1. Physics 7B Lecture 824-Feb-2010 Slide 1 of 31 Physics 7B-1 (A/B) Professor Cebra Rotational Kinematics and Angular Momentum Conservation Winter 2010 Lecture 8

2. Physics 7B Lecture 824-Feb-2010 Slide 2 of 31 Rotational Motion If the force is always perpendicular to the direction of the velocity, then only the direction changes. C entripetal Acceleration v = 2  r/T Centripetal Force F c = ma c =mv 2 /r FCFC

3. Physics 7B Lecture 824-Feb-2010 Slide 3 of 31 Angular Quantities There is an analogy between objects moving along a straight path and objects moving along a circular path. LinearRotational Position (x)↔ Angle (  ) Velocity (v)↔ Angular Velocity (  ) Acceleration (a)↔ Angular Acceleration (  ) Momentum (p)↔Angular Momentum (L) Force (F)↔ Torque (  ) Mass (m)↔Moment of Inertia (I)

4. Physics 7B Lecture 824-Feb-2010 Slide 4 of 31 Angular Displacement  By convention,  is measured clockwise from the x-axis.

5. Physics 7B Lecture 824-Feb-2010 Slide 5 of 31 Angular Velocity Angular velocity is a vector: – Right hand rule to determines direction of ω – Velocity and radius determine magnitude of ω vtvt r

6. Physics 7B Lecture 824-Feb-2010 Slide 6 of 31 Angular Velocity What’s the angular velocity of a point particle? v r┴r┴ r θ

7. Physics 7B Lecture 824-Feb-2010 Slide 7 of 31 Angular Acceleration Angular Acceleration: Rate of change in angular velocity A stationary disk begins to rotate. After 3 seconds, it is rotating at 60 rad/sec. What is the average angular acceleration?

8. Physics 7B Lecture 824-Feb-2010 Slide 8 of 31 Rotational Kinematics Rotational Motion (  = constant) Linear Motion (a = constant)  0 +  t v = v 0 + at  = (1/2)(  0 +  )t x = (1/2)(v 0 + v)t  =  0  0 t + (1/2)  t 2 x = x 0 + v 0 t + (1/2)at 2  2 =  0 2 + 2  v 2 = v 0 2 + 2ax

9. Physics 7B Lecture 824-Feb-2010 Slide 9 of 31 Rotational Kinematics A commercial jet airplane is idling at the end of the run way. At idle, the turbo fans are rotating with an angular velocity of 175 rad/sec. The Tower gives the jetliner permission to take- off. The pilot winds up the jet engines with an angular acceleration of 175 rad/s 2. The engines wind up through an angular displacement of 2000 radians. What is the final angular velocity of the turbo-fan blades? Solution:  2 =  0 2 + 2  = (175 rad/s) 2 + 2(175 rad/s 2 )(2000 rad) = 7.00 x 10 5 rad 2 /s 2  = 837 rad/s

10. Physics 7B Lecture 824-Feb-2010 Slide 10 of 31 Angular Momentum What’s the momentum of a rolling disk? Two types of motion: – Translation: – Rotation: v

11. Physics 7B Lecture 824-Feb-2010 Slide 11 of 31 Angular Momentum Angular Momentum: Product of position vector and momentum vector Why is angular momentum important? Like energy and momentum, angular momentum is conserved. Angular Impulse: Change in angular momentum vector

12. Physics 7B Lecture 824-Feb-2010 Slide 12 of 31 Angular Momentum What’s the angular momentum of a point particle? p r┴r┴ r θ

13. Physics 7B Lecture 824-Feb-2010 Slide 13 of 31 Torque Which way will the scale tip? Rotation of scale is influenced by: – Magnitude of forces – Location of forces 1 kg1.5 kg1 kg1.5 kg

14. Physics 7B Lecture 824-Feb-2010 Slide 14 of 31 Torque Which is more effective? Rotation of wrench is influenced by: – Magnitude of forces – Location of forces – Direction of forces DEMO: Torque Meter

15. Physics 7B Lecture 824-Feb-2010 Slide 15 of 31 Torque Torque: The cause or agent of angular acceleration The angular velocity of an object will not change unless acted upon by a torque The net torque on an object is equal to the rate of change of angular momentum

16. Physics 7B Lecture 824-Feb-2010 Slide 16 of 31 Newton’s Second Law - Rotation  = I  Angular impulse =  t =  L

17. Physics 7B Lecture 824-Feb-2010 Slide 17 of 31 Torque What’s the torque on a point particle? F r┴r┴ r θ

18. Physics 7B Lecture 824-Feb-2010 Slide 18 of 31 Rotational Kinetic Energy What’s the kinetic energy of a rolling disk? Two types of motion: – Translation: – Rotation: v

19. Physics 7B Lecture 824-Feb-2010 Slide 19 of 31 Rolling Bodies mg  Normal Force: F N = mg cos  Parallel Force: F P = mg sin  Frictional Force: f <  s F N  s mg cos  Angular Acceleration  = I   = fR Linear Acceleration F tot = ma F tot = mg sin  – f  = a/R

20. Physics 7B Lecture 824-Feb-2010 Slide 20 of 31 Moment of Inertia Demo: Crazy Rollers Angular Acceleration  = I   = fR Linear Acceleration F tot = ma F tot = mg sin  – f  = a/R As I increases, a decreases

21. Physics 7B Lecture 824-Feb-2010 Slide 21 of 31 Moment of Inertia Let’s use conservation of energy

22. Physics 7B Lecture 824-Feb-2010 Slide 22 of 31 Moment of Inertia Which will have the greater acceleration? Which will have the greater angular acceleration? m 1 = 1 kg F = 5 N m 2 = 2 kg F = 5 N m = 1 kg r 1 = 1 m r 2 = 2 m F = 5 N m = 1 kg F = ma Fr = mra  = mr 2    mR 2

23. Physics 7B Lecture 824-Feb-2010 Slide 23 of 31 Moment of Inertia What’s the moment of inertia of an extended object?

24. Physics 7B Lecture 824-Feb-2010 Slide 24 of 31

25. Physics 7B Lecture 824-Feb-2010 Slide 25 of 31 Moment of Inertia

26. Physics 7B Lecture 824-Feb-2010 Slide 26 of 31 Conservation of Angular Momentum Demo: Rotation with Weights Conservation of L: L = I  If moment of inertia goes own, then angular velocity must go up.

27. Physics 7B Lecture 824-Feb-2010 Slide 27 of 31 Conservation of Angular Momentum Demo: Gyroscope Demo: Gyro in a can Demo: Wheel on a string

28. Physics 7B Lecture 824-Feb-2010 Slide 28 of 31 Conservation of Angular Momentum The total angular momentum vector must be conserved. If you flip the spinning wheel, then something else must create an angular momentum vector. DEMO: Spinning Wheel and rotation chair

29. Physics 7B Lecture 824-Feb-2010 Slide 29 of 31 Center of Gravity The center of gravity (or Center of mass) is the point about which an object will rotate. For a body under goes both linear projectile motion and rotational motion, the location of the center of mass will behave as a free projectile, while the extended body rotates around the c.o.m. Parabolic trajectory of the center of mass Extended body rotating with constant angular velocity

30. Physics 7B Lecture 824-Feb-2010 Slide 30 of 31 Center of Gravity Angular impulse Parabolic trajectory of the center of mass

31. Physics 7B Lecture 824-Feb-2010 Slide 31 of 31 For point particle Summary of Angular Equations Direction of ω Magnitude of ω Same direction as Δω For extended body Point particle or extended body Same direction as ΔL For extended body

32. Physics 7B Lecture 824-Feb-2010 Slide 32 of 31 Announcements

33. Physics 7B Lecture 824-Feb-2010 Slide 33 of 31 DL Sections

34. Physics 7B Lecture 824-Feb-2010 Slide 34 of 31 Conservation of Angular Momentum

35. Physics 7B Lecture 824-Feb-2010 Slide 35 of 31 Torque