Physics 1501 Lecture 19
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1 Physcs 1501 ectue 19 Physcs 1501: ectue 19 Today s Agenda Announceents HW#7: due Oct. 1 Mdte 1: aveage 45 % Topcs otatonal Kneatcs otatonal Enegy Moents of Ineta Physcs 1501: ectue 19, Pg 1 Suay (wth copason to 1-D kneatcs) Angula nea constant " " 0 #t 1 0 " 0t # t a constant v v 0 at x x 0 v 0 t 1 at And fo a pont at a dstance fo the otaton axs: x θ v ω a α Physcs 1501: ectue 19, Pg Page 1
2 Physcs 1501 ectue 19 Exaple: Wheel And ope A wheel wth adus 0.4 otates feely about a fxed axle. Thee s a ope wound aound the wheel. Statng fo est at t 0, the ope s pulled such that t has a constant acceleaton a 4/s. How any evolutons has the wheel ade afte 10 seconds? (One evoluton π adans) a Physcs 1501: ectue 19, Pg 3 Wheel And ope... Use a α to fnd α: α a / 4/s / ad/s Now use the equatons we deved above just as you would use the kneatc equatons fo the begnnng of the seeste t t 0 0(10) (10)(10) ad 500 ad x 80 ev 1 " ev ad a α Physcs 1501: ectue 19, Pg 4 Page
3 Physcs 1501 ectue 19 otaton & Knetc Enegy Consde the sple otatng syste shown below. (Assue the asses ae attached to the otaton axs by assless gd ods). The knetc enegy of ths syste wll be the su of the knetc enegy of each pece: ω Physcs 1501: ectue 19, Pg 5 otaton & Knetc Enegy... So: K 1 v but v ω K 1 ( ) 1 " " whch we wte as: K 1 I v 4 4 ω v 1 I 3 3 v Defne the oent of neta about the otaton axs I has unts of kg. v 3 Physcs 1501: ectue 19, Pg 6 Page 3
4 Physcs 1501 ectue 19 ectue 19, Act 1 otatonal Knetc Enegy I have two basketballs. BB#1 s attached to a 0.1 long ope. I spn aound wth t at a ate of evolutons pe second. BB# s on a 0. long ope. I then spn aound wth t at a ate of evolutons pe second. What s the ato of the knetc enegy of BB# to that of BB#1? A) 1/4 B) 1/ C) 1 D) E) 4 BB#1 BB# Physcs 1501: ectue 19, Pg 7 otaton & Knetc Enegy... The knetc enegy of a otatng syste looks sla to that of a pont patcle: Pont Patcle otatng Syste K 1 v v s lnea velocty s the ass. K 1 I ω s angula velocty I s the oent of neta about the otaton axs. I Physcs 1501: ectue 19, Pg 8 Page 4
5 Physcs 1501 ectue 19 Moent of Ineta K 1 So I whee I Notce that the oent of neta I depends on the dstbuton of ass n the syste. The futhe the ass s fo the otaton axs, the bgge the oent of neta. Fo a gven object, the oent of neta wll depend on whee we choose the otaton axs (unlke the cente of ass). We wll see that n otatonal dynacs, the oent of neta I appeas n the sae way that ass does when we study lnea dynacs Physcs 1501: ectue 19, Pg 9 Calculatng Moent of Ineta We have shown that fo N dscete pont asses dstbuted about a fxed axs, the oent of neta s: I whee s the dstance fo the ass to the axs of otaton. Exaple: Calculate the oent of neta of fou pont asses () on the cones of a squae whose sdes have length, about a pependcula axs though the cente of the squae: Physcs 1501: ectue 19, Pg 10 Page 5
6 Page 6 Physcs 1501 ectue 19 Physcs 1501: ectue 19, Pg 11 Calculatng Moent of Ineta... The squaed dstance fo each pont ass to the axs s: / " # $ % & 4 I N 1 so I Usng the Pythagoean Theoe Physcs 1501: ectue 19, Pg 1 Calculatng Moent of Ineta... Now calculate I fo the sae object about an axs though the cente, paallel to the plane (as shown): I N 1 I
7 Physcs 1501 ectue 19 Calculatng Moent of Ineta... Fnally, calculate I fo the sae object about an axs along one sde (as shown): N I I Physcs 1501: ectue 19, Pg 13 Calculatng Moent of Ineta... Fo a sngle object, I clealy depends on the otaton axs I I I Physcs 1501: ectue 19, Pg 14 Page 7
8 Physcs 1501 ectue 19 ectue 19, Act Moent of Ineta A tangula shape s ade fo dentcal balls and dentcal gd, assless ods as shown. The oent of neta about the a, b, and c axes s I a, I b, and I c espectvely. Whch of the followng s coect: (a) (b) (c) I a > I b > I c I a > I c > I b I b > I a > I c a b c Physcs 1501: ectue 19, Pg 15 Calculatng Moent of Ineta... Fo a dscete collecton of pont asses we found: N I 1 Fo a contnuous sold object we have to add up the contbuton fo evey nfntesal ass eleent d. We have to do an ntegal to fnd I : I d d Physcs 1501: ectue 19, Pg 16 Page 8
9 Physcs 1501 ectue 19 Moents of Ineta Soe exaples of I fo sold objects: d Sold dsk o cylnde of ass M and adus, about a pependcula axs though ts cente. I " d d "dv "#d I 1 M I "# $ 3 d 1 "#4 " M V M # I 1 M Physcs 1501: ectue 19, Pg 17 Moents of Ineta... Soe exaples of I fo sold objects: I 5 M Sold sphee of ass M and adus, about an axs though ts cente. I 3 M Thn sphecal shell of ass M and adus, about an axs though ts cente. Physcs 1501: ectue 19, Pg 18 Page 9
10 Physcs 1501 ectue 19 Moents of Ineta Soe exaples of I fo sold objects: I M Thn hoop (o cylnde) of ass M and adus, about an axs though ts cente, pependcula to the plane of the hoop. I 1 M Thn hoop of ass M and adus, about an axs though a daete. Physcs 1501: ectue 19, Pg 19 Paallel Axs Theoe Suppose the oent of neta of a sold object of ass M about an axs though the cente of ass s known, I CM The oent of neta about an axs paallel to ths axs but a dstance away s gven by: I PAAE I CM M So f we know I CM, t s easy to calculate the oent of neta about a paallel axs. Physcs 1501: ectue 19, Pg 0 Page 10
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