Section 25 Decibing Rotational Motion What do object do and wh do the do it? We have a ve thoough eplanation in tem of kinematic, foce, eneg and momentum. Thi include Newton thee law of motion and two conevation law; eneg and linea momentum. Howeve, we have onl conideed the motion of object a a whole (tanlational motion). In geneal, object otate a well anlate. To complete ou undetanding then, we need to eviit the quetion what do object do? and wh do the do it? The good new i that we will be able to undetand otation b uing ou eplanation of tanlational motion a a model. In fact, the model i o good that we will onl need to intoduce one new law, but that come late. Fo now, let anwe the quetion, what do object do? b conideing otational motion. Section Outline 1. The Definition of the Rotational Vaiable 2. Rotational Kinematic 1. The Definition of the Rotational Vaiable Let think about an object that otate about ationa point at it cente, uch he wheel on a biccle that ha been inveted to epai a flat. If the wheel i pun, what do we need to do to decibe thi otational motion? Let focu on the tie valve tem. We alead made ome mall tep towad decibing the tanlational motion of the valve tem in tem of the tangential velocit and the centipetal acceleation. Howeve, we didn t think much about it actual poition. The location of the valve tem i mot eail decibed b the angle it make with oodinate ai,. So, let define, Angula Poition: The otational location of an object with epect to oodinate tem epeented b an angle. If the angle i changing, then we have an angula diplacement, Angula Diplacement: A change in angula poition o hange in angle. The ate at which the object pin can now be defined, Angula Velocit: The ate of angula diplacement. Finall, we can decibe the change in the pin ate b defining, Angula Acceleation: The ate of change of angula The analog to the wa we defined the quantitie to decibe tanlational motion i eactl the ame. Phicit aen t to bight if omething i woking, wh change it? Below i table compaing the quantitie we ue to decibe tanlational motion and the quantitie fo otational motion. 25-1
Tanlational Vaiable Poition: The location of an object with epect to oodinate tem. Diplacement: A change in poition. Velocit: The ate of diplacement. Acceleation: The ate of change of Rotational Vaiable Angle: The otational location of an object with epect to oodinate tem. Angula Diplacement: A change in angle. Angula Velocit: The ate of angula diplacement. Angula Acceleation: The ate of change of angula At the ight ae two image of the bike wheel. In the fit image the valve tem i at the bottom. In the econd, the bike wheel ha olled a ditance fowad. So, the ac length along the wheel i alo and the angle it ha otated though i labeled. Uing the cicumfeence of a cicle and one otation i 2π adian, we can make the atio, which can be witten in the cutoma fom, 2π = 2π, =. Thi i the elationhip between the ditance taveled long the ac (in mete) and the angula poition (in adian). We can find the peed of the point on the edge b diffeentiating with epect to time keeping in mind that the adiu i contant, d = d d () =. The quantit on the left i ou old fiend the tangential velocit, while the deivative of the angle i the angula velocit o pin ate. We ll ue a lowe cae omega and define, ω d, and ewite the tangential velocit a, = ω. Now, let uppoe the pin ate i changing. Diffeentiating both ide, d = dω. The quantit on the left i the tangential acceleation and the deivative on the ight i the ate of change of pin ate o the angula acceleation which we ll epeent with the Geek lette α, α dω and = α. 25-2
Don t foget the point one the edge alo ha entipetal acceleation which can be witten in tem of otational vaiable, = v 2 t = (ω)2 = ω 2. To ummaize, the definition needed to decibe tanlational motion ae analogou to the definition needed to decibe otational motion. The table ummaize what we ve leaned. Tanlational Vaiable Rotational Vaiable Relationhip Poition: Angle: = Diplacement: d Angula Diplacement: d d = d Velocit: v d Angula Velocit: ω d = ω Acceleation: a dv Angula Acceleation: α dω = α = ω 2 A quick wod about unit ince the ac length fomula onl wok fo angle in adian, all the othe angula quantitie mut alo be in adian. So, the angula velocit i in adian pe econd and the angula acceleation come in adian pe econd quaed. Eample 25.1: A biccle wheel 70.0cm in diamete otate at ontant 150pm. Find (a)the angula velocit, (b)the tangential velocit of a point on the im, (c)the angle it otate though in 5.00, and (d)the ditance a point on the im ha moved in 5.00. Given: = 0.350m, ω = 150pm, and t = 5.00. Find: ω =?, =?, =?, and =? (a)the angula velocit i actuall alead given but it need to be conveted fom pm to adian pe econd, ( )( )( min ) ω = 15.7 ad 150 ev min 2π ad ev 60. (b)uing the linea/angula elationhip, = ω = (0.35)(15.7) = 5.50 m. (c)uing the definition of angula velocit and the fact that i it contant, ω d ω t = ω t = (15.7)(5) = 78.5ad = 12.5ev. (d)uing the linea/angula elationhip, d = d = = (0.35)(78.5) = 27.5m. O we could have jut ued the definition of tanlational velocit, v d v = = v t = (5.5)(5) = 27.5m. t Let continuing building ou undetanding b looking at angula and linea acceleation. ω 25-3
Eample 25.2: A biccle wheel 70.0cm in diamete tat fom et eache 200pm in 1.20. Find (a)the aveage angula acceleation and (b)the tanlational acceleation when the otation ate i 100pm. Given: = 0.350m, ω i = 0, ω = 100pm = 10.5ad/, ω f = 200pm = 20.9ad/, and t = 1.20. Find: α =? and a = a c + (a)uing the definition of angula acceleation, α dω α = Δω Δt = ω f ω i = 20.9 0 α =17.4ad / 2. Δt 1.20 (b)the angula acceleation i elated to the tangential acceleation, = α = (0.35)(17.4) = 6.09m / 2. The centipetal acceleation i elated to the angula velocit, = ω 2 = (10.5) 2 (0.35) = 38.6m / 2. Thee ae the pependicula component of the total acceleation. Uing the Pthagoean Theoem, 2. Rotational Kinematic a = 2 + 2 = (38.6) 2 + (6.09) 2 a = 39.1m / 2. Recall that when and object i tanlating with ontant acceleation, we developed a et of equation to decibe the motion called the kinematic equation. Fo tanlation, v d a dv da = 0 a bunch of math v = v o + at = o + v o t + 1 2 at 2 v 2 = v o 2 + 2a( o ) o = 1 2 (v + v o)t Fo otation the mbol tand fo diffeent idea, but the mathematical elationhip between the mbol i the ame a fo tanlation. Theefo, the mathematical tep would be identical and lead to the identical kinematic equation with the otational vaiable intead of the tanlational vaiable. ω d α dω dα = 0 the ve ame math ω = ω o + αt = o + ω o t + 1 αt 2 2 ω 2 = ω 2 o + 2α( o ) o = 1 (ω + ω 2 o)t Jut he tanlational kinematic equation onl appl when the tanlational acceleation i contant, the otational kinematic equation onl appl when the otational acceleation i contant. 25-4
Eample 25.3: Fo the bike wheel of eample 2, find (a)the numbe of evolution and (b)the ditance taveled b a point on the edge of the wheel. Given: o = 0 =? ω o = 0 ω = 20.9ad/ α = 17.4ad/ 2 t = 1.20 = 35.0cm Find: =? and =? (a)uing the kinematic equation fo the angle, = o + ω o t + 1 2 αt 2 = 0 + 0 + 1 2(17.4)(1.20) 2 ev = 12.5ad 2πad = 1.99ev. (b)uing the elationhip between angle and poition, = = (35.0cm)(12.5ad) = 431cm. ω Section Summa What do object do and wh do the do it? We have now boken down the motion of object into the motion of the cente of ma (tanlation) and the otation about the cente of ma. Ou undetanding of otational motion i built upon the famewok we etablihed fo tanlational motion. We defined the ideo decibe otational motion in analog to the quantitie fom tanlational motion. The otational vaiable will be defined b analog to the tanlational vaiable. Tanlational Vaiable Poition: The location of an object with epect to oodinate tem. Diplacement: A change in poition. Velocit: The ate of diplacement. Acceleation: The ate of change of Rotational Vaiable Angle: The otational location of an object with epect to oodinate tem. Angula Diplacement: A change in angle. Angula Velocit: The ate of angula diplacement. Angula Acceleation: The ate of change of angula Tanlational Vaiable Rotational Vaiable Relationhip Poition: Angle: = Diplacement: d Angula Diplacement: d d = d Velocit: v d Angula Velocit: ω d = ω Acceleation: a dv Angula Acceleation: α dω = α = ω 2 We ealized that ince the angula vaiable ae elated to each othe in the ame wa he tanlational vaiable. Since contant tanlational acceleation led to the kinematic equation, then contant angula acceleation will lead to the ame kinematic equation uing the otational vaiable in place of the tanlational vaiable. 25-5