Oscillations and Waves (4)

1. Mr. Klapholz Shaker Heights High School Oscillations and Waves (4) Many things go back and forth; they vibrate; they oscillate. Examples include a swing, a leaf on a tree, a yo-yo, and the stock market. Without waves there would be no light, no sound, no music, no Picasso, no communication, no society, and therefore, no humanity as we know it.

2. A Pendulum and a mass on a spring.

3. A Pendulum and a mass hanging from a spring. • Why do the systems go back and forth? • When you take the system out of equilibrium, and release it, it goes back to equilibrium. Why? • In each system, what causes the restoring force? • In each system, where is the mass when the force is large? • Why doesn’t it just go back to equilibrium and stay there?

4. A Pendulum and a mass hanging from a spring. • In each case, the force is opposite in direction to the displacement. This helps define Simple Harmonic Motion. Force = -kx ma = -kx a = -(k/m)x {This always gives SHM}

5. Both systems show: • Equilibrium Position • Natural frequency (f) • How frequently the system cycles in one second. • Measured in Hertz (Hz = 1/s) • Example: the frequency of the wings of a housefly is about 200 Hz. • Amplitude • How large does the displacement ever get • Measured in meters

6. Both systems show: • Displacement • How far the mass is from equilibrium at any moment • Period (T) • How much time it takes for the motion to repeat • Measured in seconds • T = 1 / f • Angular Frequency • w = 2pf • Used for describing motion when using trig functions; one cycle is 2p radians. • Housefly Example: w = 2pf = 2p(200 Hz) ≈ 1260 Hz

7. Graph of Displacement vs. Time • You might need a separate page of paper to draw 4 graphs, one below the other, like a tower.

8. Graph of Displacement vs. Time

9. Graph of Velocity vs. Time

10. Graph of Acceleration vs. Time

11. Graph of Force vs. Time

12. Graph of Force vs. Time • Compare this graph to the defining idea: the force is opposite in direction to the displacement.

13. Review the Basic Math of SHM • The force is opposite in direction to the displacement. • Force = -kx {This always gives SHM} • ma= -kx {This always gives SHM} • a = -(k/m)x {This always gives SHM}

14. The Solution of SHM • If you take a system out of equilibrium, by a distance A (for ‘amplitude’) then the system will oscillate. • The sine and cosine functions are good descriptions of SHM. • x = A cos(wt) (This is the solution.) • velocity = -Awsin(wt) (what is the fastest that the object will move?) • acceleration = -Aw2 cos(wt)

15. Check the Solution of SHM • Recall: • a = -(k/m)x {This always gives SHM} • x = A cos(wt) and a = -Aw2 cos(wt) • Put it all together: -Aw2 cos(wt) = -(k/m) A cos(wt) w2 = (k/m) • For large restoring forces, the frequency is large. • For large masses, the frequency is small. a = -w2x

16. Connection between speed & position v2 = w2 (A2 – x2) • This equation tell us that when the displacement is equal to the amplitude, the speed is _______.

17. Connection between speed & position v2 = w(A2 – x2) • This equation tell us that when the displacement is equal to the amplitude, the speed is zero.

18. Connection between speed & position v2 = w(A2 – x2) • This equation tell us that when the displacement is equal to the amplitude, the speed is zero. • And, when the displacement is zero, the speed is

19. Connection between speed & position v2 = w(A2 – x2) • This equation tell us that when the displacement is equal to the amplitude, the speed is zero. • And, when the displacement is zero, the speed is

20. Kinetic Energy One • KE = ½mv2 • Speed depends on time: v = -Awsin(wt) • So, KE = ½m(-Awsin(wt))2 • KE = ½mA2w2sin2wt • The Kinetic energy is a varying function. What is its greatest value? • KEmax = ½mw2A2

21. Kinetic Energy Two • KE = ½mv2 • Also, speed depends on position: v2 = w2 (A2 –x2) • So KE = ½mw2(A2 – x2) • What is the greatest value that the Kinetic Energy can have? • KEmax = ½mw2A2

22. Total Energy = KE + PE • The total energy stays the same for a closed system. • As the system passes through equilibrium, the Potential Energy is zero; all of the energy is kinetic. This ‘fixes’ the value of the total energy to be the same as the maximum of the kinetic energy. • Total Energy = ½mwA2

23. Total Energy = KE + PE • PE = Total Energy - ? • PE = Total Energy – KE • We can write it as a function of position or of time. • Time View: • PE = ½mwA2 - ½mA2w2sin2wt • PE = ½mwA2{1-sin2wt}= ½mwA2 {cos2wt} • Position View: • PE = ½mwA2 - ½mw(A2 – x2) • PE = ½mwx2

24. Example: How much does this pendulum have as it passes through equilibrium? • The mass of the ‘bob’ is 200 g = 0.2 kg • The amplitude of the motion is 3 cm = 0.03 m. • The frequency of the motion is 0.5 Hz.

25. Example: How much energy does this pendulum have as it passes through equilibrium? • The KE at the equilibrium position is the greatest KE the bob will ever have. • KEmax = ½mw2A2 • w = ? • w = 2pf = 2p(0.5) = 3.14 • KEmax = ½mw2A2 • KEmax = ½ (0.2)(3.14)2(0.03)2 • KEmax = 9 x 10-4 J

26. In the previous example… • How much Gravitational Potential Energy does the bob have when it is at equilibrium? • 0 J • How much Gravitational Potential Energy does the bob have when it is at the extreme part of its motion? • 9 x 10-4 J

27. A story of damping: • Use a tea bag to make some tea. Pull the the little label of the tea bag so that the bag is suspended above the hot liquid. • The bag will begin to spin. Why? • Then the bag will spin faster. Why? • Then the bag will slow to a stop. Why? • Then the bag will spin the other way. Why? • Now change one thing…

28. Do the same experiment, but… • Suspend the bag so that it can spin, but keep the bag in the liquid, but. • What will be different? • The liquid ‘damps’ the motion. Oscillations might still occur, but with less and less amplitude. • Imagine doing the same experiment in honey. • See the figure…

29. Find the bag in water and in honey. http://www.splung.com/content/sid/2/page/damped_oscillations

30. Natural Frequency • If you disturb a system, and let it go, it will often vibrate. That vibration will be at its natural frequency. • Examples: • Pendulum • mass on spring • guitar string

31. Forced Vibration • If you push and pull on a system, then that has a big effect on what the system does. • Examples: • Pendulum • mass on spring • guitar string

32. Resonance • If you push and pull on a system, at the natural frequency of the system, then the amplitude can get really big. • Examples: • kid on a swing. • wine glass • Tacoma Narrows Bridge Disaster • St. Louis Bridge in a hotel • See the resonance curve:

33. Amplitude vs. Frequency

34. Phase • Imagine that you are keeping a nice rhythm, by clapping once per second. • Another person could clap with the same frequency, but be out of step with you, by 0.5 seconds, or 0.25 seconds, whatever. • This difference is called “phase”.

35. Waves • A wave is a disturbance that moves. • If you make disturb the surface of a swimming pool in a simple harmonic way, then at any one point on the surface the water will oscillate with SHM. • Also, the disturbances move across the pool, and that’s what we call waves. • Examples: sound, light, garden hose • What follows are properties of waves…

36. Transverse vs. Longitudinal • If the disturbance is perpendicular to the motion of the disturbance, we call the wave “Transverse”. • Examples: Classic Slinky wave, waves on the surface of a swimming pool, light. • If the disturbance is in the same direction as the motion of the disturbance, we call the wave “Longitudinal”. • Examples: Sound, waves on a slinky made by pulling and pushing.

37. Wavelength (l) http://electron9.phys.utk.edu/phys135d/modules/m10/waves.htm

38. Amplitude (A, x0)

39. Wave Speed (v) • v = fl • Practice: If the frequency is 4 Hz, and the wavelength is 0.5 m, then what is the speed of the wave? • v = fl = (4 s-1) (0.5 m) = 2 m s-1

40. Guitar Basics • If the string is thicker, then the speed of the wave is slower. What does that do to frequency? • v = fl. So f = v / l. We see that thick strings cause a low values for frequency. Thick strings make low sounds. • Tightening the tension in the string makes the speed greater. What does that do to pitch? • Putting your finger on the string makes it shorter, what does that do to frequency? Hint: f = v / l

41. Reflection • If you are standing in a pool, and you make disturbances on the surface, then when the pulses reach the wall of the pool, they will bounce back. • Waves on a slinky show this. • Mirrors reflect light. • Walls reflect sound.

42. Reflection http://electron9.phys.utk.edu/phys135d/modules/m10/waves.htm

43. Interference (Superposition) • One of the craziest things about waves is that they can can go through each other. • Examples: Slinky, Light and Sound • But while two waves are in the same spot, the result is the addition (or subtraction of the waves).

44. Constructive Interference http://electron9.phys.utk.edu/phys135d/modules/m10/waves.htm

45. Destructive Interference http://electron9.phys.utk.edu/phys135d/modules/m10/waves.htm

46. Diffraction http://learn.uci.edu/oo/getOCWPage.php?course=OC0811004&lesson=005&topic=006&page=10

47. Diffraction • Sound goes around corners. • Water waves bend around edges.

48. Diffraction depends on the size of the opening.

49. Diffraction caused by an obstacle http://mcat-review.org/waves-periodic-motion.php

50. Refraction http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/refr2.html