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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