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Simple Harmonic Motion

Section 1 Simple Harmonic Motion. Chapter 11. Simple Harmonic Motion. Simple harmonic motion =periodic motion resulting from a restoring force that is proportional to displacement. Examples Mass spring systems Pendulum. Section 2 Measuring Simple Harmonic Motion. Chapter 11.

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Simple Harmonic Motion

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  1. Section 1 Simple Harmonic Motion Chapter 11 Simple Harmonic Motion • Simple harmonic motion=periodic motion resulting from arestoring forcethat isproportional to displacement. • Examples • Mass spring systems • Pendulum

  2. Section 2 Measuring Simple Harmonic Motion Chapter 11 Amplitude, Period, and Frequency in SHM • amplitude=maximum displacement from equilibrium • Apendulum’s is angle θ (rads) • mass-spring system,=x in (m).

  3. Section 2 Measuring Simple Harmonic Motion Chapter 11 Amplitude, Period, and Frequency in SHM • Theperiod(T)=time it takes a complete cycle • SI unit is seconds (s). • The frequency(f)=number of cycles per unit of time. • SI unit is hertz (Hz). • Hz = s–1

  4. Section 2 Measuring Simple Harmonic Motion Chapter 11 Amplitude, Period, and Frequency in SHM, continued • Period and frequency areinversely related:

  5. Section 2 Measuring Simple Harmonic Motion Chapter 11 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 (for small angles).

  6. Section 2 Measuring Simple Harmonic Motion Chapter 11 Measures of Simple Harmonic Motion

  7. Section 1 Simple Harmonic Motion Chapter 11 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.

  8. Review Chapter 11 Hooke’s Law • Felastic = –kx

  9. 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?

  10. Section 1 Simple Harmonic Motion Chapter 11 Sample Problem, continued 2. Plan, continued Rearrange the equation to isolate the unknown:

  11. 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.

  12. Section 2 Measuring Simple Harmonic Motion Chapter 11 Period of a Mass-Spring System in SHM • The period of an ideal mass-spring system depends onthemassand on the spring constant. • T does not depend on the A. • only for systems that obeys Hooke’s law.

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. 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.

  18. Section 3 Properties of Waves Chapter 11 Transverse Waves

  19. 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.

  20. Section 3 Properties of Waves Chapter 11 Longitudinal Waves

  21. 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.

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

  23. 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.

  24. 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.

  25. 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.

  26. 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.

  27. 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.

  28. 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.

  29. Section 4 Wave Interactions Chapter 11 Comparing Constructive and Destructive Interference

  30. 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

  31. Section 4 Wave Interactions Chapter 11 Standing Waves

  32. 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.

  33. 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).

  34. 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.

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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. 3. In what form is the energy in the system when the mass passes through the equilibrium point? A. elastic potential energy B. gravitational potential energy C. kinetic energy D. a combination of two or more of the above

  40. 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. 3. In what form is the energy in the system when the mass passes through the equilibrium point? A. elastic potential energy B. gravitational potential energy C. kinetic energy D. a combination of two or more of the above

  41. 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. 4. In what form is the energy in the system when the mass is at maximum displacement? F. elastic potential energy G. gravitational potential energy H. kinetic energy J. a combination of two or more of the above

  42. 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. 4. In what form is the energy in the system when the mass is at maximum displacement? F. elastic potential energy G. gravitational potential energy H. kinetic energy J. a combination of two or more of the above

  43. 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. 5. Which of the following does not affect the period of the mass-spring system? A. mass B. spring constant C. amplitude of vibration D. All of the above affect the period.

  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. 5. Which of the following does not affect the period of the mass-spring system? A. mass B. spring constant C. amplitude of vibration D. All of the above affect the period.

  45. 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. 6. If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation? F. 8p s H. p s G. 4p s J.p/2 s

  46. 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. 6. If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation? F. 8p s H. p s G. 4p s J.p/2 s

  47. Chapter 11 Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. 7. What is the restoring force in the pendulum? A. the total weight of the bob B. the component of the bob’s weight tangent to the motion of the bob C. the component of the bob’s weight perpendicular to the motion of the bob D. the elastic force of the stretched string

  48. Chapter 11 Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. 7. What is the restoring force in the pendulum? A. the total weight of the bob B. the component of the bob’s weight tangent to the motion of the bob C. the component of the bob’s weight perpendicular to the motion of the bob D. the elastic force of the stretched string

  49. Chapter 11 Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. 8. Which of the following does not affect the period of the pendulum? F. the length of the string G. the mass of the pendulum bob H. the free-fall acceleration at the pendulum’s location J. All of the above affect the period.

  50. Chapter 11 Standardized Test Prep Multiple Choice, continued Base your answers to questions 7–10 on the information below. A pendulum bob hangs from a string and moves with simple harmonic motion. 8. Which of the following does not affect the period of the pendulum? F. the length of the string G. the mass of the pendulum bob H. the free-fall acceleration at the pendulum’s location J. All of the above affect the period.

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