PHYS 144 Secton 00 ecture #17 Wednesda, Oct. 9, 00 1. Rollng oton of a Rgd od. Torque. oment of Inerta 4. Rotatonal Knetc Energ 5. Torque and Vector Products Remember the nd term eam (ch 6 11), onda, Nov.! Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 1
ngular Dsplacement, Veloct, and cceleraton Usng what we have learned n the prevous slde, how θ would ou defne the angular dsplacement? How about the average angular speed? nd the nstantaneous angular speed? the same token, the average angular acceleraton nd the nstantaneous angular acceleraton? When rotatng about a fed as, ever partcle on a rgd object rotates through the same angle and has the same angular speed and angular acceleraton. Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 ω ω α α θ t f f θ t lm t 0 ω t f f lm t 0 t θ t ω ω t θ t dθ ω t dω θ f θ θ θ f
Rotatonal Knematcs The frst tpe of moton we have learned n lnear knematcs was under a constant acceleraton. We wll learn about the rotatonal moton under constant angular acceleraton, because these are the smplest motons n both cases. Just lke the case n lnear moton, one can obtan ngular Speed under constant angular acceleraton: ω f ω + αt ngular dsplacement under constant angular acceleraton: θ f One can also obtan ω f θ + ωt + ω + α 1 α t ( ) θ θ f Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00
Rollng oton of a Rgd od What s a rollng moton? To smplf the dscusson, let s make a few assumptons more generalzed case of a moton where the rotatonal as moves together wth the object rotatonal moton about the movng as 1. mt our dscusson on ver smmetrc objects, such as clnders, spheres, etc. The object rolls on a flat surface et s consder a clnder rollng wthout slppng on a flat surface R θ s srθ Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 Under what conon does ths Pure Rollng happen? The total lnear dstance the C of the clnder moved s Thus the lnear speed of the C s srθ vc ds d R θ Rω Conon for Pure Rollng 4
ore Rollng oton of a Rgd od The magntude of the lnear acceleraton of the C s a C dv C d R ω Rα P C P vc v C s we learned n the rotatonal moton, all ponts n a rgd bod moves at the same angular speed but at a dfferent lnear speed. t an gven tme the pont that comes to P has 0 lnear speed whle the pont at P has twce the speed of C C s movng at the same speed at all tmes. Wh?? rollng moton can be nterpreted as the sum of Translaton and Rotaton P C P v C vc v C + vrw P C v0 P vrw P C P v C v C Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 5
P Torque Torque s the tendenc of a force to rotate an object about an as. Torque, t, s a vector quantt. d r d oment arm F F φ ne of cton agntude of torque s defned as the product of the force eerted on the object to rotate t and the moment arm. Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 Consder an object pvotng about the pont P b the force F beng eerted at a dstance r. The lne that etends out of the tal of the force vector s called the lne of acton. The perpendcular dstance from the pvotng pont P to the lne of acton s called oment arm. When there are more than one force beng eerted on certan ponts of the object, one can sum up the torque generated b each force vectorall. The conventon for sgn of the torque s postve f rotaton s n counter-clockwse and negatve f clockwse. τ rfsnφ Fd τ τ 1 + τ Fd F 1 1 d 6
Eample for Torque one pece clnder s shaped as n the fgure wth core secton protrudng from the larger drum. The clnder s free to rotate around the central as shown n the pcture. rope wrapped around the drum whose radus s R 1 eerts force F 1 to the rght on the clnder, and another force eerts F on the core whose radus s R downward on the clnder. ) What s the net torque actng on the clnder about the rotaton as? R 1 F 1 The torque due to F 1 τ 1 R 1 F 1 and due to F τ R F R So the total torque actng on the sstem b the forces s τ τ 1 + τ R F + R 1 1 F F Suppose F 1 5.0 N, R 1 1.0 m, F 15.0 N, and R 0.50 m. What s the net torque about the rotaton as and whch wa does the clnder rotate from the rest? Usng the above result τ R 1F1 + RF 5.0 1.0 + 15.0 0.50. 5N m The clnder rotates n counter-clockwse. Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 7
What does ths mean? r Torque & ngular cceleraton F t F r m et s consder a pont object wth mass m rotatng on a crcle. What forces do ou see n ths moton? The tangental force F t and radal force F r The tangental force F t s F t ma t The torque due to tangental force F t s What do ou see from the above relatonshp? What law do ou see from ths relatonshp? Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 τ r F t τ Iα mrα ma t r mr α Iα Torque actng on a partcle s proportonal to the angular acceleraton. nalogs to Newton s nd law of moton n rotaton. How about a rgd object? The eternal tangental force df t s df t dma dmrα t df t dm The torque due to tangental force F t s dτ df t r ( r dm)α The total torque s r τ αr dm Iα What s the contrbuton due O to radal force and wh? Contrbuton from radal force s 0, because ts lne of acton passes through the pvotng pont, makng the moment arm 0. 8
Eample for Torque and ngular cceleraton unform rod of length and mass s attached at one end to a frctonless pvot and s free to rotate about the pvot n the vertcal plane. The rod s released from rest n the horzontal poston. What are the ntal angular acceleraton of the rod and the ntal lnear acceleraton of ts rght end? / g Snce the moment of nerta of the rod when t rotates about one end The onl force generatng torque s the gravtatonal force g τ Fd F g Iα I r dm λ d 0 0 0 α We obtan g I g g Usng the relatonshp between tangental and angular acceleraton a t g α What does ths mean? The tp of the rod falls faster than an object undergong a free fall. Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 9
oment of Inerta Rotatonal Inerta: easure of resstance of an object to changes n ts rotatonal moton. Equvalent to mass n lnear moton. For a group of partcles I m r For a rgd bod I r dm What are the dmenson and unt of oment of Inerta? [ ] kg m Determnng oment of Inerta s etremel mportant for computng equlbrum of a rgd bod, such as a buldng. Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 10
Eample for oment of Inerta In a sstem conssts of four small spheres as shown n the fgure, assumng the rad are neglgble and the rods connectng the partcles are massless, compute the moment of nerta and the rotatonal knetc energ when the sstem rotates about the -as at ω. I l O m m b b Thus, the rotatonal knetc energ s l Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 Wh are some 0s? Snce the rotaton s about as, the moment of nerta about as, I, s I Ths s because the rotaton s done about as, and the rad of the spheres are neglgble. 1 1 K R Iω ( l ) ω l ω Fnd the moment of nerta and rotatonal knetc energ when the sstem rotates on the - plane about the z-as that goes through the orgn O. m r l + l + mb + mb l + mb K R I ( ) l + l + m 0 + m 0 m r 1 ω 1 l ( l ) + mb ω ( ) l +mb ω 11
O r v θ m Rotatonal Knetc Energ What do ou thnk the knetc energ of a rgd object that s undergong a crcular moton s? Knetc energ of a masslet, m, movng at a tangental speed, v, s Snce a rgd bod s a collecton of masslets, the total knetc energ of the rgd object s K R Snce moment of Inerta, I, s defned as The above epresson s smplfed as K m r ω I m r K 1 K ω R I 1 m v 1 m r ω 1 1 m r ω Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 1
Calculaton of oments of Inerta oments of nerta for large objects can be computed, f we assume the object conssts of small volume elements wth mass, m. The moment of nerta for the large rgd object s lm r m It s sometmes easer to compute moments of nerta n terms How can we do ths? of volume of the elements rather than ther mass Usng the volume denst, ρ, replace dm The moments of ρ dm n the above equaton wth dv. dm ρdv I dv nerta becomes ρr dv I m 0 Eample: Fnd the moment of nerta of a unform hoop of mass and radus R about an as perpendcular to the plane of the hoop and passng through ts center. dm The moment of nerta s I r dm R dm R r dm O R What do ou notce from ths result? The moment of nerta for ths object s the same as that of a pont of mass at the dstance R. Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 1
Eample for Rgd od oment of Inerta Calculate the moment of nerta of a unform rgd rod of length and mass about an as perpendcular to the rod and passng through ts center of mass. d What s the moment of nerta when the rotatonal as s at one end of the rod. Wll ths be the same as the above. Wh or wh not? The lne denst of the rod s so the masslet s The moment of nerta s Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 I r dm I r dm λ dm λd d / / d Snce the moment of nerta s resstance to moton, t makes perfect sense for t to be harder to move when t s rotatng about the as at one end. d 4 1 1 0 [( ) 0] ( ) 0 1 14 / /
Parallel s Theorem oments of nerta for hghl smmetrc object s eas to compute f the rotatonal as s the same as the as of smmetr. However f the as of rotaton does not concde wth as of smmetr, the calculaton can stll be done n smple manner usng parallel-as theorem. I IC + D C r C D What does ths theorem tell ou? (, ) C ( C, C ) Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 oment of nerta s defned I r dm + dm Snce and are C + ' C + ' One can substtute and n Eq. 1 to obtan [( ) ( ) ] I C + ' + C + ' dm C + C dm+ C' dm+ C ' dm+ ' + ' Snce the and are the dstance from C, b defnton ' dm 0 Therefore, the parallel-as theorem I ( C + C) dm+ ( ' + ' )dm D + I C oment of nerta of an object about an arbtrar as are thesame as the sum of moment of nerta for a rotaton about the C and that of 15 the C about the rotaton as. ( ) (1) ( ) ( )dm 'dm 0
Eample for Parallel s Theorem Calculate the moment of nerta of a unform rgd rod of length and mass about an as that goes through one end of the rod, usng parallel-as theorem. C d Usng the parallel as theorem The moment of nerta about the C The lne denst of the rod s so the masslet s + D I C Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 I I C 1 r dm λ dm λd The result s the same as usng the defnton of moment of nerta. Parallel-as theorem s useful to compute moment of nerta of a rotaton of a rgd object wth complcated shape about an arbtrar as / / + d d / 1 1 4 + 4 16 1 /
z trf Torque and Vector Product et s consder a dsk fed onto the orgn O and the force F eerts on the pont p. What happens? The dsk wll start rotatng counter clockwse about the Z as O r p The magntude of torque gven to the dsk b the force F s θ F τ Frsnφ ut torque s a vector quantt, what s the drecton? How s torque epressed mathematcall? What s the drecton? τ r F The drecton of the torque follows the rght-hand rule!! The above quantt s called Vector product or Cross product C C snθ What s the result of a vector product? nother vector Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 What s another vector operaton we ve learned? Scalar product C Result? scalar cos 17 θ
Propertes of Vector Product Vector Product s Non-commutatve If the order of operaton changes the result changes Followng the rght-hand rule, the drecton changes Vector Product of two parallel vectors s 0. What does ths mean? C sn θ sn 0 0 Thus, 0 If two vectors are perpendcular to each other snθ o sn90 Vector product follows dstrbuton law ( ) d ( ) C + The dervatve of a Vector product wth respect to a scalar varable s Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 d + C + d 18
Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 19 ore Propertes of Vector Product The relatonshp between unt vectors, k j and, k k j j Vector product of two vectors can be epressed n the followng determnant form 0 j j k k j j k k k j z z k j z z z z j k + ( ) z z ( ) j z z ( )k +
Smlart etween near and Rotatonal otons ll phscal quanttes n lnear and rotatonal motons show strkng smlart. Quanttes ass ength of moton Speed cceleraton Force Work Power omentum Knetc Energ ass Dstance Force Work Knetc near Wednesda, Oct. 9, 00 PHYS 144-00, Fall 00 v a W P dr dv F f Fd ma oment of Inerta I ngle Torque Work Rotatonal Rotatonal r dm ω α τ (Radan) dθ d ω F v P τω p m v 1 mv Iα K K R I ω θ W Iω θ θ f τdθ 1 0