Vibrations and WavesVibrations and Waves

Chapter 12Chapter 12

Periodic MotionPeriodic Motion

A repeated motion is called A repeated motion is called periodic periodic motionmotion

What are some examples of periodic What are some examples of periodic motion?motion? The motion of Earth orbiting the sunThe motion of Earth orbiting the sun A child swinging on a swingA child swinging on a swing Pendulum of a grandfather clockPendulum of a grandfather clock

Simple Harmonic MotionSimple Harmonic Motion

Simple harmonic motion is a form of periodic Simple harmonic motion is a form of periodic motionmotion

The conditions for simple harmonic motion are The conditions for simple harmonic motion are as follows:as follows: The object oscillates about an equilibrium positionThe object oscillates about an equilibrium position

The motion involves a restoring force that is The motion involves a restoring force that is proportional to the displacement from equilibriumproportional to the displacement from equilibrium

The motion is back and forth over the same pathThe motion is back and forth over the same path

Earth’s OrbitEarth’s Orbit

Is the motion of the Earth orbiting the sun Is the motion of the Earth orbiting the sun simple harmonic?simple harmonic? NONO Why not?Why not? The Earth does not orbit about an equilibrium The Earth does not orbit about an equilibrium

positionposition

p. 438 of your bookp. 438 of your book The spring is stretched away from the The spring is stretched away from the

equilibrium positionequilibrium position

Since the spring is being stretched toward the Since the spring is being stretched toward the right, the spring’s restoring force pulls to the left right, the spring’s restoring force pulls to the left so the acceleration is also to the leftso the acceleration is also to the left

p. 438 of your bookp. 438 of your book

When the spring is unstretched the force When the spring is unstretched the force and acceleration are zero, but the velocity and acceleration are zero, but the velocity is maximumis maximum

p.438 of your bookp.438 of your book

The spring is stretched away from the The spring is stretched away from the equilibrium positionequilibrium position

Since the spring is being stretched toward Since the spring is being stretched toward the left, the spring’s restoring force pulls to the left, the spring’s restoring force pulls to the right so the acceleration is also to the the right so the acceleration is also to the right right

DampingDamping

In the real world, friction eventually causes In the real world, friction eventually causes the mass-spring system to stop movingthe mass-spring system to stop moving

This effect is called This effect is called dampingdamping

Mass-Spring DemoMass-Spring Demo

http://phet.colorado.edu/simulations/http://phet.colorado.edu/simulations/sims.php?sim=Masses_and_Springssims.php?sim=Masses_and_Springs

I suggest you play around with this I suggest you play around with this demo…it might be really helpful!demo…it might be really helpful!

Hooke’s LawHooke’s Law

The spring force always pushes or pulls The spring force always pushes or pulls the mass back toward its original the mass back toward its original equilibrium positionequilibrium position

Measurements show that the restoring Measurements show that the restoring force is directly proportional to the force is directly proportional to the displacement of the massdisplacement of the mass

Hooke’s LawHooke’s Law

FFelasticelastic= Spring force= Spring force k is the spring constantk is the spring constant x is the displacement from equilibriumx is the displacement from equilibrium

The negative sign shows that the direction of F The negative sign shows that the direction of F is always opposite the mass’ displacementis always opposite the mass’ displacement

kxFelastic

FlashbackFlashback

Anybody remember where we’ve seen the Anybody remember where we’ve seen the spring constant (k) before?spring constant (k) before?

PEPEelasticelastic = ½kx = ½kx22

A stretched or compressed spring has A stretched or compressed spring has elastic potential energy!!elastic potential energy!!

Spring ConstantSpring Constant

The value of the spring constant is a The value of the spring constant is a measure of the stiffness of the springmeasure of the stiffness of the spring

The bigger k is, the greater force needed The bigger k is, the greater force needed to stretch or compress the springto stretch or compress the spring

The units of k are N/m (Newtons/meter)The units of k are N/m (Newtons/meter)

Sample Problem p.441 #2Sample Problem p.441 #2

A load of 45 N attached to a spring that is A load of 45 N attached to a spring that is hanging vertically stretches the spring 0.14 hanging vertically stretches the spring 0.14 m. What is the spring constant?m. What is the spring constant?

Solving the ProblemSolving the Problem

Why do I make x Why do I make x negative?negative?

Because the Because the displacement is displacement is downdown

kxFelastic

m

N

m

N

x

Fk 321

14.0

45

Follow Up QuestionFollow Up Question

What is the elastic potential energy stored What is the elastic potential energy stored in the spring when it is stretched 0.14 m?in the spring when it is stretched 0.14 m?

Jmm

NkxPEelastic 15.314.043.321

2

1

2

1 22

The simple pendulumThe simple pendulum

The simple pendulum is a mass attached The simple pendulum is a mass attached to a stringto a string

The motion is simple harmonic The motion is simple harmonic

because the restoring force is proportional because the restoring force is proportional to the displacement and because the to the displacement and because the mass oscillates about an equilibrium mass oscillates about an equilibrium positionposition

Simple PendulumSimple Pendulum

The restoring force is a component of the The restoring force is a component of the mass’ weightmass’ weight

As the displacement increases, the As the displacement increases, the gravitational potential energy increasesgravitational potential energy increases

Simple Pendulum ActivitySimple Pendulum Activity

http://phet.colorado.edu/simulations/http://phet.colorado.edu/simulations/sims.php?sim=Pendulum_Labsims.php?sim=Pendulum_Lab

You should also play around with this You should also play around with this activity to help your understandingactivity to help your understanding

Comparison between pendulum Comparison between pendulum and mass-spring system (p. 445)and mass-spring system (p. 445)

Measuring Simple Harmonic Motion Measuring Simple Harmonic Motion (p. 447)(p. 447)

Amplitude of SHMAmplitude of SHM

Amplitude is the maximum displacement Amplitude is the maximum displacement from equilibriumfrom equilibrium

The more energy the system has, the The more energy the system has, the higher the amplitude will behigher the amplitude will be

Period of a pendulumPeriod of a pendulum

T = periodT = period

L= length of stringL= length of string

g= 9.81 m/sg= 9.81 m/s22

g

LT 2

Period of the PendulumPeriod of the Pendulum

The period of a pendulum only depends The period of a pendulum only depends on the length of the string and the on the length of the string and the acceleration due to gravityacceleration due to gravity

In other words, changing the mass of the In other words, changing the mass of the pendulum has no effect on its period!!pendulum has no effect on its period!!

Sample Problem p. 449 #2Sample Problem p. 449 #2

You are designing a pendulum clock to You are designing a pendulum clock to have a period of 1.0 s. How long should have a period of 1.0 s. How long should the pendulum be?the pendulum be?

Solving the ProblemSolving the Problem

2

22

4*

2 gT

gT

L

g

LT 2

m

sm

gTg

TL 25.

)4(

81.91

4*

2 2

2

2

2

22

Period of a mass-spring systemPeriod of a mass-spring system

T= periodT= period

m= mass m= mass

k = spring constantk = spring constant

k

mT 2

Sample Problem p. 451 #2Sample Problem p. 451 #2

When a mass of 25 g is attached to a When a mass of 25 g is attached to a certain spring, it makes 20 complete certain spring, it makes 20 complete vibrations in 4.0 s. What is the spring vibrations in 4.0 s. What is the spring constant of the spring?constant of the spring?

What information do we have?What information do we have?

M= .025 kgM= .025 kg

The mass makes 20 complete vibrations in The mass makes 20 complete vibrations in 4.0s4.0s That means it makes 5 vibrations per secondThat means it makes 5 vibrations per second So f= 5 Hz So f= 5 Hz T= 1/5 = 0.2 secondsT= 1/5 = 0.2 seconds

Solve the problemSolve the problem

k

mT 2

m

Nkgm

Tm

Tk 7.24025.

20.0

4422

2

2

22

Day 2: Properties of WavesDay 2: Properties of Waves A A wavewave is the motion of a disturbance is the motion of a disturbance

Waves transfer energy by transferring the motion of Waves transfer energy by transferring the motion of matter instead of transferring matter itselfmatter instead of transferring matter itself

A A mediummedium is the material through which a is the material through which a disturbance travelsdisturbance travels What are some examples of mediums?What are some examples of mediums? WaterWater Air Air

Two kinds of WavesTwo kinds of Waves

Mechanical WavesMechanical Waves require a material require a material mediummedium i.e. Sound wavesi.e. Sound waves

Electromagnetic WavesElectromagnetic Waves do not require a do not require a material mediummaterial medium i.e. x-rays, gamma rays, etci.e. x-rays, gamma rays, etc

Pulse Wave vs Periodic WavePulse Wave vs Periodic Wave

A A pulse wavepulse wave is a single, non periodic is a single, non periodic disturbancedisturbance

A A periodic waveperiodic wave is produced by periodic is produced by periodic motionmotion Together, single pulses form a periodic waveTogether, single pulses form a periodic wave

Transverse WavesTransverse Waves

Transverse WaveTransverse Wave: The particles move : The particles move perpendicular to the wave’s motionperpendicular to the wave’s motion

Wave moves inX direction

Particles move iny direction

Longitudinal (Compressional) WaveLongitudinal (Compressional) Wave

Longitudinal (Compressional) Waves: Longitudinal (Compressional) Waves: Particles move in same direction as wave Particles move in same direction as wave motion (Like a Slinky)motion (Like a Slinky)

Longitudinal (Compressional) WaveLongitudinal (Compressional) Wave

Troughs: Areas of Low Density becauseThe coils are stretched

Crests: Regions of High Density becauseThe coils are compressed

Wave SpeedWave Speed

The speed of a wave The speed of a wave is the product of its is the product of its frequency times its frequency times its wavelengthwavelength

f is frequency (Hz)f is frequency (Hz)

λλ (lambda) (lambda) Is Is wavelength (m)wavelength (m)

fv

Sample Problem p.457 #4Sample Problem p.457 #4

A tuning fork produces a sound with a A tuning fork produces a sound with a frequency of 256 Hz and a wavelength in frequency of 256 Hz and a wavelength in air of 1.35 mair of 1.35 m a. What value does this give for the speed of a. What value does this give for the speed of

sound in air?sound in air?

b. What would be the wavelength of the wave b. What would be the wavelength of the wave produced b this tuning fork in water in which produced b this tuning fork in water in which sound travels at 1500 m/s?sound travels at 1500 m/s?

Part aPart a

Given:Given: f = 256 Hzf = 256 Hz λλ = 1.35 m = 1.35 m v = ?v = ?

s

mmHzfv 6.345)35.1)(256(

Part bPart b

Given:Given: f = 256 Hzf = 256 Hz v =1500 m/sv =1500 m/s λλ = ? = ?

mHzsm

f

v86.5

256

1500

Wave InterferenceWave Interference

Since waves are not matter, they can Since waves are not matter, they can occupy the same space at the same timeoccupy the same space at the same time

The combination of two overlapping waves The combination of two overlapping waves is called is called superpositionsuperposition

The Superposition PrincipleThe Superposition Principle

The superposition principle: When two or The superposition principle: When two or more waves occupy the same space at the more waves occupy the same space at the same time, the resultant wave is the vector same time, the resultant wave is the vector sum of the individual wavessum of the individual waves

Constructive Interference (p.460)Constructive Interference (p.460)

When two waves are traveling in the same When two waves are traveling in the same direction, direction, constructive interference constructive interference occurs and the resultant wave is larger occurs and the resultant wave is larger than the original wavesthan the original waves

Destructive InterferenceDestructive Interference

When two waves are traveling on opposite When two waves are traveling on opposite sides of equilibrium, sides of equilibrium, destructive destructive interferenceinterference occurs and the resultant occurs and the resultant wave is smaller than the two original wave is smaller than the two original waveswaves

ReflectionReflection

When the motion of a wave reaches a When the motion of a wave reaches a boundary, its motion is changedboundary, its motion is changed

There are two types of boundariesThere are two types of boundaries Fixed BoundaryFixed Boundary Free BoundaryFree Boundary

Free BoundariesFree Boundaries

A free boundary is A free boundary is able to move with the able to move with the wave’s motion wave’s motion

At a free boundary, At a free boundary, the wave is reflectedthe wave is reflected

Fixed BoundariesFixed Boundaries

A fixed boundary A fixed boundary does not move with does not move with the wave’s motion the wave’s motion (pp. 462 for more (pp. 462 for more explanation)explanation)

Consequently, the Consequently, the wave is reflected and wave is reflected and invertedinverted

Standing WavesStanding Waves

When two waves with the same properties When two waves with the same properties (amplitude, frequency, etc) travel in (amplitude, frequency, etc) travel in opposite directions and interfere, they opposite directions and interfere, they create a create a standing wavestanding wave..

Standing WavesStanding Waves

N NA

A

ANNN

NNNNA

A

A

Standing waves have nodes and Standing waves have nodes and antinodesantinodes

Nodes: The points where the two Nodes: The points where the two waves cancelwaves cancel

Antinodes: The places where the Antinodes: The places where the largest amplitude occurslargest amplitude occurs

There is always one more node There is always one more node than antinodethan antinode

Sample Problem p.465 #2Sample Problem p.465 #2

A string is rigidly attached to a post at one A string is rigidly attached to a post at one end. Several pulses of amplitude 0.15 m end. Several pulses of amplitude 0.15 m sent down the string are reflected at the sent down the string are reflected at the post and travel back down the string post and travel back down the string without a loss of amplitude. What is the without a loss of amplitude. What is the amplitude at a point on the string where amplitude at a point on the string where the maximum displacement points of two the maximum displacement points of two pulses cross? What type of interference is pulses cross? What type of interference is this?this?

Solving the ProblemSolving the Problem

What type of boundary is involved here?What type of boundary is involved here? FixedFixed So that means the pulse will be reflected and So that means the pulse will be reflected and

invertedinverted

What happens when two pulses meet and What happens when two pulses meet and one is inverted?one is inverted? Destructive interference Destructive interference The resultant amplitude is 0.0 mThe resultant amplitude is 0.0 m

Helpful SimulationsHelpful Simulations Mass-Spring system: Mass-Spring system:

http://phet.colorado.edu/simulations/sims.php?http://phet.colorado.edu/simulations/sims.php?sim=Masses_and_Springssim=Masses_and_Springs

Pendulum: Pendulum: http://phet.colorado.edu/simulations/sims.php?http://phet.colorado.edu/simulations/sims.php?sim=Pendulum_Labsim=Pendulum_Lab

Wave on a string system: Wave on a string system: http://phet.colorado.edu/simulations/sims.php?http://phet.colorado.edu/simulations/sims.php?sim=Wave_on_a_Stringsim=Wave_on_a_String

http://www.walter-fendt.de/ph14e/stwaverefl.htmhttp://www.walter-fendt.de/ph14e/stwaverefl.htm