Dynamics of Rotational Motion Toque: the otational analogue of foce Toque = foce x moment am τ = l moment am = pependicula distance though which the foce acts a.k.a. leve am l l l l τ = l = sin φ = tan τ = Phys2C0 p Example: To loosen a difficult nut, a wench with a piece of pipe fo additional leveage is used. If the distance fom the cente of the nut to the end of the pipe is 0.80 m, and the wench and pipe make an angle of 20º with espect to the hoizontal, detemine the magnitude and diection of the toque applied by a 900 N man standing on the edge of the pipe. Phys2C0 p2
Toque and Angula Acceleation single paticle contibution,tan = m a sinceτ = τ = m α 2 i i Στ = Σm α = Iα i 2 Στ = Iα,tan,tan a,tan fo all component paticles : = α y-axis axis of otation,y,tan,ad Phys2C0 p3 Poblem solving tactics: Daw a diagam o each body, daw a fee body diagam Choose coodinates and diections (including fo otations) Relate toques to angula acceleations and foces to linea acceleations fo moe than one body, epeat and note any geometical elation between motions of bodies Phys2C0 p4
A cod is wapped aound a solid 50 kg cylinde which has a diamete of 0.20 m, and which otates (fictionlessly) about an axis though its cente. A 9.00 N foce is applied to the end of the cable, causing the cable to unwind and the dum (initially at est) to otate. What is the angula acceleation of the dum? 9.00 N A mass m is suspended by a sting wapped aound a pulley of adius R and moment of inetia I. The mass and pulley ae initially at est. Detemine the acceleation of the disk and of the suspended mass in tems of the given paametes (m, I, R). Examine the special case whee the pulley is a unifom disk of mass M. Phys2C0 p5 A mass m 2 is suspended by a sting wapped aound a pulley of adius R and moment of inetia I. The sting is then connected to a mass m which slides on a hoizontal fictionless suface. The masses and pulley ae initially at est. Detemine the acceleations of the masses and the pulley in tems of the the given paametes (m, m 2, I, R). Examine the special case whee the pulley can be teated as a thin cylindical shell of mass M. M m m 2 Phys2C0 p6
example: A pimitive yo-yo is made of a solid cylinde of mass M with sting wapped aound its adius R. What is the speed of the cylinde afte it has dopped a distance h? What is acceleation of the cylinde and the tension in the sting? example: A solid bowling bowl olls without slipping down a amp inclined at an angle β with espect to the hoizontal. What is the ball s acceleation? Phys2C0 p8
Rolling iction: defoming sufaces Stiff suface; nomal foce though cm, no toque Distoted suface; nomal foce offset fom cm, net toque slows otation Phys2C0 p9 Wok and Powe in Rotational Motion oce applied to a point on a otating object dw = W = tan = τdθ Rdθ τdθ W = τ θ when toque is constant R θ tan s tan Phys2C0 p0
dω τdθ = Iαdθ = I dθ = I dt W = τdθ = Iωdω = 2 W = Iω f Iω 2 2 dw dθ P = = τ dt dt P = τω 2 i dθ dω = Iωdω dt Wok - Enegy Powe Phys2C0 p If the powe output of an engine is 200 hp at 6000pm, what is the coesponding toque? A moto poduces a toque of 0 N-m with disk mounted to its shaft. The disk has a moment of inetia of 2.0 kg m 2. If the system stats fom est, detemine the wok done by the moto in 8.0 s, and the kinetic enegy of the disk at the end of this time. What is the aveage powe deliveed duing this 8 seconds? Phys2C0 p2
Angula momentum fo a single paticle, defined as L p = mv depends upon oigin! L = mv sinφ = mv = mv dl d dv = mv + m = v mv + ma dt dt dt dl = = τ dt L = Iω Genealize to composite object In the absence of extenal toques, angula momentum is conseved mv L mv φ ω Right hand ule fo diection Phys2C0 p3 Two disks with moments of inetia I a and I b shae a common axis of otation, but have diffeent angula velocities (ω a and ω b ). The disks ae bought togethe so that they eventually each a common final angula velocity. Deive an expession fo that final angula velocity ω. Suppose the two disks ae unifom with masses of 2.0 kg and 4.0 kg, adii of.20 m and.0 m and initial angula speeds of 50 ad/s and 200 ad/s. Detemine the final angula speed and the change in the kinetic enegy of the system. Phys2C0 p4
A bullet (0 g and initial speed 400 m/s) is fied into a doo fom a diection pependicula to the doo. The doo has a mass of 5 kg and is.0 m wide. Detemine the angula velocity of the doo afte the bullet embeds itself in the cente of the doo. Phys2C0 p5