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

PHY 113 C General Physics I 11 AM - 12:15 P M MWF Olin 101 Plan for Lecture 11: Chapter 10 – rotational motion Angular variables Rotational energy Moment of inertia Torque. Comments about Exam 1 Solutions posted online

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

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  1. PHY 113 C General Physics I • 11 AM - 12:15 PM MWF Olin 101 • Plan for Lecture 11: • Chapter 10 – rotational motion • Angular variables • Rotational energy • Moment of inertia • Torque PHY 113 C Fall 2013-- Lecture 11

  2. PHY 113 C Fall 2013-- Lecture 11

  3. Comments about Exam 1 • Solutions posted online • Please review the problems while the ideas are “fresh” in your mind • iclicker question (serious) • In future exams, would you like to have an additional “take” home component? Of course it would be assumed that all portions of the exam would be subject to strict honor code guidelines • Yes • No PHY 113 C Fall 2013-- Lecture 11

  4. Review: iclicker question In which of the following cases does gravity do positive work? yf yi a. b. yi yf PHY 113 C Fall 2013-- Lecture 11

  5. Review: iclicker question In which of the following cases does the work of gravity have a great magnitude? rf rf a. b. ri ri PHY 113 C Fall 2013-- Lecture 11

  6. Review of center of mass: Example: 1 m 1 kg 3 kg PHY 113 C Fall 2013-- Lecture 11

  7. Center of mass example from Webassign • Finding the center of mass • For a solid object composed of constant density material, the center of mass is located at the center of the object. PHY 113 C Fall 2013-- Lecture 11

  8. Another Webassign question from Assignment #9 PHY 113 C Fall 2013-- Lecture 11

  9. Another Webassign question from Assignment #9 PHY 113 C Fall 2013-- Lecture 11

  10. Examples of two-dimensional collision; balls moving on a frictionless surface PHY 113 C Fall 2013-- Lecture 11

  11. 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 11

  12. Rotation at constant angular velocity w w r vq vq = r w PHY 113 C Fall 2013-- Lecture 11

  13. r1 v1=r1w w r2 v2=r2w 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 11

  14. A wheel is initially rotating at a rate of f=30 rev/sec. R PHY 113 C Fall 2013-- Lecture 11

  15. A wheel is initially rotating at a rate of f=30 rev/sec. Because of a constant angular deceleration, the wheel comes to rest in 3 seconds. R PHY 113 C Fall 2013-- Lecture 11

  16. Example: Compact disc motion w1 w2 In a compact disk, each spot on the disk passes the laser-lens system at a constant linear speed of vq = 1.3 m/s. w1=vq/r1=56.5 rad/s w2=vq/r2=22.4 rad/s What is the average angular acceleration of the CD over the time interval Dt=4473 s as the spot moves from the inner to outer radii? a = (w2-w1)/Dt=-0.0076 rad/s2 PHY 113 C Fall 2013-- Lecture 11

  17. Object rotating with constant angular velocity (a = 0) w R v=Rw v=0 Kinetic energy associated with rotation: “moment of inertia” PHY 113 C Fall 2013-- Lecture 11

  18. Moment of inertia: iclicker exercise: Which case has the larger I? A. a B. b PHY 113 C Fall 2013-- Lecture 11

  19. Moment of inertia: PHY 113 C Fall 2013-- Lecture 11

  20. Note that the moment of inertia depends on both • The position of the rotational axis • The direction of rotation m m m m d d d d I=m(2d)2=4md2 I=2md2 PHY 113 C Fall 2013-- Lecture 11

  21. iclicker question: Suppose each of the following objects each has the same total mass M and outer radius R and each is rotating counter-clockwise at an constant angular velocity of w=3 rad/s. Which object has the greater kinetic energy? (b) (circular ring) (a) (Solid disk) PHY 113 C Fall 2013-- Lecture 11

  22. Various moments of inertia: R R R solid sphere: I=2/5 MR2 solid rod: I=1/3 MR2 solid cylinder: I=1/2 MR2 PHY 113 C Fall 2013-- Lecture 11

  23. Calculation of moment of inertia: Example -- moment of inertia of solid rod through an axis perpendicular rod and passing through center: R PHY 113 C Fall 2013-- Lecture 11

  24. iclicker exercise: 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 11

  25. PHY 113 C Fall 2013-- Lecture 11

  26. iclicker exercise: 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 11

  27. Review of rotational energy associated with a rigid body PHY 113 C Fall 2013-- Lecture 11

  28. 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 11

  29. From Webassign: PHY 113 C Fall 2013-- Lecture 11

  30. CM CM PHY 113 C Fall 2013-- Lecture 11

  31. iclicker exercise: 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 11

  32. q r How to make objects rotate. Define torque: t = r x F t = rF sin q F sin q q F Note: We will define and use the “vector cross product” next time. For now, we focus on the fact that the direction of the torque determines the direction of rotation. PHY 113 C Fall 2013-- Lecture 11

  33. Another example of torque: PHY 113 C Fall 2013-- Lecture 11

  34. PHY 113 C Fall 2013-- Lecture 11

  35. 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 11

  36. 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 11

  37. 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 11

  38. 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 11

  39. Rolling motion reconsidered: Kinetic energy associated with rotation: Distance to axis of rotation Rolling: PHY 113 C Fall 2013-- Lecture 11

  40. Rolling motion reconsidered: Kinetic energy associated with rotation: Distance to axis of rotation Rolling: PHY 113 C Fall 2013-- Lecture 11

  41. Note that rolling motion is caused by the torque of friction: Newton’s law for torque: F fs PHY 113 C Fall 2013-- Lecture 11

  42. Bicycle or automobile wheel: t fs PHY 113 C Fall 2013-- Lecture 11

  43. 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 11

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