PHY 113 C General Physics I 11 AM - 12:15 P M MWF Olin 101 Plan for Lecture 12:

1. PHY 113 C General Physics I • 11 AM - 12:15 PM MWF Olin 101 • Plan for Lecture 12: • Chapters 10 & 11 – Rotational motion, torque, and angular momentum • Torque • Angular momentum PHY 113 C Fall 2013-- Lecture 12

2. PHY 113 C Fall 2013-- Lecture 12

3. Angular motion angular “displacement”  q(t) angular “velocity”  angular “acceleration”  “natural” unit == 1 radian Relation to linear variables: sq= r (qf-qi) vq = r w aq = r a s PHY 113 C Fall 2013-- Lecture 12

4. Rotational motion angular “displacement”  q(t) angular “velocity”  angular “acceleration”  q(t) w(t) Special case of constant angular acceleration: a = a0: w(t) = wi + a0 t q(t) = qi + wi t +½ a0 t2 (w(t))2 = wi2 + 2 a0 (q(t) - qi) PHY 113 C Fall 2013-- Lecture 12

5. Webassign Assignment #10: A centrifuge in a medical laboratory rotates at an angular speed of 3800 rev/min. When switched off, it rotates 48.0 times before coming to rest. Find the constant angular acceleration of the centrifuge. Recall: Special case of constant angular acceleration: a = a0: w(t) = wi + a0 t q(t) = qi + wi t +½ a0 t2 (w(t))2 = wi2 + 2 a0 (q(t) - qi) PHY 113 C Fall 2013-- Lecture 12

6. Review of rotational energy associated with a rigid body PHY 113 C Fall 2013-- Lecture 12

7. Note that for a given center of rotation, any solid object has 3 moments of inertia; some times two or more can be equal j d d m m i k iclicker exercise: Which moment of inertia is the smallest? (A) i (B) j (C) k IA=0 IC=2md2 IB=2md2 PHY 113 C Fall 2013-- Lecture 12

8. From Webassign: PHY 113 C Fall 2013-- Lecture 12

9. Digression – use of rotational energy in energy storage http://www.beaconpower.com/products/about-flywheels.asp PHY 113 C Fall 2013-- Lecture 12

10. Beacon Power company (went bankrupt in 2011) Continuing efforts – http://www.scientificamerican.com/article.cfm?id=energy-storage-role-in-electric-grid PHY 113 C Fall 2013-- Lecture 12

11. Physics of rolling -- CM CM PHY 113 C Fall 2013-- Lecture 12

12. Rolling motion reconsidered: Kinetic energy associated with rotation: Distance to axis of rotation Rolling: PHY 113 C Fall 2013-- Lecture 12

13. Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first? C B A PHY 113 C Fall 2013-- Lecture 12

14. q How can you make objects rotate? Define torque: t = r x F t = rF sin q r F sin q q F PHY 113 C Fall 2013-- Lecture 12

15. Another example of torque: PHY 113 C Fall 2013-- Lecture 12

16. PHY 113 C Fall 2013-- Lecture 12

17. Example form Webassign #11 • iclicker exercise • When the pivot point is O, which torque is zero? • A. t1? • B. t2? • C. t3? t3 X t2 t1 PHY 113 C Fall 2013-- Lecture 12

18. Newton’s second law applied to center-of-mass motion Newton’s second law applied to rotational motion ri mi di Fi PHY 113 C Fall 2013-- Lecture 12

19. An example: A horizontal 800 N merry-go-round is a solid disc of radius 1.50 m and is started from rest by a constant horizontal force of 50 N applied tangentially to the cylinder. Find the kinetic energy of solid cylinder after 3 s. F R K = ½ I w2 t = I a w = wi + at = at In this case I = ½ m R2 and t = FR PHY 113 C Fall 2013-- Lecture 12

20. Re-examination of “Atwood’s” machine T1-m1g = m1a T2-m2g = -m2a t =T2R – T1R = I a = I a/R R I T2 T1 T1 T2 PHY 113 C Fall 2013-- Lecture 12

21. Another example: Two masses connect by a frictionless pulley having moment of inertia I and radius R, are initially separated by h=3m. What is the velocity v=v2= -v1 when the masses are at the same height? m1=2kg; m2=1kg; I=1kg m2 ; R=0.2m. h m1 m2 v1 v2 h/2 PHY 113 C Fall 2013-- Lecture 12

22. Example from Webassign #10 Tl PHY 113 C Fall 2013-- Lecture 12

23. Note that rolling motion is caused by the torque of friction: Newton’s law for torque: F fs PHY 113 C Fall 2013-- Lecture 12

24. Bicycle or automobile wheel: t fs PHY 113 C Fall 2013-- Lecture 12

25. iclicker exercise: • What happens when the bicycle skids? • Too much torque is applied • Too little torque is applied • The coefficient of kinetic friction is too small • The coefficient of static friction is too small • More than one of these PHY 113 C Fall 2013-- Lecture 12

26. Vector cross product; right hand rule PHY 113 C Fall 2013-- Lecture 12

27. For unit vectors: k j i PHY 113 C Fall 2013-- Lecture 12

28. More details of vector cross products: PHY 113 C Fall 2013-- Lecture 12

29. iclicker exercise: • What is the point of vector products • To terrify physics students • To exercise your right hand • To define an axial vector • To keep track of the direction of rotation PHY 113 C Fall 2013-- Lecture 12

30. From Newton’s second law: iclicker exercise: • Is this • Wrong? • Approximately right? • Exactly right? PHY 113 C Fall 2013-- Lecture 12

31. From Newton’s second law – continued – conservation of angular momentum: PHY 113 C Fall 2013-- Lecture 12

32. Torque and angular momentum Define angular momentum: For composite object: L = Iw Newton’s law for torque: In the absence of a net torque on a system, angular momentum is conserved. PHY 113 C Fall 2013-- Lecture 12

33. iclicker exercise: A student sits on a rotatable stool holding a spinning bicycle wheel with angular momentum Li. What happens when the wheel is inverted? counterclockwise • The student will remain at rest. • The student will rotate counterclockwise. • The student will rotate clockwise. PHY 113 C Fall 2013-- Lecture 12

34. More details: PHY 113 C Fall 2013-- Lecture 12

35. Other examples of conservation of angular momentum w1 w2 m m m m d2 d1 d1 d2 I2=2md22 I1=2md12 I1w1=I2w2 w2=w1 I1/I2 PHY 113 C Fall 2013-- Lecture 12

36. Webassign problem: • A disk with moment of inertia I1 is initially rotating at angular velocity wi. A second disk having angular momentum I2, initially is not rotating, but suddenly drops and sticks to the second disk. Assuming angular moment to be conserved, what would be the final angular velocity wf? PHY 113 C Fall 2013-- Lecture 12

37. Summary – conservation laws we have studied so far PHY 113 C Fall 2013-- Lecture 12