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Chapter 12
Oscillatory Motion
Periodic motion is motion of an object that regularly repeats
If the force is always directed toward the equilibrium position, the motion is called simple harmonic motion
Restoring Force and Hooke’s Law
The block is displaced to the right of x = 0 The position is
positive The restoring force
is directed to the left
Hooke’s Law states Fs = - k x
2
2netd xF ma m kxdt
= ⇒ = −
A solution is x(t) = A cos (ωt + φ)or x(t) = A sin (ωt + φ)
where kwm
=
cos(30 90 ) cos(120 ) sin 30cos(30 ) cos(90 60 ) sin 60
o o o o
o o o o
+ = = −
= − =
Simple Harmonic Motion –Graphical Representation A solution is x(t) = A cos (ωt + φ) A, ω, φ are all
constants A cosine or sine
curves can be used to give physical significance to these constants
Period and Frequency The frequency and period equations
can be rewritten to solve for ω
The period and frequency can also be expressed as:
Motion Equations for Simple Harmonic Motion
max2
max
v Aa A
ω
ω
=
=
SHM Example 1: cosine
Initial conditions at t = 0 are x (0)= A v (0) = 0
This means φ = 0
SHM Example 2: sine Initial conditions at
t = 0 are x (0)=0 v (0) = vi
This means φ = − π/2
2 2 2 21 1 1 12 2 2 2total o oE mv kx mv kx= + = +
Conservation of Total Energy
Chapter 13
Mechanical Waves
Terminology: Amplitude and Wavelength The crest of the wave:
This distance is called the amplitude, A The point at the negative
amplitude is called the trough
The wavelength: λThe distance from one crest to the next
y = f(x) y = f(x-a)
f(x) f(x-a)
a
Traveling Wave The brown curve
represents a snapshot of the curve at t = 0
The blue curve represents the wave at some later time, t
Wave Equations
The wave y(x,t) = A sin (k x – ωt) The speed v = λƒ If x ≠ 0 at t = 0, the wave function can
be generalized toy = A sin (k x – ωt + φ)
where φ is called the phase constant
Waves on a String
This is the speed of a wave on a string It applies to any shape pulse
Speed of Sound WavesCompression and rarefaction regions of the gas continue to move
Sound Waves as Displacement or Pressure Wave A sound wave may
be considered either a displacement wave or a pressure wave
The pressure wave is 90o out of phase with the displacement wave
Chapter 14
Superposition and Standing Waves
Superposition of Sinusoidal Waves y1 = A sin (kx - ωt) y2 = A sin (kx - ωt + φ)
y = A {sin a + sin b} == 2A cos [(a-b)/2] sin [(a+b)/2]
y = y1+y2= 2A cos (φ/2) sin (kx - ωt + φ/2)
Sinusoidal Waves with Destructive Interference
When φ = π, then cos (φ/2) = 0 Also any even multiple of π
The amplitude of the resultant wave is 0 Crests of one wave
coincide with troughs of the other wave
The waves interfere destructively
Standing Waves Assume two waves with the same
amplitude, frequency and wavelength, traveling in opposite directions in a medium
y1 = A sin (kx – ωt) and y2 = A sin (kx + ωt) They interfere according to the
superposition principle
Superposition of Two Identical Traveling Waves in Opposite Directions = Standing Wave y1 = A sin (kx - ωt) moving to the right y2 = A sin (kx + ωt) moving to the left
y = A {sin a + sin b} == 2A cos [(a-b)/2] sin [(a+b)/2]
y = y1+y2= [2A cos (ωt)] sin (kx )
Nodes and Antinodes, Photoy = y1+y2 = [2A cos (ωt)] sin (kx )
Nodes and Antinodes, cont
The diagrams above show standing-wave patterns produced at various times by two waves of equal amplitude traveling in opposite directions
In a standing wave, the elements of the medium alternate between the extremes shown in (a) and (c)
Standing Waves in a String Consider a string fixed
at both ends The string has length L Standing waves are set
up by a continuous superposition of waves incident on and reflected from the ends
There is a boundary condition on the waves
Standing Waves in a String
1/2λ1 = L so λ1 = 2L
Standing Waves in a String, 4 The second
harmonic λ = L
The third harmonic λ = 2L/3
Standing Waves on a String, Summary
n = 1, 2, 3, …
n is the nth harmonic These are the possible modes for the string
22
nn
Ln Ln
λ λ= ∴ =
vv f fλλ
= ∴ =
Standing Waves in an Open Tube
Standing Waves in a Tube Closed at One End
Beats
Temporal interference will occur when the interfering waves have slightly different frequencies
Beating is the periodic variation in amplitude at a given point due to the superposition of two waves having slightly different frequencies
Beats
The beat frequency is ƒbeat = |ƒ1 – ƒ2|
Average frequency heardƒbeat = (ƒ1 + ƒ2)/2
Quality of Sound –Tuning Fork A tuning fork
produces only the fundamental frequency
Quality of Sound –Flute The same note
played on a flute sounds differently
The second harmonic is very strong
The fourth harmonic is close in strength to the first
Quality of Sound –Clarinet The fifth harmonic is
very strong The first and fourth
harmonics are very similar, with the third being close to them
Analyzing Nonsinusoidal Wave Patterns If the wave pattern is periodic, it can be
represented as closely as desired by the combination of a sufficiently large number of sinusoidal waves that form a harmonic series
Any periodic function can be represented as a series of sine and cosine terms This is based on a mathematical technique called
Fourier’s theorem
Fourier Series A Fourier series is the corresponding
sum of terms that represents the periodic wave pattern
If we have a function y that is periodic in time, Fourier’s theorem says the function can be written as
ƒ1 = 1/T and ƒn= nƒ1 An and Bn are amplitudes of the waves
Fourier Synthesis of a Square Wave
Fourier synthesis of a square wave, which is represented by the sum of odd multiples of the first harmonic, which has frequency f
In (a) waves of frequency fand 3f are added.
In (b) the harmonic of frequency 5f is added.
In (c) the wave approaches closer to the square wave when odd frequencies up to 9f are added.
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