1. Physics 218, Lecture XX1 Physics 218 Lecture 20 Dr. David Toback

2. Physics 218, Lecture XX2 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Force  Torque  Today we’ll finish: –Mass –Momentum –Energy

3. Physics 218, Lecture XX3

4. 4 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Force  Torque  Today we’ll finish: –Mass –Momentum –Energy

5. Physics 218, Lecture XX5 Calculating Moments of Inertia

6. Physics 218, Lecture XX6 Calculate the Moment of Inertia A pulley has mass M, uniform density, radius R, and rotates around its fixed axis Calculate its moment of inertia R

7. Physics 218, Lecture XX7 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Force  Torque  Today we’ll finish: –Mass  Moment of Inertia  –Momentum –Energy

8. Physics 218, Lecture XX8 Angular Momentum Define the Angular Momentum L L = I  Is this a vector?

9. Physics 218, Lecture XX9 Newton’s Laws Momentum vs. Angular Momentum p=mv  L=I  Newton’s Laws F=dp/dt   = dL/dt

10. Physics 218, Lecture XX10 Conservation of Angular Momentum By Newton’s laws, the angular momentum of a body can change, but the angular momentum for a system cannot change  Conservation of Angular Momentum

11. Physics 218, Lecture XX11 Ice Skater This one you’ve seen on TV Try this at home in a chair that rotates Get yourself spinning with your arms and legs stretched out, then pull them in

12. Physics 218, Lecture XX12 Clutch Design Model a car clutch as two plates, each with radius R, and masses M A and M B Assume I Plate = ½MR 2 Plate A spins with speed  1 and Plate B is at rest. We close them and they spin together Find the final angular velocity of the system

13. Physics 218, Lecture XX13 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Force  Torque  Today we’ll finish: –Mass  Moment of Inertia  –Momentum  Angular Momentum L –Energy

14. Physics 218, Lecture XX14 Rotational Kinetic Energy KE = ½I  2 Conservation of Energy must take into account rotational kinetic energy

15. Physics 218, Lecture XX15 Rotating Rod A rod of mass uniform density, mass m and length l pivots at a hinge. It starts at rest at a right angle and is let go What is  when it reaches the bottom? What is the velocity of the tip at the bottom?

16. Physics 218, Lecture XX16 Rotation and Translation Objects can both Rotate and Translate Both have energy separately KE = ½ mv 2 + ½I  2 Always true when you have rolling without slipping

17. Physics 218, Lecture XX17 Rolling Down an Incline A solid ball of mass m and radius R rolls without slipping down a plane with height h. What will be the final speed of the ball? What would be the speed if the ball didn’t roll? Note: I sphere = 2/5MR 2

18. Physics 218, Lecture XX18 Person on a Disk A person with mass m stands on the edge of a disk with radius R and moment ½MR 2. Neither is moving. The person then starts moving on the disk with speed V. Find the angular velocity of the disk

19. Physics 218, Lecture XX19 Same Problem: Forces Same problem but with Forces

20. Physics 218, Lecture XX20 Next Time Chapter 11 –It’s a review of Chapter 10, but more sophisticated –Math, Torque, Angular Momentum, Energy The material will not be on the 3 rd exam, but will help with the exam. It will all be on the final. Exam 3 is Tuesday November 26 th

21. Physics 218, Lecture XX21

22. Physics 218, Lecture XX22 Hollow Cylinder Consider a hollow cylinder with uniform density, inner radius R 1, outer radius R 2 and total Mass M. Find the moment of Inertia

23. Physics 218, Lecture XX23 Parallel-Axis Theorem Quick Trick for calculating Moments I = I cm + Mh 2 Example