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Learn about Simple Harmonic Motion, Hooke's Law, and Spring Constant Calculation in Physics Chapter 12. View slides and solve sample problems.

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  1. How to Use This Presentation • To View the presentation as a slideshow with effects select “View” on the menu bar and click on “Slide Show.” • To advance through the presentation, click the right-arrow key or the space bar. • From the resources slide, click on any resource to see a presentation for that resource. • From the Chapter menu screen click on any lesson to go directly to that lesson’s presentation. • You may exit the slide show at any time by pressing the Esc key.

  2. Resources Chapter Presentation Visual Concepts Transparencies Sample Problems Standardized Test Prep

  3. Chapter 11 Vibrations and Waves Table of Contents Section 1 Simple Harmonic Motion Section 2 Measuring Simple Harmonic Motion Section 3 Properties of Waves Section 4 Wave Interactions

  4. 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.

  5. Section 1 Simple Harmonic Motion Chapter 12 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 alwaysopposite the direction of the mass’s displacement from equilibrium (x = 0).

  6. Section 1 Simple Harmonic Motion Chapter 12 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.

  7. Section 1 Simple Harmonic Motion Chapter 12 Hooke’s Law, continued • Measurements show that thespringforce, or 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. (is given or on chart).

  8. Section 1 Simple Harmonic Motion Chapter 12 Spring Constant

  9. Section 1 Simple Harmonic Motion Chapter 12 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 12 Sample Problem, continued • 1.Define • Given: • m = 0.55 kg • x = –2.0 cm = –0.20 m • g = 9.81 m/s2 • Diagram: • Unknown: • k = ?

  11. Section 1 Simple Harmonic Motion Chapter 12 Sample Problem, continued 2. Plan Choose an equation or situation: When the mass is attached to the spring,the equilibrium position changes. At the new equilibrium position, the net force acting on the mass is zero. So the spring force (given by Hooke’s law) must be equal and opposite to the weight of the mass. Fnet = 0 = Felastic + Fg Felastic = –kx Fg = –mg –kx – mg = 0

  12. Section 1 Simple Harmonic Motion Chapter 12 Sample Problem, continued 2. Plan, continued Rearrange the equation to isolate the unknown:

  13. Section 1 Simple Harmonic Motion Chapter 12 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.

  14. Chapter 12 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

  15. Chapter 12 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

  16. Chapter 12 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

  17. Chapter 12 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

  18. Chapter 12 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

  19. Chapter 12 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

  20. Chapter 12 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

  21. Chapter 12 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

  22. Section 2 Measuring Simple Harmonic Motion Chapter 12 Objectives • Identifythe amplitude of vibration. • Recognizethe relationship between period and frequency. • Calculatethe period and frequency of an object vibrating with simple harmonic motion.

  23. Section 2 Simple Harmonic Motion Chapter 12 Simple Harmonic Motion • The motion of a vibrating mass-spring system is an example ofsimple 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. (It’s “there and back again”, a Hobbits Tale!)

  24. Section 2 Simple Harmonic Motion Chapter 12 Simple Harmonic Motion

  25. Section 2 Simple Harmonic Motion Chapter 12 Force and Energy in Simple Harmonic Motion

  26. Section 2 Simple Harmonic Motion Chapter 12 The Simple Pendulum • A simple pendulumconsists of amass called a bob, which is attached to a fixed string. • At any displacement from equilibrium, theweightof the bob(Fg)can be resolved into two components. • The xcomponent (Fg,x = Fg sin q)is theonly force acting on the bobin the direction of its motion and thus is therestoring force. The forces acting on the bob at any point are the force exerted by the string and the gravitational force.

  27. Section 2 Simple Harmonic Motion Chapter 12 The Simple Pendulum, continued • The magnitude of the restoring force (Fg,x = Fg sin q)is proportional to sin q. • When themaximum angle of displacement q is relatively small(<15°), sin qis approximately equal toqin radians. • As a result,the restoring force is very nearly proportional to the displacement. • Thus, the pendulum’s motion is an excellent approximation of simple harmonic motion.

  28. Section 2 Simple Harmonic Motion Chapter 12 Restoring Force and Simple Pendulums

  29. Section 2 Simple Harmonic Motion Chapter 12 Simple Harmonic Motion

  30. Section 2 Measuring Simple Harmonic Motion Chapter 12 Amplitude, Period, and Frequency in SHM • In SHM, the maximum displacement from equilibrium is defined as theamplitude of the vibration. • Apendulum’samplitude can be measured by the angle between the (distance from 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 theradian (rad)and themeter (m).

  31. Section 2 Measuring Simple Harmonic Motion Chapter 12 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). • Thefrequency(f)is the number of cycles or vibrations per unit of time. • The SI unit of frequency is hertz (Hz). • Hz = s–1

  32. Section 2 Measuring Simple Harmonic Motion Chapter 12 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.

  33. Section 2 Measuring Simple Harmonic Motion Chapter 12 Measures of Simple Harmonic Motion

  34. Section 2 Measuring Simple Harmonic Motion Chapter 12 Measures of Simple Harmonic Motion

  35. Section 2 Measuring Simple Harmonic Motion Chapter 12 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).

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

  37. Chapter 12 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.

  38. Chapter 12 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.

  39. Chapter 12 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

  40. Chapter 12 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

  41. Chapter 12 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

  42. Chapter 12 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

  43. Chapter 12 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.

  44. Chapter 12 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.

  45. Section 3 Properties of Waves Chapter 12 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.

  46. Section 3 Properties of Waves Chapter 12 Wave Motion • Awaveis the motion of a disturbance that transfers Energy. • Amediumis a physical environment (substance) 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.

  47. Section 3 Properties of Waves Chapter 12 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 (repetitive) 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.

  48. Section 3 Properties of Waves Chapter 12 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.

  49. Section 3 Properties of Waves Chapter 12 Wave Types “styles”, continued • Atransverse waveis a wave whose particles vibrate perpendicularlyto the direction of the wave motion. • Thecrestis thehighest point above the equilibrium position, and thetroughis thelowest point below the equilibrium position. • Thewavelength (l)is the distance between two adjacent similar points of a wave.

  50. Section 3 Properties of Waves Chapter 12 Transverse Waves

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