1. Physics 218, Lecture XXV1 Physics 218 Lecture 25 Dr. David Toback

2. Physics 218, Lecture XXV2 Checklist for Today Things due earlier this week –Chapter 15 & 16 in WebCT –Chapter 18 reading –Chapter 18 problems for Recitation Today: LAST LECTURE Things due Monday –Chapter 18 in WebCT

3. Physics 218, Lecture XXV3 The Schedule This Week (4/21) Monday: Chapter 15 & 16 due in WebCT Tues: Reading & Lecture: Chapter 18 Wed: Recitation on Chapter 18, Lab Thurs: Last lecture, Chapter 18 –Not covering Chapter 19 Next Week (4/28) No lectures or recitations Chapter 18 HW due in WebCT Example final problems at http://faculty.physics.tamu.edu/toback/218/Lectures/FinalReview.pdf Finish your HW on WebCT and do the Mini-Practice Exam Week after that (5/5) Final: Monday May 5 th, 1PM-3PM in this room

4. Physics 218, Lecture XXV4 Next Semester Some of you continue to ask about next semester: I am NOT teaching 208 However I am planning on teaching a new course: “Big Bang, Black Holes, No Math” –Intro to Cosmology (PHYS289) –Based on “A Brief History of Time” by Stephen Hawking –If you can’t take it, you might pass this info along to friends

5. Physics 218, Lecture XXV5 Chap 18 Overview: Oscillations Oscillations and vibrations Equations of motion Simple Model: Mass on a spring Simple Harmonic Motion Energy

6. Physics 218, Lecture XXV6

7. 7 Review: Simple Harmonic Motion Mass on a Spring Amplitude: Max distance Period: Time it takes to get back to here

8. Physics 218, Lecture XXV8 Summary: Equation of Motion Mass m on a spring with spring constant k: x = A sin(  t +  ) Where  2 = k/m A is the Amplitude  is the “phase” (phase just allows us to set t=0 when we want)

9. Physics 218, Lecture XXV9 Questions Simple Harmonic Motion x = A sin(  t +  ) Can you answer the following: What is the amplitude? What is the phase? What is the angular frequency? What is the velocity at the end points? What is the velocity at the middle?

10. Physics 218, Lecture XXV10 Simple Harmonic Motion At some level sinusoidal motion is the definition of Simple Harmonic Motion A system that undergoes simple harmonic motion is called a simple harmonic oscillator

11. Physics 218, Lecture XXV11 Check: This looks like a cosine. Makes sense… Spring and Mass Paper which tells us what happens as a function of time

12. Physics 218, Lecture XXV12 Mass on a spring with gravity h Initial Final Take an un-stretched spring. Gently add a mass and let go. The total stretch from equilibrium is a distance h before it starts to rise again. It then oscillates What is the equation of motion?

13. Physics 218, Lecture XXV13 Energy in SHO’s Potential Energy for a spring U = ½kx 2 Kinetic Energy for a block KE = ½mv 2 Total Energy E = U+KE = ½kx 2 +½mv 2

14. Physics 218, Lecture XXV14 Energy Example A box of mass m is attached to a spring of constant k, stretched to a distance X o, held in place and then and released. Ignore gravity and friction. As a function of time: 1.What is the position? 2.What is the velocity? 3.What is the kinetic energy? 4.What is the potential energy? 5.What is the total energy of the system?

15. Physics 218, Lecture XXV15 Understanding Phase: Initial Conditions A block with mass m is attached to the end of a spring, with spring constant k. The spring is stretched a distance D and let go at t=0 –What is the position of the mass at all times? –Where does the maximum speed occur? –What is the maximum speed?

16. Physics 218, Lecture XXV16 Example: Spring with a Push We have a spring system –Spring constant: K –Mass: M –Initial position: X 0 –Initial Velocity: V 0 Find the position at all times

17. Physics 218, Lecture XXV17 Challenge Exam Announcement of this semester’s 218 Challenge Exam: Monday April 28 th at 6:00PM –Not required (just for fun) –Does not (will not!!!) count as part of your final grade Test your skills against the best Aggies from all the Physics 218 sections (not just this lecture) on Physics 218 material Students who perform well will be recognized as Mechanics Scholars, and honored at a banquet in their honor. Other prizes, including cash. Handout information on my WebSite

18. Physics 218, Lecture XXV18 That’s it! Next Week: –No classes or Recitations –HW for Chapter 18 due –Example final problems at http://faculty.physics.tamu.edu/toback/218/Lectures/FinalReview.pdf –Finish your HW on WebCT, do the Mini-Practice Exam Final: Monday May 5 th, 1PM-3PM in this room –Chapters 14 -16 and 18 will be on the final –I guarantee a problem from each Good luck studying for Finals!

19. Physics 218, Lecture XXV19

20. Physics 218, Lecture XXV20 Uniform Circular Motion It turns out Uniform Circular Motion is really Simple Harmonic Motion but in two dimensions

21. Physics 218, Lecture XXV21 Uniform Circular Motion in 2D A ball is moving anti-clockwise in uniform circular motion with a radius R and speed V. At t=0 we set  0 =0. 1.What is the equation of motion for  ? 2.What is the equation of motion in the x direction? 3.What is the displacement at all times? 4.What is the velocity at all times? 5.What is the speed at all times?

22. Physics 218, Lecture XXV22 Pendulums Pendulums can be a pretty good Simple Harmonic Oscillator The amount of swing has to be small

23. Physics 218, Lecture XXV23 Example: Pendulum For small swing (  small) show that the motion of a ball on a pendulum is simple harmonic

24. Physics 218, Lecture XXV24 Period of a Pendulum Show that the period of a pendulum doesn’t depend on the mass.

25. Physics 218, Lecture XXV25 Coming up… Thursday: Last class meeting –Review for the final, we’ll skip Chapter 15 –Do all the problems on the handout before class –Be prepared to vote for which type of problem you want to review most Monday December 11 th Final Exam, 1PM. Here –The material in Chapters 12 and 13 will be on the final –I guarantee there will be at least one problem from each chapter on the final Homework 13 and FCI II are due Wednesday A 100% on all homework is required in order to pass the class

26. Physics 218, Lecture XXV26 Next Lecture Notes on the final: –Chapter 14 will be on the final. –Nothing from Chapter 15 on the final. –I guarantee there will be at least one problem from these last two weeks on the final –Final is Monday Dec 15 th, 1:00PM- 3:00PM, here Next time –Review –Come prepared with questions

27. Physics 218, Lecture XXV27 Draw position as a function of Time? Spring and Mass Paper which tells us what happens as a function of time

28. Physics 218, Lecture XXV28 Ballistic Pendulum A bullet plows into a block of wood which is part of a pendulum. We know m, M and v. How high, h, does the block of wood go? What is the equation of motion after the collision?

29. Physics 218, Lecture XXV29 Some Useful Relations and Weird Results

30. Physics 218, Lecture XXV30 Doing a mass on a spring with g Take an un-stretched spring with constant k. Add a mass, m. Slowly stretch to -x 0, and bring to rest and let go. What is the equation of motion? Add a mass Stretch