Transcript Unit 7 Power Point

Oscillations About Equilibrium
7.1 Periodic Motion
Periodic Motion – repeat, same time, same
path
Period (T) – time required for one complete
cycle (seconds) or seconds/cycle
Frequency (f) – the number of oscillations per
second (s-1 or hertz)
1
T
f
7.2 Simple Harmonic Motion
7.2 Simple Harmonic Motion
A form of Periodic Motion
Simple Harmonic Motion
A restoring force is applied proportional to the
distance from equilibrium
So Hooke’s Law
F  kx
7.2 Simple Harmonic Motion
If a graph of simple harmonic motion is
created
And spread out over time
We get a wave pattern
Amplitude – maximum
displacement
7.2 Simple Harmonic Motion
The mathematical relationship between
displacement and position is
 2 
x  A cos
t
 T 
When given in this form, you can determine
the Amplitude, and Period of a particle
undergoing simple harmonic motion
7.2 Simple Harmonic Motion
7.3 The Period of a Mass on a Spring
The period of a spring is given by the equation
m
T  2
k
A larger mass would have greater inertia –
longer period
A larger spring constant would produce more
acceleration, so a shorter period
The period is independent of amplitude
7.3 The Period of a Mass on a Spring
If a spring is hung vertically
Only the equilibrium position changes
At equilibrium
Fy  kx  mg
The weight is a constant force, so the
motion is still dependant on just
the force of the spring
7.3 The Period of a Mass on a Spring
7.4 Energy Conservation in Oscillatory Motion
If we ignore friction, then he energy of a spring
moving horizontally is
E mv
K  U kx
E
1
2
2
L
1
2 s
2
The maximum value
for energy is
E  kA
1
2
2
7.4 Energy Conservation in Oscillatory Motion
7.5 The Pendulum
A simple Pendulum
The potential energy
is
U
U mg
 mgy
( L  L cos q )
Lcosq
So potential energy
L-Lcosq
is zero at
equilibrium (like SHM)
7.5 The Pendulum
q
L
The period of a pendulum is given as
L
T  2
g
Independent of the mass of the bob
7.5 The Pendulum
Restoring Force
Forces
Components
T
A pendulum does not act as a
Simple Harmonic Oscillator,
but at small angles
mgsinq
(<30o) it approximates SHM
7.5 The Pendulum
mgcosq
W
7.6 Damped Oscillations
The amplitude of a real oscillating object will
decrease with time – called damping
Underdamped – takes several swing before
coming to rest (above)
7.6 Damped Oscillation
Overdamped – takes a long time to reach
equilibrium
Critical damping – equalibrium reached in the
shortest time
7.6 Damped Oscillation
7.7 Driven Oscillations and Resonance
Natural Frequency – depends on the variables
(m,k or L,g) of the object
Forced Vibrations –
caused by an
external force
7.7 Driven Oscillations and Resonance
Resonant Frequency – the natural vibrating
frequency of a system
Resonance – when the external frequency is
near the natural frequency and damping is
small
Tacoma Narrow Bridge
7.7 Driven Oscillations and Resonance
7.8 Types of Waves
Mechanical Waves – travels through a
medium
The wave travels through the medium, but the
medium undergoes simple harmonic
motion
Wave motion
Particle motion
7.8 Types of Waves
Waves transfer energy, not
particles
A single bump of a wave is called a pulse
A wave is formed when a force is applied to
one end
Each successive particle is moved by the one
next to it
7.8 Types of Waves
Parts of a wave
Transverse wave
– particle
motion
perpenduclar to wave motion
Wavelength (l) measured in meters
Frequency (f) measured in Hertz (Hz)
Wave Velocity (v) meters/second
v fl
7.8 Types of Waves
Longitudinal (Compressional) Wave
Particles move
parallel to the
direction of wave motion
Rarefaction – where
particles are spread
out
Compression – particles
are close
7.8 Types of Waves
Earthquakes
S wave – Transverse
P wave – Longitudinal
Surface Waves – can travel along the
boundary
Notice the circular motion of the particles
7.8 Types of Waves
7.9 Reflection and Transmission of Waves
When a wave comes to a
boundary (change in
medium) at least some of
the wave is reflected
The type of reflection depends
on if the boundary is fixed
(hard) - inverted
7.9 Reflection and Transmission of Waves
When a wave comes to a
boundary (change in
medium) at least some of
the wave is reflected
Or movable (soft) – in phase
7.9 Reflection and Transmission of Waves
For two or three dimensional we think in terms
of wave fronts
A line drawn perpendicular to the wave front is
called a ray
When the waves get far from their source and
are nearly straight, they are called plane
waves
7.9 Reflection and Transmission of Waves
Law of Reflection – the angle of reflection
equals the angle of incidence
qi  q r
Angles are always measured from
the normal
7.9 Reflection and Transmission of Waves
Law of Reflection – the angle of reflection
equals the angle of incidence
qi  q r
Angles are always measured from
the normal
7.9 Reflection and Transmission of Waves
7.10 Characteristics of Sound
Sound is a longitudinal wave
Caused by the vibration of a medium
The speed of sound depends on the medium
it is in, and the temperature
For air, it is calculated as
TK
vs  331.5
273.15
7.10 Characteristics of Sound
Loudness – sensation of intensity
Pitch – sensation of frequency
Range of human hearing – 20Hz to 20,000 Hz
ultrasonic – higher than human hearing
dogs hear to 50,000 Hz,
bats to 100,000 Hz
infrasonic – lower than human hearing
7.10 Characteristics of Sound
Often called pressure waves
Vibration produces areas of higher pressure
These changes in pressure are recorded by
the ear drum
7.10 Characteristics of Sound
7.11 Intensity of Sound
Loudness – sensation
Relative to surrounding and intensity
Intensity – power per unit area
Humans can detect intensities
as low as 10-12 W/m2
The threshold of pain
is 1 W/m2
7.11 Intensity of Sound
P
I
A
Sound intensity is usually
in
Source ofmeasured
Sound
Sound Level
(dB)
decibels (dB)
Jet Plane at 30 m
140
Sound level is given Threshold
as
of Pain
120
Loud Rock Concert
120
I
  10
logat 30 m
Siren
100
I
Auto Interior at 090 km/h
75
Busy Street Traffic
I – intensity of the sound
Conversation at-12
0.50 m
I0 – threshold of hearing (10 W/m2)
Quiet Radio
– sound level in dB Whisper
Rustle of Leaves
Threshold
of Hearing
Some common relative
intensities
7.11 Intensity of Sound
70
65
40
20
10
0
7.12 The Ear
Steps in sound transmission
7.12 The Ear
7.13 Sources of Sound: Strings and Air Columns
Vibrations in strings
Fundamental frequency
l  2L
v
f1 
2L
lL
v f 2  2 f1
f2 
L
Next Harmonic
7.13 Sources of Sound
Vibrations in strings
Next Harmonic
l L
2
3
v f 3  3 f1
f3  2
3 L
Strings produce all harmonics – all whole
number multiples of the fundamental
frequency
7.13 Sources of Sound
Vibrations in an open ended tube (both ends)
Fundamental frequency
l  2L
v
f1 
2L
lL
v f 2  2 f1
f2 
L
Next Harmonic
7.13 Sources of Sound
Vibrations in open ended tubes
Next Harmonic
l L
2
3
v f 3  3 f1
f3  2
3 L
Open ended tubes produce all harmonics – all
whole number multiples of the fundamental
frequency
7.13 Sources of Sound
Vibrations in an closed end tube (one end)
Fundamental frequency
l  4L
v
f1 
4L
l L
v f 3  3 f1
f3  4
3 L
Next Harmonic
4
3
7.13 Sources of Sound
Vibrations in open ended tubes
Next Harmonic
l  54 L
v f 5  5 f1
f5  4
5 L
Closed end tubes produce only odd
harmonics
7.13 Sources of Sound
7.14 Interference of Sound Waves: Beats
If waves are produced by two identical
sources
A pattern of constructive and destructive
interference forms
Applet
7.14 Interference of Sound Waves: Beats
7.15 The Doppler Effect
Doppler Effect – the change in pitch due to the
relative motion between a source of sound
and the receiver
Doppler Effect
Applies to all wave phenomena
Light Doppler
Objects moving toward you have a higher
apparent frequency
Objects moving away have a lower apparent
frequency
7.15 The Doppler Effect
If an object is stationary the equation for the
wave velocity is
v  fl
Sound waves travel outward evenly in all
directions
Doppler Applet
If the object moves toward the observed, the
waves travel at the same velocity, but each
new vibration is created closer to the observer
7.15 The Doppler Effect
The general equation is
 V  V0 

f  f s 
 V  Vs 
The values of Vo (speed of observer) and Vs
(speed of source) is positive when they
approach each other
Radar Gun
7.15 The Doppler Effect
7.16 Interference
Interference – two waves pass through the
same region of space at the same time
The waves pass through each other
Principle of Superposition – at the point where
the waves meet the displacement of the
medium is the algebraic sum of their
separate displacements
7.16 Interference
Phase – relative position of the wave crests
If the two waves are “in phase”
Constructive interference
If the two waves are “out of phase”
Destructive Interference
7.16 Interference
For a wave (instead of a single phase)
Interference is
calculated by adding
amplitude
In real time this looks
like
7.16 Interference