7/1/2008. Adhi Harmoko S. a c = v 2 /r. F c = m x a c = m x v 2 /r. Ontang Anting Moment of Inertia. Energy

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1 7//008 Adh Haoko S Ontang Antng Moent of neta Enegy Passenge undego unfo ccula oton (ccula path at constant speed) Theefoe, thee ust be a: centpetal acceleaton, a c. Theefoe thee ust be a centpetal foce, c. t s dected towad the ccle s cente t depends on speed and sze of ccle: When Ontang Antng s tunng: acceleaton s towad the cente of ccle (nwad) so fcttous foce s outwad. Ths s fcttous foce s called centfugal foce Centfugal foce s an expeence of neta: t s NOT a eal foce. a c = v / c = x a c = x v / 3 4 Dufan duduk d bangku palng lua pada peanan ontang antng, and Duf duduk d tengah tengah antaa dufan dan ttk tengah. Ontang antng begeak ebuat satu putaan penuh setap dua detk. Kecepatan angula Duf saa dengan (a) Dufan (b) Dua kal Dufan (c) Setengah Dufan Kecepatan angula pada sebaang ttk pada benda pejal yang beputa pada pusat yang tetap adalah saa. Dufan & Duf beputa sekal (π adans) selaa dua detk (Kecepatan lne dufan dan duf v akan bebeda kaena v = ). V Duf = V Dufan 5 6

2 7//008 otaton about a fxed axs: Consde a dsk otatng about an axs though ts cente: st, ecall what we leaned about Unfo Ccula Moton: d = dt dx (Analogous to v = ) dt 7 Now suppose can change as a functon of te: We defne the angula acceleaton: Consde the case when α s constant. α We can ntegate ths to fnd and as a functon of te: d d α = = dt dt α = constant = + αt 0 = 0 + 0t + αt 8 α = constant = 0 + αt = 0 + 0t + αt α v x Angula ne α = constant a = constant + αt = 0 v v + at = 0 ecall also that fo a pont at a dstance away fo the axs of otaton: x = v = And takng the devatve of ths we fnd: a = α = t + αt x = x 0 + v 0 t + at And fo a pont at a dstance fo the otaton axs: 9 0 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: So: K v but v = K ( ) = whch we wte as: K = v 4 Defne the oent of neta about the otaton axs v 3 3 v v

3 7//008 The knetc enegy of a otatng syste looks sla to that of a pont patcle: Pont Patcle K = v v s lnea velocty. s the ass otatng Syste K = s angula velocty s the oent of neta about the otaton axs K = Notce that the oent of neta depends on the dstbuton of ass n the syste. The futhe the ass s fo the otaton axs, the bgge the oent of neta. o 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 appeas n the sae way that ass does when we study lnea dynacs! 3 4 We have shown that fo N dscete pont asses dstbuted about a fxed axs, the oent of neta s: 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: The squaed dstance fo each pont ass to the axs s: so = = N Usng the Pythagoean Theoe = = 4 = / 5 6 Now calculate fo the sae object about an axs though the cente, paallel to the plane (as shown): = = 4 = N nally, calculate fo the sae object about an axs along one sde (as shown): N = =

4 7//008 o a sngle object, clealy depends on the otaton axs!! o a dscete collecton of pont asses we found: o 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 : = d d 9 0 Soe exaples of fo sold objects: Thn hoop (o cylnde) of ass M and adus, about an axs though ts cente, pependcula to the plane of the hoop. = M Soe exaples of fo sold objects: Sold sphee of ass M and adus, about an axs though ts cente. = M 5 Thn hoop of ass M and adus, about an axs though a daete. = M Sold dsk o cylnde of ass M and adus, about a pependcula axs though ts cente. = M Soe exaples of fo sold objects: Thn od of ass M and length, about a pependcula axs though ts cente. = M Two sphees have the sae adus and equal asses. One s ade of sold alunu, and the othe s ade fo a hollow shell of gold. Whch one has the bggest oent of neta about an axs though ts cente? (a) sold alunu (b) hollow gold (c) sae Thn od of ass M and length, about a pependcula axs though ts end. = M 3 sold hollow sae ass & adus 3 4 4

5 7//008 Moent of neta depends on ass (sae fo both) and dstance fo axs squaed, whch s bgge fo the shell snce ts ass s located fathe fo the cente. The sphecal shell (gold) wll have a bgge oent of neta Suppose the oent of neta of a sold object of ass M about an axs though the cente of ass, CM, s known. The oent of neta about an axs paallel to ths axs but a dstance D away s gven by: SOD < SHE sold hollow So f we know CM, t s easy to calculate the oent of neta about a paallel axs. sae ass & adus 5 6 Consde a thn unfo od of ass M and length D. gue out the oent of neta about an axs though the end of the od. PAAE = CM + MD We know CM = M END = M + M = END D=/ x CM CM So M 3 whch agees wth the esult on a pevous slde. M Suppose a foce acts on a ass constaned to ove n a ccle. ^ Consde ts acceleaton n the decton at soe nstant: a = α ^ Now use Newton s nd aw n the decton: = a = α Multply by : = α α a ^ ^ 7 8 = α = Ι α use Defne toque: τ =. τ s the tangental foce tes the leve a. τ = α Toque has a decton: + zf t tes to ake the syste spn CCW. z f t tes to ake the syste spn CW. α a ^ ^ So fo a collecton of any patcles aanged n a gd confguaton: Snce the patcles ae connected gdly, they all have the sae α τ = α τ α = NET {, = { α τ Ι

6 7//008 τ NET = α Ths s the otatonal analogue of NET = a Toque s the otatonal analogue of foce: The aount of twst povded by a foce. Moent of neta s the otatonal analogue of ass. f s bg, oe toque s equed to acheve a gven angula acceleaton. Toque has unts of kg /s = (kg /s ) = N ecall the defnton of toque: τ = = snφ = snφ τ = p p = dstance of closest appoach Equvalent defntons φ φ φ p 3 3 So f φ = 0 o, then τ = 0 τ = snφ n whch of the cases shown below s the toque povded by the appled foce about the otaton axs bggest? n both cases the agntude and decton of the appled foce s the sae. And f φ = 90 o, then τ = axu (a) case (b) case (c) sae axs case case Toque = x (dstance of closest appoach) The appled foce s the sae. The dstance of closest appoach s the sae Toque s the sae! case case Consde the wok done by a foce actng on an object constaned to ove aound a fxed axs. o an nfntesal angula dsplaceent d: dw =. d = d cos(β) = d cos(90 φ) = d sn(φ) = sn(φ) d dw = τ d We can ntegate ths to fnd: Analogue of W = Δ axs W = τ W wll be negatve f τ and have opposte sgns! d β φ d = d

7 7//008 ecall the Wok/Knetc Enegy Theoe: ΔK = W NET Ths s tue n geneal, and hence apples to otatonal oton as well as lnea oton. So fo an object that otates about a fxed axs: A assless stng s wapped 0 tes aound a dsk of ass M = 40 g and adus = 0 c. The dsk s constaned to otate wthout fcton about a fxed axs though ts cente. The stng s pulled wth a foce = 0 N untl t has unwound. (Assue the stng does not slp, and that the dsk s ntally not spnnng). How fast s the dsk spnnng afte the stng has unwound? ( f ) WNET ΔK = = M The wok done s W = τ The toque s τ = (snce φ = 90 o ) The angula dsplaceent s π ad/ev x 0 ev So W = (. )(0 N)(0π ad) = 6.8 J τ M W NET = W = 6.8 J= ΔK = ecall that fo a dsk about ts cental axs s gven by = M ΔK = M = W M = 4W = M 4( 6.8 J) (.04kg)(. ) = 79.5 ad/s Stngs ae wapped aound the ccufeence of two sold dsks and pulled wth dentcal foces fo the sae dstance. Dsk has a bgge adus, but both have the sae oent of neta. Both dsks otate feely aound axes though the centes, and stat at est. Whch dsk has the bggest angula velocty afte the pull? (a) dsk (b) dsk (c) sae The wok done on both dsks s the sae! W = d The change n knetc enegy of each wll theefoe also be the sae snce W = ΔK But we know So snce = ΔK = d 4 4 7

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