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Chapter Presentation

Transparencies Sample Problems

Visual Concepts

Standardized Test Prep

Resources

(3)

Vibrations and Waves

Chapter 11 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 11 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.

(5)

Chapter 11 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

(F_elastic) is always

opposite the direction of the mass’s displacement from equilibrium (x = 0).

(6)

Chapter 11 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.

(7)

Chapter 11 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:

F_elastic = –kx

spring force = –(spring constant  displacement)

• The quantity k is a positive constant called the spring

constant.

(8)

Chapter 11 Spring Constant

(9)

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)

Chapter 11 Sample Problem, continued

Unknown:

k = ?

1. Define Given:

m = 0.55 kg

x = –2.0 cm = –0.20 m g = 9.81 m/s

²

Diagram:

(11)

Chapter 11 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.

F

_net

= 0 = F

_elastic

+ F

_g

F

_elastic

= –kx

F

_g

= –mg

(12)

Chapter 11 Sample Problem, continued

2. Plan, continued

Rearrange the equation to isolate the unknown:

kx  mg  0 kx  mg k   mg

x

(13)

Chapter 11 Sample Problem, continued

3. Calculate

Substitute the values into the equation and solve:

k   mg

x   (0.55 kg)(9.81 m/s

²

) –0.020 m

k  270 N/m 4. Evaluate

The value of k implies that 270 N of force is

(14)

Chapter 11 Simple Harmonic Motion

• The motion of a vibrating mass-spring system is an example of simple harmonic motion.

• Simple harmonic motion describes any periodic motion that is the result of a restoring force that is proportional to displacement.

• Because simple harmonic motion involves a restoring force, every simple harmonic motion is a back-

and-forth motion over the same path.

(15)

Chapter 11 Simple Harmonic Motion

(16)

Chapter 11 Force and Energy in Simple Harmonic Motion

(17)

Chapter 11 The Simple Pendulum

• A simple pendulum consists of a mass 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

• At any displacement from

equilibrium, the weight of the

bob (F

_g

) can be resolved into two components.

• The x component (F

_g,x

= F

_g

sin

 ) is the only force acting on the

bob in the direction of its motion

(18)

Chapter 11 The Simple Pendulum, continued

• The magnitude of the restoring force

(Fg,x = Fg sin



) is proportional to sin



.

• When the maximum angle of

displacement  is relatively small

(<15°), sin 

is approximately equal to

 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.

(19)

Chapter 11 Restoring Force and Simple Pendulums

(20)

Chapter 11 Simple Harmonic Motion

(21)

Chapter 11 Objectives

• Identify the amplitude of vibration.

• Recognize the relationship between period and frequency.

• Calculate the period and frequency of an object vibrating with simple harmonic motion.

Section 2 Measuring Simple Harmonic Motion

(22)

Chapter 11 Amplitude, Period, and Frequency in SHM

• In SHM, the maximum displacement from equilibrium is defined as the amplitude of the vibration.

– A pendulum’s amplitude can be measured by the angle between the pendulum’s equilibrium position and its

maximum displacement.

– For a mass-spring system, the amplitude is 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).

(23)

Chapter 11 Amplitude, 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

(24)

Chapter 11 Amplitude, Period, and Frequency in SHM, continued

• Period and frequency are inversely related:

f  1

T or T  1 f

• Thus, any time you have a value for period or

frequency, you can calculate the other value.

(25)

Chapter 11 Measures of Simple Harmonic Motion

(26)

Chapter 11 Measures of Simple Harmonic Motion

(27)

Chapter 11 Period of a Simple Pendulum in SHM

• The period of a simple pendulum depends on the length and on the free-fall acceleration.

T  2  ^L a

_g

• The period does not depend on the mass of the bob or on the amplitude (for small angles).

period  2  ^length

free-fall acceleration

(28)

Chapter 11 Period of a Mass-Spring System in SHM

• The period of an ideal mass-spring system

depends on the mass and on the spring constant.

T  2  ^m k

• The period does not depend on the amplitude.

• This equation applies only for systems in which the spring obeys Hooke’s law.

period  2  ^mass

spring constant

(29)

Chapter 11 Objectives

• Distinguish local particle vibrations from overall wave

motion.

• Differentiate between pulse waves and periodic waves.

• Interpret waveforms of transverse and longitudinal

waves.

• Apply the relationship among wave speed, frequency,

and wavelength to solve problems.

• Relate energy and amplitude.

Section 3 Properties of Waves

(30)

Chapter 11 Wave Motion

• A wave is the motion of a disturbance.

• A medium is 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 called mechanical waves. Water waves and

sound waves are mechanical waves.

• Electromagnetic waves such as visible light do not require a medium.

(31)

Chapter 11 Wave Types

• A wave that consists of a single traveling pulse is called a pulse 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, a periodic wave is produced.

• A wave whose source vibrates with simple harmonic motion is called a sine wave. Thus, a sine wave is a special case of a periodic wave in which the periodic motion is simple harmonic.

(32)

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.

(33)

Chapter 11 Wave Types, continued

• A transverse wave is 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 () is the distance between two adjacent similar points of a wave.

(34)

Chapter 11 Transverse Waves

(35)

Chapter 11 Wave Types, continued

• 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.

(36)

Chapter 11 Longitudinal Waves

(37)

Chapter 11 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.

• The period of a wave describes 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 is inversely related to its frequency.

(38)

Chapter 11 Characteristics of a Wave

(39)

Chapter 11 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 = f



wave speed = frequency  wavelength

• This equation applies to both mechanical and electromagnetic waves.

(40)

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 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.

(41)

Chapter 11 Objectives

• Apply the superposition principle.

• Differentiate between constructive and destructive interference.

• Predict when a reflected wave will be inverted.

• Predict whether specific traveling waves will produce a standing wave.

• Identify nodes and antinodes of a standing wave.

Section 4 Wave Interactions

(42)

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 called superposition.

(43)

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.

(44)

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.

(45)

Chapter 11 Comparing Constructive and Destructive Interference

(46)

Chapter 11 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

(47)

Chapter 11 Standing Waves

(48)

Chapter 11 Standing Waves

• A standing wave is 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 node is a point in a standing wave that maintains zero displacement.

– An antinode is a point in a standing wave, halfway between two nodes, at which the largest

displacement occurs.

(49)

Chapter 11 Standing Waves, continued

• Only certain wavelengths produce standing wave patterns.

• The ends of the string must be

nodes because 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 include 2L (b), L (c), and 2/3L (d).

(50)

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.

(51)

Multiple Choice

Base your answers to questions 1–6 on the information below.

Chapter 11

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

(52)

Multiple Choice

Chapter 11

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

(53)

Multiple Choice, continued

Chapter 11

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

(54)

Multiple Choice, continued

Chapter 11

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

(55)

Multiple Choice, continued

Chapter 11

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

(56)

Multiple Choice, continued

Chapter 11

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

(57)

Multiple Choice, continued

Chapter 11

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

(58)

Multiple Choice, continued

Chapter 11

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

(59)

Multiple Choice, continued

Chapter 11

5. Which of the following does not affect the period of the mass-spring system?

A. mass

B. spring constant

C. amplitude of vibration

(60)

Multiple Choice, continued

Chapter 11

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.

(61)

Multiple Choice, continued

Chapter 11

6. If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation?

F. 8



s H.



s G. 4



s J.



 s

(62)

Multiple Choice, continued

Chapter 11

6. If the mass is 48 kg and the spring constant is 12 N/m, what is the period of the oscillation?

F. 8



s H.



s G. 4 s J.



 s

(63)

Multiple Choice, continued

Chapter 11

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

(64)

Multiple Choice, continued

Chapter 11

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

(65)

Multiple Choice, continued

Chapter 11

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.

(66)

Multiple Choice, continued

Chapter 11

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.

(67)

Multiple Choice, continued

Chapter 11

9. If the pendulum completes exactly 12

cycles in 2.0 min, what is the frequency of the pendulum?

A. 0.10 Hz B. 0.17 Hz C. 6.0 Hz D. 10 Hz

(68)

Multiple Choice, continued

Chapter 11

9. If the pendulum completes exactly 12

cycles in 2.0 min, what is the frequency of the pendulum?

A. 0.10 Hz B. 0.17 Hz C. 6.0 Hz D. 10 Hz

(69)

Multiple Choice, continued

Chapter 11

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. 1.76 H. 106

G. 21.6 J. 239

(70)

Multiple Choice, continued

Chapter 11

10. If the pendulum’s length is 2.00 m and a_g = 9.80 m/s², how many complete

oscillations does the pendulum make in 5.00 min?

F. 1.76 H. 106 G. 21.6 J. 239

(71)

Multiple Choice, continued

Base your answers to questions 11–13 on the graph.

Chapter 11

11. What kind of wave does this graph represent?

A. transverse wave C. electromagnetic wave B. longitudinal wave D. pulse wave

(72)

Multiple Choice, continued

Chapter 11

11. What kind of wave does this graph represent?

A. transverse wave C. electromagnetic wave B. longitudinal wave D. pulse wave

(73)

Multiple Choice, continued

Chapter 11

12. Which letter on the graph represents wavelength?

F. A H. C

G. B J. D

(74)

Multiple Choice, continued

Chapter 11

12. Which letter on the graph represents wavelength?

F. A H. C

G. B J. D

(75)

Multiple Choice, continued

Chapter 11

13. Which letter on the graph is used for a trough?

A. A C. C

B. B D. D

(76)

Multiple Choice, continued

Chapter 11

13. Which letter on the graph is used for a trough?

A. A C. C

B. B D. D

(77)

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.

Chapter 11

14. What is the amplitude of the resultant wave if the interference is constructive?

F. 0.22 m G. 0.53 m H. 0.75 m J. 1.28 m

(78)

Multiple Choice, continued

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.

Chapter 11

14. What is the amplitude of the resultant wave if the interference is constructive?

F. 0.22 m G. 0.53 m H. 0.75 m J. 1.28 m

(79)

Multiple Choice, continued

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.

Chapter 11

15. What is the amplitude of the resultant wave if the interference is destructive?

A. 0.22 m B. 0.53 m C. 0.75 m D. 1.28 m

(80)

Multiple Choice, continued

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.

Chapter 11

15. What is the amplitude of the resultant wave if the interference is destructive?

A. 0.22 m B. 0.53 m C. 0.75 m D. 1.28 m

(81)

Multiple Choice, continued

Chapter 11

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. 0.667 m/s G. 0.800 m/s H. 1.80 m/s J. 9.60 m/s

(82)

Multiple Choice, continued

Chapter 11

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. 0.667 m/s G. 0.800 m/s H. 1.80 m/s J. 9.60 m/s

(83)

Short Response

Chapter 11 17. Green light has a wavelength of 5.20  10

^–7

m and a speed in air of 3.00  10

⁸

m/s. Calculate the

frequency and the period of the light.

(84)

Short Response

Chapter 11 17. Green light has a wavelength of 5.20  10

^–7

m and a speed in air of 3.00  10

⁸

m/s. Calculate the

frequency and the period of the light.

Answer: 5.77  10

¹⁴

Hz, 1.73  10

^–15

s

(85)

Short Response, continued

Chapter 11 18. What kind of wave does not need a medium through

which to travel?

(86)

Short Response, continued

Chapter 11 18. What kind of wave does not need a medium through which to travel?

Answer: electromagnetic waves

(87)

Short Response, continued

Chapter 11 19. List three wavelengths that could form standing

waves on a 2.0 m string that is fixed at both ends.

(88)

Short Response, continued

Chapter 11 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 n  =

4.0 m (where n is a positive integer).

(89)

Extended Response

Chapter 11 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.

(90)

Extended Response

Chapter 11 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

(91)

Extended Response, continued

Chapter 11 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.

(92)

Extended Response, continued

Chapter 11 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: 0.319 m

(93)

Chapter 11 Hooke’s Law

(94)

Chapter 11 Transverse Waves

(95)

Chapter 11 Longitudinal Waves

(96)

Chapter 11 Constructive Interference

(97)

Chapter 11 Destructive Interference

(98)

How to Use This Presentation

How to Use This Presentation

Resources

Chapter 11

Table of Contents

Section 1 Simple Harmonic Motion

Section 2 Measuring Simple Harmonic Motion Section 3 Properties of Waves

Section 4 Wave Interactions

Chapter 11

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.

Chapter 11

Hooke’s Law

Chapter 11

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.

Chapter 11

Hooke’s Law, continued

• Measurements show that the spring force, or

• This relationship is known as Hooke’s Law:

• The quantity k is a positive constant called the spring

Chapter 11

Spring Constant

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?

Chapter 11

Sample Problem, continued

1. Define Given:

m = 0.55 kg

x = –2.0 cm = –0.20 m g = 9.81 m/s

Diagram:

Chapter 11

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.

F

= 0 = F

+ F

F

= –kx

F

= –mg

Chapter 11

Sample Problem, continued

2. Plan, continued

Rearrange the equation to isolate the unknown:

kx  mg  0 kx  mg k   mg

x

Chapter 11

Sample Problem, continued

3. Calculate

Substitute the values into the equation and solve:

k   mg

x   (0.55 kg)(9.81 m/s

) –0.020 m

k  270 N/m 4. Evaluate

The value of k implies that 270 N of force is

Chapter 11

Simple Harmonic Motion

• The motion of a vibrating mass-spring system is an example of simple harmonic motion.

• Simple harmonic motion describes any periodic motion that is the result of a restoring force that is proportional to displacement.

• Because simple harmonic motion involves a restoring force, every simple harmonic motion is a back-

and-forth motion over the same path.

Chapter 11

Simple Harmonic Motion

Chapter 11

Force and Energy in Simple Harmonic Motion

Chapter 11

T  2  ^L a

period  2  ^length

T  2  ^m k

period  2  ^mass