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2. Units of Chapter 8
3. 8-1 Angular Quantities In purelyrotationalmotion, all points on the object move incirclesaround theaxisof rotation ( O ). Theradiusof the circle isr . All points on a straight line drawn through the axis move through the sameanglein the sametime .When P (at radius r) travels an arc length , OP sweeps out an angle . Angular Displacement of the body 4. 8-1 Angular Quantities
In doing problems in this chapter,need calculators inRADIAN MODE!!!! 5.
8-1 Angular Quantities 6. 3 10 -4rad = ? r = 100 m,= ? a)= (3 10 -4rad) [(360/2 )/rad] = 0.017 For small angles:The chordarc length b)= r = (100)(3 10 -4 ) = 0.03 m = 3 cm MUSTbe in radians in partb Example 8-2 : 7. 8-1 Angular Quantities Angulardisplacement: The average angularvelocityis defined as the total angulardisplacementdivided bytime : Theinstantaneousangular velocity: (8-2a) (8-2b) (Units =rad/s ), ValidONLYif is inrad ! 8. 8-1 Angular Quantities The angularaccelerationis the rate at which the angular velocity changes with time: Theinstantaneousacceleration: (8-3a) (8-3b) (Units =rad/s 2 ) ValidONLY if is inrad & is inrad/s ! 9.
10. Connection Between Angular & Linear Quantities v = ( / t), = r v = r( / t) = r Radians! v = r Depends on r ( is the same for all points!) v A= r A A v B= r B B v B> v Asince r B> r A Therefore, objectsfartherfrom the axis of rotation will movefaster . 11. 8-1 Angular Quantities If the angular velocity of a rotating objectchanges , it has atangentialacceleration: a tan= ( v/ t), v =r = r ( / t) a tan= r Even if the angular velocity is constant, each point on the object has acentripetalacceleration: 12. 8-1 Angular Quantities Here is thecorrespondencebetweenlinearandrotationalquantities: 13. 8-1 Angular Quantities Thefrequencyis the number of completerevolutionsper second: or= 2 f (angular frequency) Frequencies are measured inhertz .Theperiodis the time one revolution takes: 14. 8-2 Constant Angular Acceleration The equations of motion forconstantangular acceleration are the same as those forlinearmotion, with the substitution of theangularquantities for thelinearones. NOTE: These areONLY VALIDif all angular quantities are in radian units!! 15. 8-3 Rolling Motion (Without Slipping) In (a), a wheel isrollingwithout slipping. The point P, touching the ground, is instantaneously atrest , and the center moves with velocity v. In (b) the same wheel is seen from areference framewhere C is at rest. Now point P is moving with velocity v. Thelinear speedof the wheel is related to itsangular speed : 16. Example 8-7
c)= ( 2 0 2)/(2 )Stopped = 0 = -0.902 rad/s 2 d)t = ( 0 )/ Stopped = 0 t = 27.4 s 17. 8-4 Torque To make an objectstartrotating, aforceis needed; thepositionanddirectionof the force matter as well. Theperpendiculardistance from the axis of rotation to the line along which the force acts is called thelever arm . 18. 8-4 Torque Alongerlever arm is very helpful in rotating objects. 19.
Lower caseGreek tau 8-4 Torque 20. 8-4 Torque Here, the lever arm for F Ais the distance from theknobto thehinge ; the lever arm for F Diszero ; and the lever arm for F Cis as shown. 21. 8-4 Torque Thetorqueis defined as: F =F sin F = F cos = rF sin Units of torque: Newton-meters (N m) 22. Example 8-9 r =r 2 sin60 2 = -r 2 F 2 sin60 1 = r 1 F 1 = 1 + 2= -6.7 m N Always use the following sign convention ! Counterclockwise rotation+ torque Clockwise rotation- torque 23. Section 8-5: Rotational Dynamics
24. Simplest Possible Case A massmmoving in aCircle of radiusr , one forceF TANGENTIAL tothe circle = rF Newtons 2 ndLaw + relation (a = r ) between tangential & angular accelerations F = ma = mr So = mr 2 Newtons 2 ndLaw for Rotations Proportionality constant between&ismr 2 (point mass only!) 25.
26.
27. 8-5 Rotational Dynamics; Torque and Rotational Inertia The quantityis called therotational inertiaof an object. Thedistributionof mass matters here these two objects have the same mass, but the one on the left has a greaterrotational inertia , as so much of its mass is far from the axis of rotation. 28. 8-5 Rotational Dynamics; Torque and Rotational Inertia Therotational inertiaof an object depends not only on itsmass distributionbut also the location of theaxisof rotation compare (f) and (g), for example. 29. Example 8-10
30. 8-6 Solving Problems in Rotational Dynamics
31. 5. Apply Newtons second law forrotation . If therotational inertiais not provided, you need to find itbeforeproceeding with this step. 6. Apply Newtons second law fortranslationand other laws and principles as needed. 7.Solve . 8.Checkyour answer for units and correct order of magnitude. 8-6 Solving Problems in Rotational Dynamics 32. Example 8-11
33. 8-7 Rotational Kinetic Energy Thekinetic energyof a rotating object is given byBy substituting the rotational quantities, we find that the rotational kinetic energy can be written: An object that has bothtranslationalandrotationalmotion also has both translational and rotationalkinetic energy : (8-15) (8-16) 34. Example 8-13:
35. 8-7 Rotational Kinetic Energy When usingconservationof energy, bothrotationalandtranslationalkinetic energy must be taken into account. All these objects have the samepotentialenergy at the top, but the time it takes them to get down the incline depends on how muchrotationalinertiathey have. 36. 8-7 Rotational Kinetic Energy The torque doesworkas it moves the wheel through an angle : (8-17) Torque:= Fr Work:W = F = Fr = 37. 8-8 Angular Momentum and Its Conservation In analogy with linear momentum, we can defineangular momentum L :
38. 8-8 Angular Momentum and Its Conservation Therefore, systems that canchangetheir rotational inertia through internal forces will also change theirrateof rotation: 39. Example 8-15
40. Translation-Rotation
41. Summary of Chapter 8
42. Summary of Chapter 8, cont.
43. Summary of Chapter 8, cont.