 7 N)(.3 m) + ( N)(.38 m) + ( 4. N)(.m) 7 N + N + 4. N.8 m W x W + Wx + L + W + L 7 8

3 /4/ 9.3 Cente Gaty Newtn s n Law Rtatna tn Cnceptua Exampe 7 Oeang a Cag ane Ths accent ccue because the pane was eae twa the ea. Hw a sht n the cente gaty the pane cause the accent? τ ( m )α Net extena tque ment neta 9 T ma T τ T a T α τ ( m )α ment Ineta, I Net extena tque ment neta τ I α I ngua acceeatn ( m ) ngua acceeatn must be expesse n aans/s. 9.4 Newtn s Secn Law Rtatna tn but a xe xs Exampe 9 The ment Ineta Depens n Whee the xs Is. (b) Tw patces each hae mass an ae xe at the ens a thn g. The ength the s L. n the mment neta when ths bject tates eate t an axs that s pepencua t the at (a) ne en an (b) the cente. I m m + m m L + m L (a) ( m ) m + m m( ) m( L) I + m m m I ml L ( ) ( ) ( ) m m m L L I ml Newtn s n Law Rtatna tn Exampe Hstng a Cate The cmbne mment neta the ua puey s 5. kg m. The cate weghs 44 N. tensn 5 N s mantane n the cabe attache t the mt. n the angua acceeatn the ua puey. τ T T Iα a y ( mg + ma ) Iα T y α ( mg + m α ) Iα T T mg ma y α T mg + ma y T mg I + m ( 5 N)(. m) ( 45kg)( 9.8m s )(. m) 4. kg m + ( 45kg)(. m).3a s y 3

4 /4/ 9.5 Rtatna Wk an Enegy DEINITION O ROTTIONL WORK s θ Rtatna wk ne by a cnstant tque n tunng an bject thugh an ange s W R τθ The ange must be n aans. 9.5 Rtatna Wk an Enegy DEINITION O ROTTIONL KINETIC ENERGY KE T m ω m T ω The tatna knetc enegy a g tatng bject s KE R Iω W s θ τ Unts: jue (J) KE ( m ω )) ( m ) ω Iω The angua spee must be n a/s. Unts: jue (J) W τθ Rtatna Wk an Enegy Exampe 3 Rng Cynes thn-wae hw cyne (mass m h, aus h ) an a s cyne (mass m s, aus s ) stat m est at the tp an ncne. Detemne whch cyne has the geatest tansatna spee upn eachng the bttm. E m + Iω + mgh ENERGY CONSERVTION m ω + I + mgh + Iω + Iω m + mgh m mgh ω m + I mgh mgh m + I 9. ngua mentum DEINITION O NGULR OENTU The angua mmentum L by s mment neta * angua ecty L Iω The angua spee must be n a/s. Unts: kg m /s CONSERVTION O NGULR OENTU ngua mmentum emans cnstant (s cnsee) net extena tque s ze. Cnceptua Exampe 4 Spnnng Skate n ce skate s spnnng wth bth ams an a eg utstetche. She pus he ams an eg nwa an he spnnng mtn changes amatcay. Use the pncpe cnseatn angua mmentum t expan hw an why he spnnng mtn changes. The cyne wth the smae mment neta w hae a geate na tansatna spee. 5 4

5 /4/ 9. ngua mentum Exampe 5 Satete n Eptca Obt satete n eptca bt aun eath. Its csest appach s 8.37x m m eath-cente. Its geatest stance s 5.x m m eath-cente. Spee satete at pegee s 845 m/s. n spee at apgee. angua mmentum cnseatn I ω I ω I m ω m m ( 8.37 m )( 845 m s ) 5. m L Iω 8 m s 7 5