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Objectives

Section 1 Simple Harmonic Motion. Chapter 12. Objectives. Identify the conditions of simple harmonic motion. Explain how force, velocity, and acceleration change as an object vibrates with simple harmonic motion. Calculate the spring force using Hooke’s law.

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Objectives

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  1. Section 1 Simple Harmonic Motion Chapter 12 Objectives • Identifythe conditions of simple harmonic motion. • Explainhow force, velocity, and acceleration change as an object vibrates with simple harmonic motion. • Calculatethe spring force using Hooke’s law.

  2. Section 1 Simple Harmonic Motion Simple Harmonic Motion • Motion that repeats in a regular pattern over and over again is called periodic motion.

  3. Section 1 Simple Harmonic Motion Hooke’s Law • One type ofperiodic motionis the motion of a mass attached to a spring. • The direction of the force acting on the mass (Felastic) is always opposite the direction of the mass’s displacement from equilibrium (x = 0).

  4. Section 1 Simple Harmonic Motion Hooke’s Law, continued At equilibrium: • The spring force and the mass’s acceleration become zero. • The speedreaches amaximum. At maximum displacement: • The spring forceand the mass’saccelerationreach amaximum. • The speedbecomeszero.

  5. Section 1 Simple Harmonic Motion Hooke’s Law, continued • Measurements show that thespringforce, Felasticor restoring force,isdirectly proportionalto thedisplacementof the mass. • This relationship is known asHooke’s Law: Felastic = –kx spring force = –(spring constant  displacement) • The quantitykis a positive constant called thespring constant.

  6. Section 1 Simple Harmonic Motion Chapter 11 Sample Problem Hooke’s Law If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?

  7. Section 1 Simple Harmonic Motion Chapter 11 Sample Problem, continued 3. Calculate Substitute the values into the equation and solve: 4. Evaluate The value of k implies that 270 N of force is required to displace the spring 1 m.

  8. Section 1 Simple Harmonic Motion Chapter 11 Spring Constant https://www.youtube.com/watch?v=nMQxqsyuJKE https://sites.google.com/site/physicsflash/home/shm

  9. Hooke’s Law • The spring constant can be determined experimentally • Measure the displacement (x) and the corresponding restoring force (F) • Gather several data points • Plot Force vs Displacement • Draw a trend line • Calculate the slope

  10. Hooke’s Law • The slope of the line on a Force vs Displacement graph is the spring constant (k) • It must be a positive number, greater than zero • If the data shows something else, there is a problem

  11. Section 1 Simple Harmonic Motion Chapter Simple Harmonic Motion • Simple harmonic motiondescribes any periodic motion that is the result of arestoring forcethat isproportional to displacement. • Because simple harmonic motion involves a restoring force,every simple harmonic motion is a back-and-forth motion over the same path.

  12. Section 1 Simple Harmonic Motion Chapter Simple Harmonic Motion

  13. Section 1 Simple Harmonic Motion Chapter The Simple Pendulum • A simple pendulumconsists of amass called a bob, which is attached to a fixed string. The forces acting on the bob at any point are the force exerted by the string and the gravitational force.

  14. Section 2 Measuring Simple Harmonic Motion Chapter Objectives • Identifythe amplitude of vibration. • Recognizethe relationship between period and frequency. • Calculatethe period and frequency of an object vibrating with simple harmonic motion.

  15. Section 2 Measuring Simple Harmonic Motion Chapter Amplitude, Period, and Frequency in SHM • In SHM, the maximum displacement from equilibrium is defined as theamplitude of the vibration. • Apendulum’s amplitude can be measured by the angle between the pendulum’s equilibrium position and its maximum displacement. • For amass-spring system,the amplitude is the maximum amount the spring is stretched or compressed from its equilibrium position. • TheSI unitsof amplitude are themeter (m).

  16. Section 2 Measuring Simple Harmonic Motion Chapter Amplitude, Period, and Frequency in SHM • Theperiod(T)is the time that it takes a complete cycle to occur. • The SI unit of period is seconds (s). • The frequency(f)is the number of cycles or vibrations per second (1/s, s-1) The SI unit of frequency is hertz (Hz). Hz = s–1

  17. Section 2 Measuring Simple Harmonic Motion Chapter Amplitude, Period, and Frequency in SHM, continued • Period and frequency areinversely related: • Thus, any time you have a value for period or frequency, you can calculate the other value.

  18. Section 2 Measuring Simple Harmonic Motion Chapter Measures of Simple Harmonic Motion

  19. Section 2 Measuring Simple Harmonic Motion Chapter Big Ben’s pendulum click here for video

  20. Section 2 Measuring Simple Harmonic Motion Chapter Period of a Simple Pendulum in SHM • The period of a simple pendulum depends on thelengthand on the free-fall acceleration. • The period doesnot depend on the mass of the bob or on the amplitude.

  21. Section 2 Measuring Simple Harmonic Motion Chapter Period of a Simple Pendulum in SHM • The length of a pendulum is measured from the pivot point to the center of gravity of the bob

  22. Section 3 Properties of Waves Chapter 11 Objectives • Distinguishlocal particle vibrations from overall wave motion. • Differentiatebetween pulse waves and periodic waves. • Interpret waveforms of transverse and longitudinal waves. • Applythe relationship among wave speed, frequency, and wavelength to solve problems. • Relateenergy and amplitude.

  23. Section 3 Properties of Waves Chapter 11 Wave Motion • Awaveis the motion of a disturbance. • Amediumis a physical environment through which a disturbance can travel. For example, water is the medium for ripple waves in a pond. • Waves that require a medium through which to travel are calledmechanical waves.Water waves and sound waves are mechanical waves. • Electromagnetic wavessuch as visible light do not require a medium.

  24. Section 3 Properties of Waves Chapter 11 Wave Types • A wave that consists of a single traveling pulse is called apulse wave. • Whenever the source of a wave’s motion is a periodic motion, such as the motion of your hand moving up and down repeatedly, aperiodic waveis produced. • A wave whose source vibrates with simple harmonic motion is called asine wave.Thus, a sine wave is a special case of a periodic wave in which the periodic motion is simple harmonic.

  25. Section 3 Properties of Waves Chapter 11 Relationship Between SHM and Wave Motion As the sine wave created by this vibrating blade travels to the right, a single point on the string vibrates up and down with simple harmonic motion.

  26. Section 3 Properties of Waves Chapter 11 Wave Types, continued • Atransverse waveis a wave whose particles vibrateperpendicularlyto the direction of the wave motion. • Thecrestis the highestpoint above the equilibrium position, and thetroughis thelowestpoint below the equilibrium position. • Thewavelength (l)is the distance between two adjacent similar points of a wave.

  27. Section 3 Properties of Waves Chapter 11 Transverse Waves

  28. Section 3 Properties of Waves Chapter 11 Wave Types, continued • Alongitudinal wave is a wave whose particles vibrate parallel to the direction the wave is traveling. • A longitudinal wave on a spring at some instant t can be represented by a graph. Thecrestscorrespond to compressed regions, and thetroughscorrespond to stretched regions. • The crests are regions of high density and pressure (relative to the equilibrium density or pressure of the medium), and the troughsare regions of low density and pressure.

  29. Section 3 Properties of Waves Chapter 11 Longitudinal Waves

  30. Section 3 Properties of Waves Chapter 11 Period, Frequency, and Wave Speed • The frequency of a wavedescribes the number of waves that pass a given point in a unit of time. • Theperiod of a wavedescribes the time it takes for a complete wavelength to pass a given point. • The relationship between period and frequency in SHM holds true for waves as well; the period of a wave isinversely relatedto its frequency.

  31. Section 3 Properties of Waves Chapter 11 Characteristics of a Wave

  32. Section 3 Properties of Waves Chapter 11 Period, Frequency, and Wave Speed, continued • The speed of a mechanicalwave is constant for any given medium. • Thespeed of a waveis given by the following equation: v = fl wave speed = frequency  wavelength • This equation applies to both mechanical and electromagnetic waves.

  33. Section 3 Properties of Waves Chapter 11 Waves and Energy Transfer • Waves transfer energy by the vibration of matter. • Waves are often able to transport energy efficiently. • The rate at which a wave transfers energy depends on theamplitude. • The greater the amplitude, the more energy a wave carries in a given time interval. • For a mechanical wave, the energy transferred is proportional to the square of the wave’s amplitude. • The amplitude of a wave gradually diminishes over time as its energy is dissipated.

  34. Section 4 Wave Interactions Chapter 11 Objectives • Apply the superposition principle. • Differentiatebetween constructive and destructive interference. • Predictwhen a reflected wave will be inverted. • Predictwhether specific traveling waves will produce a standing wave. • Identifynodes and antinodes of a standing wave.

  35. Section 4 Wave Interactions Chapter 11 Wave Interference • Two different material objects can never occupy the same space at the same time. • Because mechanical waves are not matter but rather are displacements of matter, two waves can occupy the same space at the same time. • The combination of two overlapping waves is calledsuperposition.

  36. Section 4 Wave Interactions Chapter 11 Wave Interference, continued In constructive interference,individual displacements on the same side of the equilibrium position are added together to form the resultant wave.

  37. Section 4 Wave Interactions Chapter 11 Wave Interference, continued In destructive interference,individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave.

  38. Section 4 Wave Interactions Chapter 11 Reflection • What happens to the motion of a wave when it reaches a boundary? • At afree boundary, waves arereflected. • At afixedboundary, waves arereflectedand inverted. Free boundary Fixed boundary

  39. Section 4 Wave Interactions Chapter 11 Standing Waves • A standing waveis a wave pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere. • Standing waves have nodes and antinodes. • A nodeis a point in a standing wave that maintainszero displacement. • Anantinodeis a point in a standing wave, halfway between two nodes, at which thelargest displacementoccurs.

  40. Section 4 Wave Interactions Chapter 11 Standing Waves, continued • Only certain wavelengthsproduce standing wave patterns. • Theendsof the string must benodesbecause these points cannot vibrate. • A standing wave can be produced for any wavelength that allows both ends to be nodes. • In the diagram, possible wavelengths include2L(b),L(c), and2/3L(d).

  41. Section 4 Wave Interactions Chapter 11 Standing Waves This photograph shows four possible standing waves that can exist on a given string. The diagram shows the progression of the second standing wave for one-half of a cycle.

  42. Chapter 11 Standardized Test Prep Multiple Choice Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 1. In what direction does the restoring force act? A. to the left B. to the right C. to the left or to the right depending on whether the spring is stretched or compressed D. perpendicular to the motion of the mass

  43. Chapter 11 Standardized Test Prep Multiple Choice Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 1. In what direction does the restoring force act? A. to the left B. to the right C. to the left or to the right depending on whether the spring is stretched or compressed D. perpendicular to the motion of the mass

  44. Chapter 11 Standardized Test Prep Multiple Choice, continued Base your answers to questions 1–6 on the information below. A mass is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface. 2. If the mass is displaced –0.35 m from its equilibrium position, the restoring force is 7.0 N. What is the spring constant? F. –5.0  10–2 N/m H. 5.0  10–2 N/m G. –2.0  101 N/m J. 2.0  101 N/m

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