Important Dates: Post Test: Dec during recitations. If you have taken the post test, don t come to recitation!

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1 Important Dates: Post Test: Dec. 8 0 durng rectatons. If you have taken the post test, don t come to rectaton! Post Test Make-Up Sessons n ARC 03: Sat Dec. 6, 0 AM noon, and Sun Dec. 7, 8 PM 0 PM. Post Survey Deadlne: Frday Dec., :59 PM. Last Homework Due: Frday Dec., :59 PM. Don t forget to do the student evaluatons: There wll be tutorng sessons n ARC on Dec. 5 and 6. Fnal Exam: Tuesday, Dec. 6, 4 7 PM. Locatons: CAC Gym and Scott Hall. Fnal Exam Make-Up (wth permsson): Frday, Dec. 9, :00 3:00 PM, SEC 03.

2 TORQUE AND ROTATIONAL DYNAMICS REVIEW Center of Mass: x cm m x M Moton of CM Rgd Body Rotaton av av t t t t r cm F ; ; ext d dt m r M Ma d dt cm d dt dp dt s r For Constant Angular Acceleraton: ω 0 0 αt ω t ω ω ω o 0 αt α 0

3 -Clcker Choose orgn x = 0 at CM (black X): m x cm m xcm 0 m x m cm cm x

4 ENERGY OF ROTATIONAL MOTION Rotatng Rgd Body Consder mass pont : K m v v r m ( r ) m r For a dfferent pont j: where r Total Knetc Energy: K K j m j r j s dstance from axs of rotaton ) mr K K ( m r Moment of Inerta: I K I Rotatonal Knetc Energy [NOTE: I s taken about axs of rotaton]

5 EXAMPLE: Isolated Masses Close together: I m L L m 4 4 m m Far apart: I m L L m m m Unform rod of length L I I m r L L r dr r r λ = mass/unt length I r 3 3 L / L / L 3 3 L L ML

6 Moment of Inerta for Common Shapes ML ML 3 I I I M a b I Ma 3 I M R R MR I MR I MR 5 I MR 3 I Sold Shell (Useful to have on a formula sheet) (, hnt, hnt )

7 -Clcker Three flat objects (rng, dsc, and square loop) have the same mass M and the same outer dmenson. The axs of rotaton passes through the CM and s perpendcular to the plane of the objects. Rank the moments of nerta of these objects about ths axs. A. A > B > C B. A = B > C C. C > B = A D. C > A > B E. The moments of nerta are all the same

8 -Clcker The three objects shown here all have the same mass M and outer radus R. Each object s rotatng about ts axs of symmetry (shown). All three have the same rotatonal knetc energy. Whch one s rotatng fastest? K I

9 PARALLEL AXIS THEOREM If the moment of nerta around an axs passng through the CM s known, what s the moment of nerta around a dfferent (but parallel) axs? I' I cm Md

10 Applcaton of Parallel Axs Theorem: ML I I ML 3 d M I 3 ML ML ML Md L M

11 r TORQUE Tendency of force to cause rotaton Force appled to pont away from axs of rotaton = Vector from axs to pont where force s appled Moment Arm: Smplest Case: Then: General Case: Axs of Rotaton r dstance from axs to lne of force F l torque : Fl Moment arm l = r snf Sgn of : = moment arm Unts: N.m ft.lbs Only component of to causes rotaton r r (component to rf snf but l = rsnf (+) Produce CCW Rotaton (-) Produce CW Rotaton F s radal) Fr snf Fl

12 -Clcker r, r 3,4 Fr snf

13 -Clcker Two pulleys wth dfferent rad are attached to one another so that they rotate together. Each pulley has a strng wrapped around t wth a weght hangng from t. The two weghts are the same. What s the drecton of the net torque on the system? A. Clockwse B. Counterclockwse C. The net torque s zero D. None of the above

14 -Clcker Three forces of equal magntude are appled to a 3 x m rectangle. Forces F and F act at 45 angles to the vertcal as shown, whle F3 acts horzontally. What s the drecton of the net torque abound pont B? B A. Clockwse B. Counterclockwse C. Zero D. It cannot be determned wth the nformaton gven

15 AND FOR RIGID BODIES Suppose tangental force, F t, causes m to rotate about axs through 0 ( But at r z ) I z z z Ft r matr mr z I (compare F = ma) z Suppose multple torques on rgd body: z I z

16 EXAMPLE: A dsk of mass M s wrapped wth massless strng attached to mass M that falls under gravty. What s acceleraton of M? TR TR I I Ia R T Ia R R a but mg T ma mg ma T ma Ia R g a REALITY CHECK! I I a g mr mr M I MR a g m If M m a 0 If M m a g

17 -Clcker Two pulleys wth dfferent rad are attached to one another so that they rotate together. Each pulley has a strng wrapped around t wth a weght hangng from t. When the wheel s released t s found to have an angular acceleraton vector that s drected nto the page. Whch mass s heaver? A B A. Mass A B. Mass B C. They are the same. D. Impossble to tell wthout more nformaton.

18 Combned rotatonal and translatonal moton Moton can be descrbed as Moton of CM + Moton about CM K Mv cm I cm Important Example: Rollng Wthout Slppng s R v cm ds dt R d dt R vcm a cm R R [NOTE: Pont of contact s nstantaneously at rest]

19 EXAMPLE: Racng Down the Inclne: A ball and a cylnder of equal radus roll wthout slppng down an nclne. Who wns? Rolls due to statc frcton No sldng No work done by frcton CONSERVATION OF MECHANICAL ENERGY K U K g g U 0 Mgh Mv cm I cm Icylnd shell. MR I sold cylnder MR I sold sphere Summarze as: I bmr b,,,...) ( 5 MR 5 Mgh Mv cm ( ) Mvcm Mgh b bmr v cm vcm R gh b

20 -Clcker A sold sphere rolls wthout slppng along a track shaped as shown at rght. It starts from rest at pont A and s movng vertcally when t leaves the track at pont B. B At ts hghest pont whle n the ar, the sphere wll be A. Above the heght at pont A B. Below the heght at pont A C. Equal to the heght at pont A D. Not enough nformaton gven

21 -Clcker A ball s rollng clockwse (wthout slppng) up a ramp, slowng down. What are the forces on the ball? A. Normal force and weght B. Normal force, weght, and force of frcton actng down the ramp C. Normal force, weght, and force of frcton actng up the ramp D. The forces depend on how fast the ball s rotatng E. None of the above

22 Important Dates: Post Test: Dec. 8 0 durng rectatons. If you have taken the post test, don t come to rectaton! Post Test Make-Up Sessons n ARC 03: Sat Dec. 6, 0 AM noon, and Sun Dec. 7, 8 PM 0 PM. Post Survey Deadlne: Frday Dec., :59 PM. Last Homework Due: Frday Dec., :59 PM. Don t forget to do the student evaluatons: There wll be tutorng sessons n ARC on Dec. 5 and 6. Fnal Exam: Tuesday, Dec. 6, 4 7 PM. Locatons: CAC Gym and Scott Hall. Fnal Exam Make-Up (wth permsson): Frday, Dec. 9, :00 3:00 PM, SEC 03.