Vibrations and Waves

1. Vibrations and Waves Simple Harmonic Motion Wave Interactions Mechanical Waves (eg, Sound) Electromagnetic Waves (eg, Light)

2. Wavelength Amplitude Equilibrium Position Properties of Waves • Amplitude – Maximum displacement from equilibrium • Period – Time to complete one cycle (wavelength) of motion. Represented by T; Units of sec. • Frequency – Number of cycles (wavelengths) per unit time. Represented by f; Units of Hz or sec-1 kHz (AM radio station), MHz (FM radio station), GHz (radar, microwaves), etc.. • Wavelength – Distance between two adjacent corresponding points on a wave (e.g., crests, troughs, etc.). Represented by lambda, l; Units of length (m, ft, etc.) Note: Period = 1 / Frequency and Frequency = 1 / Period

3. Direction of Medium (Spring) Direction of Wave Waves and Wave Motion • What is a Wave?? • The motion of a disturbance! • Example: • One person on each end of a long spring (or rope) • A pulse is produced in the spring…. • Wave pulse moves from one end of the spring to the other, • BUT no part of the spring is being carried from one person to the other.

4. Periodic Motion • Definition: Back and forth motion over the same path • Examples: • Mass - Spring System • Bungee Jumping • Shock Absorbers on Vehicles • Pendulums: • Child on a swing; • Trapeze Artists • Pendulum of a grandfather clock • Wrecking Ball

5. Simple Harmonic Motion • Definition: • Vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium • Sine waves describe particles vibrating with SHM • Examples: • Mass – Spring System (Hooke’s Law) • Pendulum (small angles, <15 degrees) • Examples: Visible light, radio waves, microwaves, x-rays, etc.

6. Fel x = displacement in meters Fg Question--- Is there a direct relationship between the displacement of a mass on a spring and the elastic force of that spring?Is Felastic proportional to x?... i.e, Felastic= Constant * x? At equilibrium, • Net force is zero • So, Fg + Felastic = 0 Fg = - Felastic Fg = - Constant * x

7. Mass-Spring SystemWorksheet Fel x = displacement in meters Fg • NOTE: • Fg = force due to gravity (Fg = m*g) • Fel= Elastic Force of the spring Use 250g, 500g, 1000g masses. Eight combinations, including 0 mass.

8. Mass-Spring SystemPlot Fg vs. x Fg, (in Newtons) Displacement, x (in meters)

9. Hooke’s Law For a Spring-Mass System, Robert Hooke established the relationship between Force and Displacement: Felastic = - kx where, k is known as the “Spring Constant”, measuring the “stiffness” of the spring. Units for k is N/m.

10. x = -0.02 m Hooke’s Law (con’t) Example 1: If a mass of 0.55kg attached to a vertical spring stretches the spring 2 cm from its equilibrium position, what is the spring constant? Given: m = 0.55 kg x = -0.02 m g = -9.8 m/s2 Solution: Fnet = 0 = Felastic + Fg 0 =- kx + mg or, kx = mg k = mg/x = (0.55 g)(-9.8 m/s2)/(-0.02 m) = 270 N/m Fel Fel Fg Fg

11. m k Mass-Spring System Period of a Mass-Spring System: T = 2p√ Where, k is the spring constant and m is the mass NOTE: Changing the amplitude of the vibration (x) does NOT affect the period or frequency of vibration.

12. Example: Mass-Spring System A body of a 1275 kg car is supported on a frame by four springs, each of which has a spring constant of 2.0 x 104 N/m. Two people riding in the car have a combined mass of 153 kgs. Find the period of vibration of the car when it is driven over a pothole in the road. Solution: k = 2 x 104 N/m m = 1275 kg + 153 kg = 1428 kg But themass is evenly distributed over 4 springs, so meff = 1428/4 = 357 kgs T = 2 * p *(357 kgs/2 x 104 N/m)1/2 = 2 * p* ( 0.01785 s2)1/2 = 0.84 s

13. L g Pendulum System Period of a Pendulum System: T = 2p√ Where, L is the length of the pendulum arm g is the acceleration due to gravity NOTE: Changing the amplitude of the vibration (q) does NOT affect the period or frequency of vibration.

14. Pendulums and Spring-Mass Systems The period and frequency of motion for each of these systems is INDEPENDENT of: Pendulum: Amplitude (q) Mass on swinging arm Mass-Spring System: Amplitude (x)

15. Example: Pendulum You need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and that its period is 12 s. How tall is the tower? Given: T = 12 s g = 9.8 m/s2 Solution: Use the equation for the period of the pendulum and solve for L. T = 2 * p * (L / g )1/2 T2 = 4 * p2 * (L / g) (T2 * g) / (4 * p2) = L ((12 s2)2 * 9.8 m/s2) / (4 * p2) = 35.7 m tall

16. You are sightseeing in Europe…and curious about the architectural structures…. What would be other ways to determine the height of a tower given minimal pieces of data?? Review and Revisit…

17. Wave Speed, v Speed of wave (v) depends upon: • Medium • Frequency, f • Wavelength, l Wave Speed = wavelength x frequency or Equation:v = lf

18. Think About It…. • Given the equation for the speed of waves: v = lf Does this mean, for example, that high frequency sounds (high pitches), travel faster than low frequency sounds???? NO!!! Wavelength and frequency vary inversely to produce the same speed of all sounds

19. Velocity, v A B Doppler Effect MOVING SOUND-GENERATING OBJECT STATIONARY SOUND-GENERATING OBJECT Waves are created at point source and radiate outward creating a wave front with the same frequency as that of the source. Although the frequency of the sound generating object remains constant, wave fronts reach the observer at Point B more frequently than Point A.

20. Doppler Effect Doppler Effect: The frequency shift that is the result of relative motion between the source of waves and an observer. Higher frequency: Object approaching Lower frequency: Object receding Some Applications: Echolocation (e.g., Submarines, Dolphins, Bats, etc.) Police Radar Weather Tracking

21. Resonance Every object (all matter!!) vibrates at a characteristic frequency – resonant (“natural”) frequency. Resonance: A condition that exists when the frequency of a force applied to a system matches the natural frequency of the system. Examples: • Pushing a swing • Tuning a radio station • Voice-shattered glass. • Tacoma Narrows Bridge Collapse in 1940.—High winds set up standing waves in the bridge, causing the bridge to oscillate at one of its natural frequencies.

22. Wave Interactions • Unlike Matter, more than one wave or vibration can exist • at the same time and • in the same space. • This is known as SUPERPOSITION. • Superposition Principle: • The method of summing the displacements (amplitudes) of 2 or more waves to produce a resultant wave. • Applies to all waves types – mechanical and electromagnetic.

23. = + = + Interference Patterns The individual waves can overlap and produce interference patterns. The resultant wave is the sum of the displacements from equilibrium (ie the amplitude) at each point for the individual waves. Constructive Interference = Reinforcement Destructive Interference = Cancellation

24. = + = + Superposition Principle

25. Constructive and Destructive Interference Definitions • Constructive Interference: • Interference in which individual displacements on the SAME SIDE of the equilibrium position are added together to form the resultant wave. • Destructive Interference: • Interference in which individual displacements on OPPOSITE sides of the equilibrium position are added together to form the resultant wave.

26. Wave Superposition-- Demo Demo1: (1) Using a long coiled spring, generate a transverse pulse wave(s). a. First, from one end while other end fixed. b. Then from both ends of the spring simultaneously and in the same direction. Observe that the amplitudes of traveling waves add as the waves pass one another.

27. Wave Superposition– Activities Using a long coiled spring, generate transverse pulse wave(s) from each end of the spring simultaneously. Observe the pulse that reaches your hand after the pulses have passed through one another. Experiment with the following variables: a. Displacements in opposite directions; same directions b. Pulses of different amplitudes c. Combinations of a. and b. What did you observe? Which examples were constructive and which were destructive? What can you conclude? Observe that the pulses that pass through from one side to the other are unaffected by the presence of the other pulse!!

28. Beat Frequency equals: fbeat = f1 – f2 f1 f2 BEATS

29. Standing Waves Standing Waves: Resultant wave created by the interference of two waves traveling at the same frequency, amplitude and wavelength in opposite directions. Standing Waves have Nodes and Antinodes

30. Nodes and Antinodes • Nodes: • Points in the standing wave where the two waves cancel – complete destructive interference– creating a stationary point! • Antinodes • Point in the standing wave, halfway between the nodes, at which the largest amplitude occurs.

31. Standing Waves Wavelength, l l1 = 2L l2 = L l3 = 2L/3 l4 = 2L/4 or ½ L Only certain frequencies of vibration produce standing waves for a given string length!!! ….More later when we get to SOUND… The wavelength of each of the standing waves depends on the string length, L ln = 2L/n

32. Standing Waves on a Vibrating String Wavelength, l l1 = 2L l2 = L l3 = 2L/3 l4 = 2L/4 or ½ L Frequency, f f1 = v / l1 f2= 2 f1 f3 = 3 f1 f4 = 4f1 A N N Fundamental Frequency or 1st Harmonic A A N N N 2nd Harmonic A A A N N 3rd Harmonic N N A A A A N 4th Harmonic N N N N ln = 2L/n fn = n v/2L n = 1, 2, 3, …

33. Standing Waves on a Vibrating String • Fundamental Frequency: • The lowest frequency of vibration of a standing wave: f1 = v / l1 = v / 2L Where, v is the speed of waves on the vibrating string (NOT the speed of the resultant waves in air!!!!) L is the portion of the string that is vibrating

34. Harmonic Series of Standing Waves on a Vibrating String A series of frequencies that includes the fundamental frequency and integral multiples of the fundamental frequency. fn = n v / 2L, n = 1,2,3,…. Frequency = harmonic number x (speed of wave on the string) / (2 x length of the vibrating string)

35. A A N N N A A A N N N N Standing Waves in an Air ColumnOPEN at BOTH ENDS Frequency, f f1 = v / l1 f2= 2 f1 f3 = 3 f1 Wavelength, l l1 = 2L l2 = L l3 = 2L/3 Fundamental Frequency or 1st Harmonic 2nd Harmonic A A A A 3rd Harmonic N N N N N ln = 2L/n fn = n v/2L n = 1, 2, 3, … Example: Organ Pipes; Flute

36. A A N N A A A N N N N A A A A N N N N N Standing Waves in an Air ColumnCLOSED at ONE END Frequency, f f1 = v / l1 f3= 3 f1 f5 = 5 f1 Wavelength, l l1 = 4L l3 = 4L/ 3 l5 = 4L/5 N Fundamental Frequency or 1st Harmonic 3rd Harmonic 5th Harmonic fn = n v/4L n = 1, 3, 5,… ln = 4L/n Example*: Clarinet, Saxophone, Trumpet

37. Waves Types • Pulse Waves – A Single non-periodic disturbance • Periodic Waves -- A wave whose source is a form of periodic motion • Transverse Waves– A wave whose particles vibrate perpendicular to the direction of wave motion. • Longitudinal Waves – A wave whose particles vibrate parallel to the direction of the wave motion

38. Wave Motion • Mechanical Waves– Propagation requires a medium • Examples: Sound waves; ripples in water, etc • Electromagnetic Waves – Propagation does NOT require a medium; can travel in a vacuum

39. LIGHT Characteristics of “Light” • Electromagnetic Wave: • A TRANSVERSE wave • Consisting of alternating electric and magnetic fields at right angles to each other, • Travels through a vacuum • At the speed of light, c (3 x 108 m/s) • Wave-Particle Duality (more later!)– Light can also be described as a “Particle” See Holt T63

40. Visible Light • Visible Light: • Small Part of EM Spectrum • Wavelengths: 700 nm (red) > l > 400 nm (violet) • Frequencies: 4.3 x 1014 Hz < f < 7.5 x 1014 Hz

41. Electromagnetic Spectrum

42. Speed of Light • All EM radiation travel at the speed of light in a vacuum… • but their wavelengths and frequencies will vary! • Wave Speed Equation: c = l * f Speed of light = wavelength x frequency

43. Interactions of EM Radiation with Matter • Radiation interacts with matter in 3 principal ways: • Scattered ….from the material’s surface • Absorbed ….by the material • Transmitted ….through the material, often changing direction in the process.

44. Polarization of Light • Unpolarized light: Randomly oscillating charges (electric and magnetic fields) • Linear Polarization: The alignment of the electromagnetic waves in such a way that the vibrations of the electric fields in each of the waves are parallel to each other. For example, certain processes can separate waves with electric field oscillations in the vertical direction from those in the horizontal direction. • Light can be linearly polarized through: • Transmission, and/or • Reflection and Scattering See Holt T 70, 71

45. Polarization of Light via Transmission The transmission axis of the substance is parallel to the plane of polarization of the light– Light passes through freely and “brightly”! Direction of Wave Transmission Axis

46. Polarization of Light via Transmission As the angle between the plane of polarization for the light and the transmission axis of the substance increases from 0 to 90 degrees, amount of light passing through decreases from 100% to 0%, The transmission axis of the substance is perpendicular to the plane of polarization of the light– NO Light passes through X Direction of Wave Transmission Axis

47. Polarization of Light via Reflection • When light is reflected a certain angle from a surface, the reflected light is completely polarized parallel to the reflecting surface. • For example, if the reflecting surface is parallel to the ground, then the light is polarized horizontally. • Eg, roadways, car hoods, bodies of water • Sunglasses application… • filter out horizontally polarized “glare” with a “vertical” polarizer.

48. Polarization of Light by Scattering • Scattering of light (the absorption and re-radiation of light) by particles in the atmosphere can also cause polarization. • Example: SUNLIGHT: • When unpolarized beam of sunlight strikes air molecules in the atmosphere, the electrons in the molecules begin to vibrate in the same plane as the electric field of the incoming wave. • The re-radiated light is polarized in the direction of the electron oscillations.

49. Physics of Color • All kinds of interactions of light with matter (scattering, absorption, and transmission) depend on the wavelength of the EM radiation. • Rules governing the scattering of EM Waves: • If the object causing the scattering is much smaller than the wavelength of radiation, then shorterwavelengths are scattered much more strongly than longer ones. • If the object causing the scattering is much larger than the wavelength of incoming radiation, then all wavelengths are scattered equally.

50. Why is the Sky Blue?? • Sunlight is scattered by air molecules in the atmosphere. • Since the size of molecules (tenths of nanometers) is much less than the wavelength of visible light (hundreds of nms), we expect short wavelengths (blue light) to be much more scattered than longer ones (red light). • The sky appears blue!