John G. Cramer Professor of Physics B451 PAB cramer@phys.washington

1. Physics 121C - MechanicsLecture 23 (Knight: 14.1 - 14.6)Simple Harmonic MotionDecember 7, 2005 (29 Slides) John G. Cramer Professor of Physics B451 PAB cramer@phys.washington.edu

2. Announcements • Tycho is Fixed! Homework Assignment #9 is due on Midnight on Friday, December 9. • Tycho Survey: Please complete the “Survey” in the Homework section of Tycho as soon as possible. Its completion is required for this course. • The Final Exam will be held in this room on Monday, December 12 at 8:30 AM (Note: NOT 10:30 AM!). • Exam #3 will be returned on Friday. The Exam 3 grades and solutions are now on the web. If your Exam 3 scores are missing, see Laura Clement (C136 PAB). Exam #3 regrade requests will be accepted until noon on Monday, December 12. Physics 121C - Lecture 23

3. 1.7 2.7 3.7 Exam 3 Statistics Average = 57.83 Std Dev = 13.91 High = 88 Median = 58 Low = 14 Physics 121C - Lecture 23

4. Lecture Schedule (Part 4) We are here! Physics 121C - Lecture 23

5. Oscillations • Oscillations (whether sinusoidal or otherwise) have some common characteristics: • They take place around an equilibrium position; • The motion is periodic and repeats with each cycle. Physics 121C - Lecture 23

6. Frequency and Period Physics 121C - Lecture 23

7. Example: Radio Station Frequency and Period What is the oscillation period of an FM radio station that broadcasts at 100 MHz? Note that 1/Hz = s Physics 121C - Lecture 23

8. Simple Harmonic Motion The special case of an oscillation that has smooth sinusiodal motion is called “Simple Harmonic Motion”, abbreviated SHM. The motion can be described by a period (the time required for one complete oscillation) and an amplitude (the distance from the equilibrium position to a turning point). An example is a glider attached to a spring and moving on an air track. Physics 121C - Lecture 23

9. Kinematics ofSimple Harmonic Motion Physics 121C - Lecture 23

10. Example: A System in SHM • An air-track glider is attached to a spring,pulled 20 cm to the right, and releasedat t-=0. It makes 15 completeoscillations in 10 s. • What is the period of oscillation? • What is the object’s maximum speed? • What is its position and velocity at t=0.80 s? Physics 121C - Lecture 23

11. Example: Finding the Time A mass, oscillating in simple harmonic motion, starts at x=A and has period T. At what time, as a fraction of T, does the mass first pass through x=½A? Physics 121C - Lecture 23

12. SHM and Circular Motion Uniform circular motion projected into one dimension is simple harmonic motion. Physics 121C - Lecture 23

13. SHM and Circular Motion Physics 121C - Lecture 23

14. The Phase Constant f0 Physics 121C - Lecture 23

15. The Phase Constant f0 Physics 121C - Lecture 23

16. Example:Using Initial Conditions A mass on a spring oscillates with a period of 0.80 s and an amplitude of 10 cm. At t=0 s it is 5.0 cm to the left of equilibrium and moving to the left. What are its position and direction of motion at t=2.0 s? Physics 121C - Lecture 23

17. Clicker Question 1 Shown are four harmonic oscillators at t=0. All are oscillating with amplitude A as shown. Their velocity directions are shown by the green arrows. Which one has the phase constant f0=p/4? Physics 121C - Lecture 23

18. Energy and SHM Physics 121C - Lecture 23

19. Energy and SHM Physics 121C - Lecture 23

20. Example:Using Conservation of Energy A 500 g block on a spring is pulled a distance of 20 cm and released. The subsequent oscillations are measured to have a period of 0.80 s. At what position (or positions) is the speed of the block 1.0 m/s? Physics 121C - Lecture 23

21. Clicker Question 2 Four springs have been compressed from their equilibrium positions at x=0 cm. Which system has the largest maximum speed in its oscillation? Physics 121C - Lecture 23

22. The Dynamics of SHM Physics 121C - Lecture 23

23. The Dynamics of SHM Physics 121C - Lecture 23

24. The Equation of Motion Physics 121C - Lecture 23

25. Solving the SHMEquation of Motion What is x? Let’s make a wild guess: Physics 121C - Lecture 23

26. Example:Analyzing an Oscillator • At t=0, a 500 g block oscillating on a spring is observed to be moving to the right at x=15 cm. It reaches a maximum displacement of 25 cm at t=0.300s. • Draw a position vs. time graph for one cycle of motion. • At what times during the cycle will the mass pass through 20 cm? Physics 121C - Lecture 23

27. Vertical Oscillations Physics 121C - Lecture 23

28. Vertical Oscillations Physics 121C - Lecture 23

29. Example:A Vertical Oscillation A 200 g block hangs from a spring with spring constant 10 N/m. The block is pulled down to a point where the spring is 30 cm longer than it’s unstretched length, then released. Where is the block and what is its velocity 3.0 s later? Physics 121C - Lecture 23

30. End of Lecture 23 • Tycho is Fixed! Homework Assignment #9 is due on Midnight on Friday, December 9. • Tycho Survey: Please complete the “Survey” in the Homework section of Tycho as soon as possible. • The Final Exam will be held in this room on Monday, December 12 at 8:30 AM (Note: NOT 10:30 AM!). • Exam #3 should be graded and returned by Friday. The Exam 3 solutions are on the web. Exam #3 regrade requests will be accepted until noon on Monday, December 12. Physics 121C - Lecture 23