Physcs 07 Lectue 6 Goals: Lectue 6 Chapte Extend the patcle odel to gd-bodes Undestand the equlbu of an extended object. Analyze ollng oton Undestand otaton about a fxed axs. Eploy consevaton of angula oentu concept Assgnent: HW7 due Mach 5 th Afte Spng Beak Tuesday: Catch up Physcs 07: Lectue 6, Pg otatonal Dynacs: A chld s toy, a physcs playgound o a student s nghtae A ey-go-ound s spnnng and we un and jup on t. What does t do? What pncples would apply? We ae standng on the and ou fends spn t faste. What happens to us? We ae standng on the a walk towads the cente. Does anythng change? Physcs 07: Lectue 6, Pg Page
Physcs 07 Lectue 6 otatonal Vaables otaton about a fxed axs: Consde a dsk otatng about an axs though ts cente: θ How do we descbe the oton: dθ dt π (ad/s) T v Tangental dx (Analogous to the lnea case v ) dt / Physcs 07: Lectue 6, Pg 3 otatonal Vaables... ecall: At a pont a dstance away fo the axs of otaton, the tangental oton: x θ v a α v θ x α constant + α t 0 θ θ0 + 0 t + α t (angula accelaton n ad/s (angula velocty n ad/s) (angula poston n ad) ) α Physcs 07: Lectue 6, Pg 4 Page
Physcs 07 Lectue 6 Copason to -D kneatcs Angula α constant 0 + α t θ θ α 0 + 0 t + t v x x Lnea a constant v 0 + a t 0 + v0 t + a t And fo a pont at a dstance fo the otaton axs: x θ v a T α Hee a T efes to tangental acceleaton Physcs 07: Lectue 6, Pg 5 Execse otatonal Defntons A fend at a paty (pehaps a lttle tpsy) sees a dsk spnnng and says Ooh, look! Thee s a wheel wth a negatve and postve α! Whch of the followng s a tue stateent about the wheel?? A. The wheel s spnnng counte-clockwse and slowng down. B. The wheel s spnnng counte-clockwse and speedng up. C. The wheel s spnnng clockwse and slowng down. D. The wheel s spnnng clockwse and speedng up Physcs 07: Lectue 6, Pg 6 Page 3
Physcs 07 Lectue 6 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 T 4 /s. How any evolutons has the wheel ade afte 0 seconds? (One evoluton π adans) a T Physcs 07: Lectue 6, Pg 7 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 T 4 /s. How any evolutons has the wheel ade afte 0 seconds? (One evoluton π adans) evolutons (θ θ 0 ) / π and a T α θ θ 0 + 0 t + ½ α t (θ θ 0 ) / π 0 + ½ (a T /) t / π (0.5 x 0 x 00) / 6.8 a T Physcs 07: Lectue 6, Pg 8 Page 4
Physcs 07 Lectue 6 Syste of Patcles (Dstbuted Mass): Untl now, we have consdeed the behavo of vey sple systes (one o two asses). But eal objects have dstbuted ass! Fo exaple, consde a sple otatng dsk and equal ass plugs at dstances and. Copae the veloctes and knetc eneges at these two ponts. Physcs 07: Lectue 6, Pg 9 Syste of Patcles (Dstbuted Mass): ½ v ½ ( ) ½ (v) ½ ( ) Twce the adus, fou tes the knetc enegy The otaton axs attes too! v ) ( Physcs 07: Lectue 6, Pg 0 Page 5
Physcs 07 Lectue 6 A specal pont fo otaton Syste of Patcles: Cente of Mass (CM) If an object s not held then t wll otate about the cente of ass. Cente of ass: Whee the syste s balanced! Buldng a oble s an execse n fndng centes of ass. + + oble Physcs 07: Lectue 6, Pg Syste of Patcles: Cente of Mass How do we descbe the poston of a syste ade up of any pats? Defne the Cente of Mass (aveage poston): Fo a collecton of N ndvdual pont lke patcles whose asses and postons we know: CM N M CM y x (In ths case, N ) Physcs 07: Lectue 6, Pg Page 6
Physcs 07 Lectue 6 Saple calculaton: Consde the followng ass dstbuton: CM N M X CM (,6) Y CM ( x 0 + x + x 0 )/4 etes CM î + Y CM X CM ( x 0 + x + x 4 )/4 etes ĵ (,) + Z CM X CM etes Y CM 6 etes kˆ (0,0) (4,0) Physcs 07: Lectue 6, Pg 3 Syste of Patcles: Cente of Mass Fo a contnuous sold, convet sus to an ntegal. y x d CM d d d M whee d s an nfntesal ass eleent. Physcs 07: Lectue 6, Pg 4 Page 7
Physcs 07 Lectue 6 Connecton wth oton... So fo a gd object whch otates about ts cente of ass and whose CM s ovng: N TOTAL TOTAL + otaton + otaton Fo a pont p otatng: p Tanslaton MV CM CM M pvp p( p ) V CM Physcs 07: Lectue 6, Pg 5 otaton & netc 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: ½ v + ½ v + ½ 3 v 3 + ½ 4 v 4 4 4 v 4 3 3 Physcs 07: Lectue 6, Pg 6 Page 8
Physcs 07 Lectue 6 otaton & netc Enegy Notce that v, v, v 3 3, v 4 4 So we can ewte the suaton: 4 4 4 ] v [ We ecognze the quantty, oent of neta o I, and wte: 4 otatonal I 4 I N 3 3 Physcs 07: Lectue 6, Pg 7 Calculatng Moent of Ineta I N 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 L, about a pependcula axs though the cente of the squae: L Physcs 07: Lectue 6, Pg 8 Page 9
Physcs 07 Lectue 6 Calculatng Moent of Ineta... Fo a sngle object, I depends on the otaton axs! Exaple: I 4 4 ( / L / ) I L I L I L L Physcs 07: Lectue 6, Pg 9 Hoe Execse 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: I (A) I a > I b > I c (B) (C) I a > I c > I b I b > I a > I c a b c Physcs 07: Lectue 6, Pg 0 Page 0
Physcs 07 Lectue 6 Hoe Execse Moent of Ineta I a (L) I b 3 L I c (L) Whch of the followng s coect: (A) I a > I b > I c (B) (C) I a > I c > I b I b > I a > I c L L a b c Physcs 07: Lectue 6, Pg Moents of Ineta Fo a contnuous sold object we have to add up the contbuton fo evey nfntesal ass eleent d. An ntegal s equed to fnd I : I Soe exaples of I fo sold objects: d d L d Sold dsk o cylnde of ass M and adus, about pependcula axs though ts cente. Use the table I ½ M Physcs 07: Lectue 6, Pg Page
Physcs 07 Lectue 6 Moents of Ineta Soe exaples of I fo sold objects: Thn hoop (o cylnde) of ass M and adus, about an axs though t cente, pependcula to the plane of the hoop s just M Thn hoop of ass M and adus, about an axs though a daete. Physcs 07: Lectue 6, Pg 3 Execse otatonal netc Enegy We have two balls of the sae ass. Ball s attached to a 0. long ope. It spns aound at evolutons pe second. Ball s on a 0. long ope. It spns aound at evolutons pe second. What s the ato of the knetc enegy of Ball to that of Ball? A. ¼ B. ½ C. D. E. 4 Ball Ball I I Physcs 07: Lectue 6, Pg 4 Page
Physcs 07 Lectue 6 Execse otatonal netc Enegy / ½ / ½ 0. / 0. 4 What s the ato of the knetc enegy of Ball to that of Ball? (A) /4 (B) / (C) (D) (E) 4 Ball Ball Physcs 07: Lectue 6, Pg 5 otaton & netc Enegy... The knetc enegy of a otatng syste looks sla to that of a pont patcle: Pont Patcle v v s lnea velocty s the ass. otatng Syste I I s angula velocty I s the oent of neta about the otaton axs. Physcs 07: Lectue 6, Pg 6 Page 3
Physcs 07 Lectue 6 So Moent of Ineta and otatonal Enegy I whee 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. I Fo a gven object, the oent of neta depends on whee we choose the otaton axs (unlke the cente of ass). In otatonal dynacs, the oent of neta I appeas n the sae way that ass does n lnea dynacs! Physcs 07: Lectue 6, Pg 7 Execse Wok & Enegy Stngs ae wapped aound the ccufeence of two sold dsks and pulled wth dentcal foces, F, fo the sae lnea dstance, d. Dsk has a bgge adus, but both ae dentcal ateal (.e. the densty ρ M / V s the sae). Both dsks otate feely aound axes though the centes, and stat at est. Whch dsk has the bggest angula velocty afte the dop? W F d ½ I (A) Dsk (B) Dsk (C) Sae stat fnsh F d F Physcs 07: Lectue 6, Pg 8 Page 4
Physcs 07 Lectue 6 Execse Wok & Enegy Stngs ae wapped aound the ccufeence of two sold dsks and pulled wth dentcal foces fo the sae lnea dstance. Dsk has a bgge adus, but both ae dentcal ateal (.e. the densty ρ M/V s the sae). Both dsks otate feely aound axes though the centes, and stat at est. Whch dsk has the bggest angula velocty afte the dop? W F d ½ I ½ I (I / I ) ½ (A) Dsk (B) Dsk (C) Sae and I < I stat fnsh F d F Physcs 07: Lectue 6, Pg 9 TOTAL TOTAL Lectue 6 + otatonal + otatonal Tanslatonal MV CM Assgnent: HW7 due Mach 5 th Fo the next Tuesday: Catch up otatonal I I Physcs 07: Lectue 6, Pg 30 Page 5
Physcs 07 Lectue 6 Lectue 6 Assgnent: HW7 due Mach 5 th Afte Spng Beak Tuesday: Catch up Physcs 07: Lectue 6, Pg 3 Page 6