1. Physics 218, Lecture XXI1 Physics 218 Lecture 21 Dr. David Toback

2. Physics 218, Lecture XXI2 Exam Next Tuesday not Today’s Topics not covered on the exam, but are a review

3. Physics 218, Lecture XXI3 General Rotation Chapter 11 Today’s lecture will be about extending the angular stuff we did last week. In many ways it’s a summary –Some math –Torque –Angular Momentum –Kepler’s 2 nd Law

4. Physics 218, Lecture XXI4

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6. 6 Vector Cross Product This is the last way of multiplying vectors we will see Pneumonic: Matrix

7. Physics 218, Lecture XXI7 Torque Redefined A better way to define Torque This gives us the magnitude and the direction

8. Physics 218, Lecture XXI8 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 

9. Physics 218, Lecture XXI9 Angular Momentum Again we use the Cross Product:

10. Physics 218, Lecture XXI10 Angular Motion of a Particle Determine the angular momentum, L, of a particle,with mass m and speed v, moving in uniform circular motion with radius r.

11. Physics 218, Lecture XXI11 Kepler’s 2 nd Law 2 nd Law: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out area in equal periods of time

12. Physics 218, Lecture XXI12 A bullet strikes a cylinder A bullet of speed V and mass m strikes a solid cylinder of mass M and inertia ½MR 2, at radius R and sticks. The cylinder is anchored at point 0 and is initially at rest. What is  of the system after the collision? Is energy Conserved?

13. Physics 218, Lecture XXI13 Atwood’s Machine A pulley with a fixed center (at point O), radius R  and moment of inertia I, has a massless rope wrapped around it (no slipping). The rope has two masses, m 1 and m 2 attached to its ends. Assume m 2 >m 1 What is the acceleration of the system? Do some checks.

14. Physics 218, Lecture XXI14 Next Period: Exam There is an exam here, next class period (Tuesday) Same as usual: –Problems from the Textbook, Homework and Lecture –Formula sheet handed out –All formulas you use must be derived from those on the formula sheet Topics are: Chapters 8-10 –Conservation of Energy –Momentum and Collisions –Rotational Motion notToday’s Topics not covered, but are a review

15. Physics 218, Lecture XXI15 Next Tuesday: Exam There is an exam here, not next class period, but Tuesday Same as usual: –Problems from the Textbook, Homework and Lecture –Formula sheet handed out –All formulas you use must be derived from those on the formula sheet Topics are: Chapters 8-10 –Conservation of Energy –Momentum and Collisions –Rotational Motion notToday’s Topics not covered, but are a review Next week’s topics: Oscillations

16. Physics 218, Lecture XXI16 Why does the Bicycle Wheel Turn to the Right?

17. Physics 218, Lecture XXI17 Angular Momentum Again we use the Cross Product: Derivation of  = dL/dt

18. Physics 218, Lecture XXI18 L for a system of many bodies Have to be careful with Angular Momentum –  = dl/dt for a single particle –  =  dl/dt) for a system of many particles –All internal torques cancel because of Newton’s law (all internal forces are equal and opposite) Reference Frame matters. Only true for: –The origin is an inertial Reference Frame –The center of mass

19. Physics 218, Lecture XXI19 L for a Rigid Body Find the angular momentum, L, for this body given that it is rotating around the Z axis with angular velocity 