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Vibrations and WavesChapter 12

CHAPTER 12

Vibrations and Waves



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.

CH 12



• 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.



• 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.



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).

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.




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.




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?




Chapter 12

Sample Problem, continued


Unknown:

k = ?

1. DefineGiven: m = 0.55 kg x = –2.0 cm = –0.20 m g = 9.81 m/s2

Diagram:



Chapter 12



2. PlanChoose 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



Chapter 12



2. Plan, continued Rearrange the equation to isolate the unknown:

kx mg 0

kx mg

k mg

x



Chapter 12



3. Calculate Substitute the values into the equation and

solve:

k mg

x

(0.55 kg)(9.81 m/s2 )

–0.020 m

k 270 N/m

4. Evaluate The value of k implies that 270 N of force is

required to displace the spring 1 m.



Chapter 12

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 gravitational force.

• At any displacement from equilibrium, the weight of the bob (Fg) can be resolved into two components.

• The x component (Fg,x = Fg sin ) is the only force acting on the bob in the direction of its motion and thus is the restoring force.



Chapter 12

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.



Chapter 12

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).

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




Chapter 12

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.



Chapter 12

Measures of 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:


T 2L

ag

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

period 2length

free-fall acceleration



Chapter 12

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

• The period of an ideal mass-spring system is calculated by:


T 2m

k

• The period does not depend on the amplitude.• This equation applies only for systems in which the

spring obeys Hooke’s law.

period 2mass

spring constant



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.

Section 3 Properties of Waves



Chapter 12

Wave Types

• 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.




Chapter 12

Wave Types, continued

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




Chapter 12

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.




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




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 = fwave speed = frequency wavelength

• This equation applies to both mechanical and electromagnetic 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.




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.

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.




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.




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



• 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.



Multiple Choice

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

Standardized Test PrepChapter 12

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 leftB. to the rightC. to the left or to the right depending on whether the

spring is stretched or compressedD. perpendicular to the motion of the mass



Multiple Choice, continued




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/mG. –2.0 101 N/m J. 2.0 101 N/m







3. In what form is the energy in the system when the mass passes through the equilibrium point?

A. elastic potential energyB. gravitational potential energyC. kinetic energyD. a combination of two or more of the above







4. In what form is the energy in the system when the mass is at maximum displacement?

F. elastic potential energyG. gravitational potential energyH. kinetic energyJ. a combination of two or more of the above







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

A. mass B. spring constant C. amplitude of vibrationD. All of the above affect the period.







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






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 bobC. the component of the bob’s weight perpendicular to the

motion of the bobD. the elastic force of the stretched string







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 bobH. the free-fall acceleration at the pendulum’s locationJ. All of the above affect the period.







9. If the pendulum completes exactly 12cycles in 2.0 min, what is the frequency ofthe pendulum?

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







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. 106G. 21.6 J. 239




Base your answers to questions 12–13 on the graph.


12. What kind of wave does this graph represent?

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






11. What kind of wave does this graph represent?

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






12. Which letter on the graph represents wavelength?

F. A H. CG. B J. D






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

A. A C. CB. B D. D




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. 0.22 m G. 0.53 m H. 0.75 mJ. 1.28 m




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. 0.22 m B. 0.53 m C. 0.75 mD. 1.28 m





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/sG. 0.800 m/sH. 1.80 m/sJ. 9.60 m/s



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.



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



Short Response, continued


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





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

Answer: electromagnetic waves





19.List three wavelengths that could form standing waves on a 2.0 m string that is fixed at both ends.





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).



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.



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



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.



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: 0.319 m



Chapter 12

Hooke’s Law




Chapter 12

Transverse Waves




Chapter 12

Longitudinal Waves




Chapter 12

Constructive Interference




Chapter 12

Destructive Interference




Chapter 12

Reflection of a Pulse Wave


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