CHAPTER 10. Rotation Motion. Dr. Abdallah M. Azzeer Angular velocity and angular acceleration

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1 CHAPTER 10 Rotaton Moton D. Abdallah M. Azzee Angula velocty and angula acceleaton A compact dsc otatng about a fxed axs though O pependcula to the plane of the fgue. (a) In ode to defne angula poston fo the dsc, a fxed efeence lne s chosen. A patcle at P s located at a dstance fom the otaton axs at O. (b) As the dsc otates, pont P moves though an ac length s on a ccula path of adus. Angle s θ = The SI unt of θ s adan (ad), whch s a pue numbe because t s a ato. D. Abdallah M. Azzee Page 1

2 ONE adan s the angle subtended by an ac length equal to the adus of the ac π 1 ev = 360 o = = π ad 1 ad = 57.3 o = ev π θ ( ad ) = θ (deg) o 180 The angle n ad s postve f t s counteclockwse wth espect to the postve x axs. The angula dsplacement s : θ = θ θ f D. Abdallah M. Azzee 3 The aveage angula speed s : θ θ θ f ω = = t f t t The nstantaneous angula velocty s : θ d θ ω = lm = t 0 t dt The SI unt of ω s ad/s The angula velocty ω s postve f the otaton s counteclockwse (when θ s nceasng). D. Abdallah M. Azzee 4 Page

3 The aveage angula acceleaton s : α ω ω ω t t t f = = f The nstantaneous angula velocty s : ω d ω α = lm = t 0 t d t The SI unt of α s ad/s D. Abdallah M. Azzee otatonal knematcs Rotatonal Moton wth Constant acceleaton Analogy between lnea and angula quanttes: x θ v ω a α D. Abdallah M. Azzee 6 Page 3

4 Example 10.1 A wheel otates wth a constant angula acceleaton of 3.50 ad/s. (a) If the angula speed of the wheel s.00 ad/s at t =0, though what angle does the wheel otate n.00s?and (b) how many ev has done dung ths nteval? 1 θf θ = ωt + αt = (.00) 11.0 = ev = 1.75 ev π o (c) What s the angula speed =630at t=.00 s? ω = ω + αt f = = 9.00 ad / s READ the est of example D. Abdallah M. Azzee Relatonshp Between Angula and Lnea Quanttes The poston : s = θ ( adan measue ) Note that fo all lnea-angula elatons, we must use the adan unt. The speed : ds dθ = dt dt v = ω ( adan measue ) The peod of evoluton T s π T = v π T = ω ( adan measue) D. Abdallah M. Azzee 8 Page 4

5 The acceleaton dv dω = dt dt a = α ( adan measue) t Fo a patcle n a ccula path, the centpetal acceleaton s : v a = = ω ( adan measue) a ponts adally nwad As a gd object otates about a fxed axs though O, the pont P expeences a tangental component of lnea acceleaton at and a adal component of lnea acceleaton a. The total lnea acceleaton of ths pont s a = a t + a. D. Abdallah M. Azzee 9 Dffeences between a and a t a s known as the adal component of lnea acceleaton, a t s known as the tangental component of the lnea acceleaton. a = ω s ponted adally nwad. It s non-zeo even f thee s no angula acceleaton. a = α s tangental to the otatonal path of the patcle, t s zeo f the t angula velocty s constant. Total lnea acceleaton s ( ) ( ) 4 t a = a + a = α + ω = α + ω D. Abdallah M. Azzee 10 Page 5

6 Example 10. Audo nfomaton on compact dscs ae tansmtted dgtally though the eadout system consstng of lase and lenses. The dgtal nfomaton on the dsc ae stoed by the pts and flat aeas on the tack. Snce the speed of eadout system s constant, t eads out the same numbe of pts and flats n the same tme nteval. In othe wods, the lnea speed s the same no matte whch tack s played. (a) Assumng the lnea speed s 1.3 m/s, fnd the angula speed of the dsc n evolutons pe mnute when the nne most (=3 mm) and oute most tacks (=58mm) ae ead. Usng the elatonshp between angula and tangental speed v = ω D. Abdallah M. Azzee 11 = 3 mm = 58 mm v 1.3 m / s 1.3 ω = = = = 56.5 ad / s = 9.00 ev / s = ev / mn 3 3 mm m / s 1.3 ω = = =.4 ad / s =.1 10 ev / mn 3 58 mm (b) The maxmum playng tme of a standad musc CD s 74 mnutes and 33 seconds. How many evolutons does the dsk make dung that tme? ( + ) ( + ) ω ω ev / mn 375 / mn θf = θ + ω t = 0 + ev / s 4473 s =.8 10 ev 60 f ω = = = ev (c) What s the total length of the tack past though the eadout mechansm? l = v t = 1.3 m/ s 4473 s= m t (d) What s the angula acceleaton of the CD ove the 4473 s tme nteval, assumng constant α? ( ω ω ) ( ) ad / s f 3 α = = = ad s t 4473s / D. Abdallah M. Azzee 1 Page 6

7 10-4. Rotatonal Knetc Enegy We teat the gd body as a collecton of patcles wth dffeent speeds, m s the mass of the th patcle and v s ts speed. The patcles move wth dffeent v but the same ω. Knetc enegy of a masslet, m, movng at a tangental speed, v, s 1 K = m v Snce a gd body s a collecton of masslets, the total knetc enegy of the gd object s K = m1v1 + mv + m3v = mv D. Abdallah M. Azzee K = K = m v = m ω = m ω 1 R 1 K R == m ω Snce moment of Ineta, I, s defned as I m Moment of Ineta The above expesson s smplfed as KR 1 = Iω Rotatonal Knetc Enegy: D. Abdallah M. Azzee 14 Page 7

8 Example 10.3 Oxygen Molecule d = m m = kg 1 1 ( ) ( ) 46 I = m = m d + m d = kg m d ω = ad/sec. 1 1 K R = Iω = J Cf) Aveage lnea knetc enegy at RT: 1 1 K L = Mv = J = 3K R D. Abdallah M. Azzee 15 Example 10.4 In a system conssts of fou small sphees as shown n the fgue, assumng the ad ae neglgble and the ods connectng the patcles ae massless, compute the moment of neta and the otatonal knetc enegy when the system otates about the y-axs at ω. Snce the otaton s about y axs, the moment of neta about y axs, I y, s I = m = Ma + Ma + m 0 + m 0 = Ma y 1 1 K R = Iω = ( Ma ) ω = Ma ω D. Abdallah M. Azzee 16 Page 8

9 Fnd the moment of neta and otatonal knetc enegy when the system otates on the x-y plane about the z-axs that goes though the ogn O. ( ) I = m = Ma + Ma + mb + mb = Ma + mb z 1 1 K R = Iω = ( Ma + mb ) ω = ( Ma + mb ) ω READ THE REST OF THE EXAMPLE D. Abdallah M. Azzee Calculaton of Moment of Ineta We defned the moment of neta as I m Moments of neta fo lage objects can be computed, f we assume the object conssts of small volume elements wth mass, m. The moment of neta fo the lage gd object s I = lm m = dm m 0 Usng the volume densty, ρ, eplace dm n the above equaton wth dv. dm ρ = dm = ρdv dv UNIT; kg.m D. Abdallah M. Azzee 18 Page 9

10 The moments of neta becomes I = ρ dv Example 10.5 Fnd the moment of neta of a unfom hoop of mass M and adus R about an axs pependcula to the plane of the hoop and passng though ts cente. The moment of neta s I = dm= R dm= MR What do you notce fom ths esult? The moment of neta fo ths object s the same as that of a pont of mass M at the dstance R. D. Abdallah M. Azzee 19 Example 10.6 Calculate the moment of neta of a unfom gd od of length L and mass M about an axs pependcula to the od and passng though ts cente of mass. The lne densty of the od s M λ = L M so the masslet s dm = λdx = dx L The moment of neta s L / x M M 1 3 I y = dm = dx = x L / L L M L L = 3L M L ML = = 3L L / L / D. Abdallah M. Azzee 0 Page 10

11 What s the moment of neta when the otatonal axs s at one end of the od. L x M M 1 3 I ' = dm dx x y = = 0 L L 3 0 M 3 M 3 = ( L ) 0 = ( L ) 3L 3L ML = 3 Wll ths be the same as the above. Why o why not? Snce the moment of neta s esstance to moton, t makes pefect sense fo t to be hade to move when t s otatng about the axs at one end. L D. Abdallah M. Azzee 1 Example 10.7 Ineta of Moment of a unfom sold cylnde Densty ρ M ρ πr l = M ρ = πr l l R di = dm = ρ π d l = πρl 3 d R 3 I πρl d = = ρπlr = MR 0 I = 1 MR D. Abdallah M. Azzee Page 11

12 Moments of Ineta of Homogeneous Rgd Objects wth Dffeent Geometes D. Abdallah M. Azzee 3 Page 1