Dynamics of Rotational Motion

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1 Dynamcs of Rotatonal Moton Torque: the rotatonal analogue of force Torque = force x moment arm = Fl moment arm = perpendcular dstance through whch the force acts a.k.a. leer arm l F l F l F l F = Fl = F r sn = F tan r τ r F PhysC0 p

2 Example: To loosen a dffcult nut, a wrench wth a pece of ppe for addtonal leerage s used. If the dstance from the center of the nut to the end of the ppe s 0.80 m, and the wrench and ppe make an angle of 0º wth respect to the horzontal, determne the magntude and drecton of the torque appled by a 900 N man standng on the edge of the ppe. PhysC0 p

3 Torque and Angular Acceleraton sngle partcle contrbuton F,tan snce m r m a m r I,tan r F,tan I a,tan for all component partcles : r z-axs axs of rotaton F,z r r F,tan F,rad PhysC0 p3

4 Problem solng tactcs: Draw a dagram For each body, draw a free body dagram Choose coordnates and drectons (ncludng for rotatons) Relate torques to angular acceleratons and forces to lnear acceleratons for more than one body, repeat and note any geometrcal relaton between motons of bodes PhysC0 p4

5 A cord s wrapped around a sold 50 kg cylnder whch has a dameter of 0.0 m, and whch rotates (frctonlessly) about an axs through ts center. A 9.00 N force s appled to the end of the cable, causng the cable to unwnd and the drum (ntally at rest) to rotate. What s the angular acceleraton of the drum? 9.00 N PhysC0 p5

6 A mass m s suspended by a strng wrapped around a pulley of radus R and moment of nerta I. The mass and pulley are ntally at rest. Determne the acceleraton of the dsk and of the suspended mass n terms of the gen parameters (m, I, R). Examne the specal case where the pulley s a unform dsk of mass M. PhysC0 p6

7 A mass m s suspended by a strng wrapped around a pulley of radus R and moment of nerta I. The strng s then connected to a mass m whch sldes on a horzontal frctonless surface. The masses and pulley are ntally at rest. Determne the acceleratons of the masses and the pulley n terms of the the gen parameters (m, m, I, R). Examne the specal case where the pulley can be treated as a thn cylndrcal shell of mass M. M m m PhysC0 p7

8 PhysC0 p8 More on Combnng Translaton and Rotaton I M I m K m r m m m m m K K m m m m K ext τ a F 0 ) ( ) ( ) ( ) ( r F,y y-axs axs axs of rotaton relate to! elocty of s m

9 example: A prmte yo-yo s made of a sold cylnder of mass M wth strng wrapped around ts radus R. What s the speed of the cylnder after t has dropped a dstance h? What s acceleraton of the cylnder and the tenson n the strng? PhysC0 p9

10 example: A sold bowlng bowl rolls wthout slppng down a ramp nclned at an angle wth respect to the horzontal. What s the ball s acceleraton? PhysC0 p0

11 Rollng Frcton: deformng surfaces Stff surface; normal force through, no torque Dstorted surface; normal force offset from, net torque slows rotaton PhysC0 p

12 Work and Power n Rotatonal Moton Force appled to a pont on a rotatng object dw F tan d Rd R F tan s F tan W W d when torque s constant PhysC0 p

13 d d Id I d dt W d Id W I f I dw d P dt dt P I d d Id dt Work - Energy Power PhysC0 p3

14 If the power output of an engne s 00 hp at 6000rpm, what s the correspondng torque? A motor produces a torque of 0 N-m wth dsk mounted to ts shaft. The dsk has a moment of nerta of.0 kg m. If the system starts from rest, determne the work done by the motor n 8.0 s, and the knetc energy of the dsk at the end of ths tme. What s the aerage power delered durng ths 8 seconds? PhysC0 p4

15 Angular momentum for a sngle partcle, defned as L r p = r m depends upon orgn! L mr sn= mr = m r dl dr d m r m dt dt dt dl r F τ dt L Iω m Generalze to composte object In the absence of external torques, angular momentum s consered r ma m r L m r Rght hand rule for drecton PhysC0 p5

16 Two dsks wth moments of nerta I a and I b share a common axs of rotaton, but hae dfferent angular eloctes ( a and b ). The dsks are brought together so that they eentually reach a common fnal angular elocty. Dere an expresson for that fnal angular elocty. Suppose the two dsks are unform wth masses of.0 kg and 4.0 kg, rad of.0 m and.0 m and ntal angular speeds of 50 rad/s and 00 rad/s. Determne the fnal angular speed and the change n the knetc energy of the system. PhysC0 p6

17 A bullet (0 g and ntal speed 400 m/s) s fred nto a door from a drecton perpendcular to the door. The door has a mass of 5 kg and s.0 m wde. Determne the angular elocty of the door after the bullet embeds tself n the center of the door. PhysC0 p7

18 Gyroscopes and Precesson τ r F dl τdt L geometry d dt dl L dt L wr I w PhysC0 p8

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