1. Physics 218, Lecture XVII1 Physics 218 Lecture 17 Dr. David Toback

2. Physics 218, Lecture XVII2 Before we begin Chapter 8HW was due Monday Ch. 10 reading assignment: due today Already covered most of chapter 9 in lecture, but I’m changing the Chapter 9 HW due date to next Wednesday Ch. 10HW still due the following Monday

3. Physics 218, Lecture XVII3 Rotational Motion Chapters 9 and 10 in four combined lectures This is the 2 nd of the 4 lectures Concentrate on the relationship between linear and angular variables Last time started kinematics… Move to dynamics just like earlier this semester

4. Physics 218, Lecture XVII4 Angular Quantities Last time: Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  This time we’ll start by discussing the vector nature of the variables and then move forward on the others: –Force –Mass –Momentum –Energy

5. Physics 218, Lecture XVII5

6. 6 Angular Velocity and Acceleration Are  and  vectors?  and  clearly have magnitude Do they have direction?

7. Physics 218, Lecture XVII7 Right-Hand Rule Yes! Define the direction to point along the axis of rotation Right-hand Rule This is true for  and 

8. Physics 218, Lecture XVII8 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Moving forward: –Force –Mass –Momentum –Energy

9. Physics 218, Lecture XVII9 Torque Torque is the analogue of Force Take into account the perpendicular distance from axis –Same force further from the axis leads to more Torque

10. Physics 218, Lecture XVII10 Slamming a door We know this from experience: –If we want to slam a door really hard, we grab it at the end –If we try to push in the middle, we aren’t able to make it slam nearly as hard

11. Physics 218, Lecture XVII11 Torque Continued What if we change the angle at which the Force is applied? What is the “Effective Radius?”

12. Physics 218, Lecture XVII12 Slamming a door We also know this from experience: –If we want to slam a door really hard, we grab it at the end and “throw” perpendicular to the hinges –If we try to pushing towards the hinges, the door won’t even close

13. Physics 218, Lecture XVII13 Torque Torque is our “slamming” ability Write Torque as  To find the direction of the torque, wrap your fingers in the direction the torque makes the object twist

14. Physics 218, Lecture XVII14 Torque and Force Torque problems are like Force problems 1.Draw a force diagram 2.Then, sum up all the torques to find the total torque Is torque a vector?

15. Physics 218, Lecture XVII15 Example: Composite Wheel Two forces, F 1 and F 2, act on different radii of a wheel, R 1 and R 2, at different angles  1 and  2.  1  is a right angle. If the axis is fixed, what is the net torque on the wheel? F1F1 F2F2  

16. Physics 218, Lecture XVII16 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Moving forward: –Force  Torque  –Mass –Momentum –Energy

17. Physics 218, Lecture XVII17 Analogue of Mass The analogue of Mass is called Moment of Inertia Example: A ball of mass m moving in a circle of radius R around a point has a moment of inertia F=ma   = 

18. Physics 218, Lecture XVII18 Calculate Moment of Inertia Calculate the moment of inertia for a ball of mass m relative to the center of the circle R

19. Physics 218, Lecture XVII19 Moment of Inertia To find the mass of an object, just add up all the little pieces of mass  To find the moment of inertia around a point, just add up all the little moments

20. Physics 218, Lecture XVII20 Torque and Moment of Inertia Force vs. Torque F=ma   = I  Mass vs. Moment of Inertia

21. Physics 218, Lecture XVII21 Pulley and Bucket A heavy pulley, with radius R, and known moment of inertia I starts at rest. We attach it to a bucket with mass m. The friction torque is  fric. Find the angular acceleration 

22. Physics 218, Lecture XVII22 Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with a constant force F T. It accelerates to an angular speed of  in time t. What is the moment of inertia of the pulley? R

23. Physics 218, Lecture XVII23 Less Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with constant force F T. It accelerates to final angular speed  in time t. A better estimate takes into account that there is friction in the system. This gives a torque (due to the axel) we’ll call this  fric. What is this better estimate of the moment of Inertia? R

24. Physics 218, Lecture XVII24 Next Time The rest of Chapter 10 –More on angular “Stuff” –Angular Momentum –Energy Get caught up on your homework!!! Chapter 8 was due Monday, Chapter 9 homework due Wednesday, Chapter 10 is due the Monday after that

25. Physics 218, Lecture XVII25

26. Physics 218, Lecture XVII26 Example of Cross Product The location of a body is length r from the origin and at an angle  from the x-axis. A force F acts on the body purely in the y direction. What is the Torque on the body? z x y 

27. Physics 218, Lecture XVII27 Calculate Moment of Inertia 1.Calculate the moment of inertia for a ball of mass m relative to the center of the circle R 2.What about lots of points? For example a wheel

28. Physics 218, Lecture XVII28 Rotating Rod A uniform rod of mass m, length l, and moment of inertia I = ml 2 /3 rotates around a pivot. It is held horizontally and released. Find the angular acceleration  and the linear acceleration a at the end. Where, along the rod, is a = g?

29. Physics 218, Lecture XVII29 Two weights on a bar Find the moment of inertia for the two different Axes middle

30. Physics 218, Lecture XVII30 Schedule Changes Please see the handout for schedule changes New Exam 3 Date: Exam 3 Tuesday Nov. 26th

31. Physics 218, Lecture XVII31 Moments of Inertia