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**CHAPTER 12 Vibrations and Waves**

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**Periodic Motion and Simple Harmonic Motion (SHM)**

CH 12 Periodic Motion and Simple Harmonic Motion (SHM) Periodic Motion is motion repeated in equal intervals of time. Periodic motion is performed, for example, by a rocking chair, a bouncing ball, a vibrating tuning fork, a swing in motion, the Earth in its orbit around the Sun, and a water wave.

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**Simple Harmonic Motion is a special case of periodic motion The requirements are:**

Repetitive movement back and forth along the same line through an equilibrium, or central position. The maximum displacement on one side of this position is equal to the maximum displacement on the other side. The time interval of each complete vibration is the same. The force responsible for the motion is always directed toward the equilibrium position and is directly proportional to the distance from it.

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In the examples given above, the rocking chair, the tuning fork, the swing, and the water wave execute simple harmonic motion, but the bouncing ball and the Earth in its orbit do not.

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**Section 1 Simple Harmonic Motion**

Chapter 12 Hooke’s Law One type of periodic motion is 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).

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**Chapter 12 Hooke’s Law, continued At equilibrium:**

Section 1 Simple Harmonic Motion Chapter 12 Hooke’s Law, continued At equilibrium: The spring force and the mass’s acceleration become zero. The speed reaches a maximum. At maximum displacement: The spring force and the mass’s acceleration reach a maximum. The speed becomes zero.

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**spring force = –(spring constant displacement)**

Section 1 Simple Harmonic Motion Chapter 12 Hooke’s Law, continued Measurements show that the spring force, or restoring force, is directly proportional to the displacement of the mass. This relationship is known as Hooke’s Law: Felastic = –kx spring force = –(spring constant displacement) The quantity k is a positive constant called the spring constant.

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**Chapter 12 Sample Problem Hooke’s Law**

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?

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**Sample Problem, continued**

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 = ?

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**Sample Problem, continued**

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

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**Sample Problem, continued**

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

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**Sample Problem, continued**

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.

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**Chapter 12 The Simple Pendulum**

Section 1 Simple Harmonic Motion Chapter 12 The Simple Pendulum A simple pendulum consists of a mass called a bob, which is attached to a fixed string. At any displacement from equilibrium, the weight of the bob (Fg) can be resolved into two components. The x component (Fg,x = Fg sin q) is the only force acting on the bob in the direction of its motion and thus is the restoring force. The forces acting on the bob at any point are the force exerted by the string and the gravitational force.

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**The Simple Pendulum, continued**

Section 1 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 the maximum angle of displacement q is relatively small (<15°), sin q is approximately equal to q in 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.

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**Simple Harmonic Motion**

Section 1 Simple Harmonic Motion Chapter 12 Simple Harmonic Motion

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**Chapter 12 Amplitude in SHM**

Section 2 Measuring Simple Harmonic Motion Chapter 12 Amplitude in SHM In SHM, the maximum displacement from equilibrium is defined as the amplitude of the vibration. A pendulum’s amplitude: Measured by the angle between the pendulum’s equilibrium position and its maximum displacement. A mass-spring system amplitude: Measured by the maximum amount the spring is stretched or compressed from its equilibrium position. The SI units of amplitude are the radian (rad) and the meter (m).

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**Period and Frequency in SHM**

Section 2 Measuring Simple Harmonic Motion Chapter 12 Period and Frequency in SHM The period (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 unit of time. The SI unit of frequency is hertz (Hz). Hz = s–1

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**Period and Frequency in SHM, continued**

Section 2 Measuring Simple Harmonic Motion Chapter 12 Period and Frequency in SHM, continued Period and frequency are inversely related: Thus, any time you have a value for period or frequency, you can calculate the other value.

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**Measures of Simple Harmonic Motion**

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

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**Calculating the Period(T) of a Simple Pendulum in SHM**

Section 2 Measuring Simple Harmonic Motion Chapter 12 Calculating the Period(T) of a Simple Pendulum in SHM The period (T) of a simple pendulum is calculated by: The period does not depend on the mass of the bob or on the amplitude (for small angles).

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**Calculating the Period(T) of a Mass-Spring System in SHM**

Section 2 Measuring Simple Harmonic Motion Chapter 12 Calculating the Period(T) of a Mass-Spring System in SHM The period of an ideal mass-spring system is calculated by: The period does not depend on the amplitude. This equation applies only for systems in which the spring obeys Hooke’s law.

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**Chapter 12 Wave Motion A wave is the motion of a disturbance.**

Section 3 Properties of Waves Chapter 12 Wave Motion A wave is the motion of a disturbance. Medium- A physical environment through which a disturbance can travel. For example, water is the medium for ripple waves in a pond. Mechanical Waves- Waves that require a medium through which to travel. Water waves and sound waves are mechanical waves. Electromagnetic waves such as visible light do not require a medium.

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**Chapter 12 Wave Types Section 3 Properties of Waves**

Pulse Wave -A wave that consists of a single traveling pulse. No oscillations Periodic 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. Sine Wave -A wave whose source vibrates with simple harmonic motion. Travelling Wave- A wave in which the medium moves in the direction of propagation of the wave. (Ocean Wave) Standing wave- A wave form which occurs in a limited, fixed medium in such a way that the reflected wave coincides with the produced wave. A common example is the vibration of the strings on a musical stringed instrument.

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**Chapter 12 Wave Types, continued Section 3 Properties of Waves**

Transverse Wave - A wave whose particles vibrate perpendicularly to the direction of the wave motion. The crest is the highest point above the equilibrium position, and the trough is the lowest point below the equilibrium position. The wavelength (l) is the distance between two adjacent similar points of a wave.

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**Chapter 12 Wave Types, continued Section 3 Properties of Waves**

A longitudinal 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. The crests correspond to compressed regions, and the troughs correspond to stretched regions. The crests are regions of high density and pressure (relative to the equilibrium density or pressure of the medium), and the troughs are regions of low density and pressure.

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**Period, Frequency, and Wave Speed**

Section 3 Properties of Waves Chapter 12 Period, Frequency, and Wave Speed The frequency of a wave describes the number of waves that pass a given point in a unit of time. f = 1/T The period of a wave describes the time it takes for a complete wavelength to pass a given point. T= 1/T

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**Period, Frequency, and Wave Speed, continued**

Section 3 Properties of Waves Chapter 12 Period, Frequency, and Wave Speed, continued The speed of a mechanical wave is constant for any given medium. The speed of a wave is given by the following equation: v = fl wave speed = frequency wavelength This equation applies to both mechanical and electromagnetic waves.

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**Waves and Energy Transfer**

Section 3 Properties of Waves Chapter 12 Waves and Energy Transfer Waves transfer energy by the vibration of matter. The rate at which a wave transfers energy depends on the amplitude. 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.

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**Chapter 12 Wave Interference**

Section 4 Wave Interactions Chapter 12 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 called superposition.

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**Wave Interference, continued**

Section 4 Wave Interactions Chapter 12 Wave Interference, continued In constructive interference, individual displacements on the same side of the equilibrium position are added together to form the resultant wave.

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**Wave Interference, continued**

Section 4 Wave Interactions Chapter 12 Wave Interference, continued In destructive interference, individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave.

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**Section 4 Wave Interactions**

Chapter 12 Reflection What happens to the motion of a wave when it reaches a boundary? At a free boundary, waves are reflected. At a fixed boundary, waves are reflected and inverted. Free boundary Fixed boundary

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Refraction- the change in direction of a propagating wave (light or sound) when passing from one medium to another. Diffraction- Diffraction refers to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings.

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**Chapter 12 Multiple Choice**

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

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**Chapter 12 Multiple Choice**

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

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**Multiple Choice, continued**

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

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**Multiple Choice, continued**

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

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**Multiple Choice, continued**

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

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**Multiple Choice, continued**

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

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**Multiple Choice, continued**

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

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**Multiple Choice, continued**

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

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**Multiple Choice, continued**

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.

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**Multiple Choice, continued**

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.

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**Multiple Choice, continued**

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

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**Multiple Choice, continued**

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

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**Multiple Choice, continued**

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

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**Multiple Choice, continued**

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

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**Multiple Choice, continued**

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.

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**Multiple Choice, continued**

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.

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**Multiple Choice, continued**

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. 9. If the pendulum completes exactly 12 cycles in 2.0 min, what is the frequency of the pendulum? A Hz B Hz C. 6.0 Hz D. 10 Hz

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**Multiple Choice, continued**

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. 9. If the pendulum completes exactly 12 cycles in 2.0 min, what is the frequency of the pendulum? A Hz B Hz C. 6.0 Hz D. 10 Hz

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**Multiple Choice, continued**

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. 10. If the pendulum’s length is 2.00 m and ag = 9.80 m/s2, how many complete oscillations does the pendulum make in 5.00 min? F H. 106 G J. 239

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**Multiple Choice, continued**

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. 10. If the pendulum’s length is 2.00 m and ag = 9.80 m/s2, how many complete oscillations does the pendulum make in 5.00 min? F H. 106 G J. 239

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued Base your answers to questions 12–13 on the graph. 12. What kind of wave does this graph represent? A. transverse wave C. electromagnetic wave B. longitudinal wave D. pulse wave

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued Base your answers to questions 12–13 on the graph. 11. What kind of wave does this graph represent? A. transverse wave C. electromagnetic wave B. longitudinal wave D. pulse wave

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued Base your answers to questions 11–13 on the graph. 12. Which letter on the graph represents wavelength? F. A H. C G. B J. D

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued Base your answers to questions 11–13 on the graph. 12. Which letter on the graph represents wavelength? F. A H. C G. B J. D

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued Base your answers to questions 11–13 on the graph. 13. Which letter on the graph is used for a trough? A. A C. C B. B D. D

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued Base your answers to questions 11–13 on the graph. 13. Which letter on the graph is used for a trough? A. A C. C B. B D. D

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued Base your answers to questions 14–15 on the passage. A wave with an amplitude of 0.75 m has the same wavelength as a second wave with an amplitude of 0.53 m. The two waves interfere. 14. What is the amplitude of the resultant wave if the interference is constructive? F m G m H m J m

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued Base your answers to questions 14–15 on the passage. A wave with an amplitude of 0.75 m has the same wavelength as a second wave with an amplitude of 0.53 m. The two waves interfere. 14. What is the amplitude of the resultant wave if the interference is constructive? F m G m H m J m

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued Base your answers to questions 14–15 on the passage. A wave with an amplitude of 0.75 m has the same wavelength as a second wave with an amplitude of 0.53 m. The two waves interfere. 15. What is the amplitude of the resultant wave if the interference is destructive? A m B m C m D m

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued Base your answers to questions 14–15 on the passage. A wave with an amplitude of 0.75 m has the same wavelength as a second wave with an amplitude of 0.53 m. The two waves interfere. 15. What is the amplitude of the resultant wave if the interference is destructive? A m B m C m D m

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued 16. Two successive crests of a transverse wave 1.20 m apart. Eight crests pass a given point 12.0 s. What is the wave speed? F m/s G m/s H m/s J m/s

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**Multiple Choice, continued**

Chapter 12 Standardized Test Prep Multiple Choice, continued 16. Two successive crests of a transverse wave 1.20 m apart. Eight crests pass a given point 12.0 s. What is the wave speed? F m/s G m/s H m/s J m/s

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**Chapter 12 Short Response**

Standardized Test Prep Short Response 17. Green light has a wavelength of 5.20 10–7 m and a speed in air of 3.00 108 m/s. Calculate the frequency and the period of the light.

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**Chapter 12 Short Response**

Standardized Test Prep Short Response 17. Green light has a wavelength of 5.20 10–7 m and a speed in air of 3.00 108 m/s. Calculate the frequency and the period of the light. Answer: 5.77 1014 Hz, 1.73 10–15 s

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**Short Response, continued**

Chapter 12 Standardized Test Prep Short Response, continued 18. What kind of wave does not need a medium through which to travel?

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**Short Response, continued**

Chapter 12 Standardized Test Prep Short Response, continued 18. What kind of wave does not need a medium through which to travel? Answer: electromagnetic waves

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**Short Response, continued**

Chapter 12 Standardized Test Prep Short Response, continued 19. List three wavelengths that could form standing waves on a 2.0 m string that is fixed at both ends.

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**Short Response, continued**

Chapter 12 Standardized Test Prep Short Response, continued 19. List three wavelengths that could form standing waves on a 2.0 m string that is fixed at both ends. Answer: Possible correct answers include 4.0 m, 2.0 m, 1.3 m, 1.0 m, or other wavelengths such that nl = 4.0 m (where n is a positive integer).

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**Chapter 12 Extended Response**

Standardized Test Prep Extended Response 20. A visitor to a lighthouse wishes to find out the height of the tower. The visitor ties a spool of thread to a small rock to make a simple pendulum. Then, the visitor hangs the pendulum down a spiral staircase in the center of the tower. The period of oscillation is 9.49 s. What is the height of the tower? Show all of your work.

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**Chapter 12 Extended Response**

Standardized Test Prep Extended Response 20. A visitor to a lighthouse wishes to find out the height of the tower. The visitor ties a spool of thread to a small rock to make a simple pendulum. Then, the visitor hangs the pendulum down a spiral staircase in the center of the tower. The period of oscillation is 9.49 s. What is the height of the tower? Show all of your work. Answer: 22.4 m

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**Extended Response, continued**

Chapter 12 Standardized Test Prep Extended Response, continued 21. A harmonic wave is traveling along a rope. The oscillator that generates the wave completes 40.0 vibrations in 30.0 s. A given crest of the wave travels 425 cm along the rope in a period of 10.0 s. What is the wavelength? Show all of your work.

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**Extended Response, continued**

Chapter 12 Standardized Test Prep Extended Response, continued 21. A harmonic wave is traveling along a rope. The oscillator that generates the wave completes 40.0 vibrations in 30.0 s. A given crest of the wave travels 425 cm along the rope in a period of 10.0 s. What is the wavelength? Show all of your work. Answer: m

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**Section 1 Simple Harmonic Motion**

Chapter 12 Hooke’s Law

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**Section 3 Properties of Waves**

Chapter 12 Transverse Waves

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**Section 3 Properties of Waves**

Chapter 12 Longitudinal Waves

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**Constructive Interference**

Section 4 Wave Interactions Chapter 12 Constructive Interference

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**Destructive Interference**

Section 4 Wave Interactions Chapter 12 Destructive Interference

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**Reflection of a Pulse Wave**

Section 4 Wave Interactions Chapter 12 Reflection of a Pulse Wave

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Objectives Identify the conditions of simple harmonic motion.

Objectives Identify the conditions of simple harmonic motion.

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