Vibrations and Waves
Chapter 12
Periodic Motion
A
repeated motion is called periodic
motion
 What
are some examples of periodic
motion?



The motion of Earth orbiting the sun
A child swinging on a swing
Pendulum of a grandfather clock
Simple Harmonic Motion

Simple harmonic motion is a form of periodic
motion

The conditions for simple harmonic motion are
as follows:



The object oscillates about an equilibrium position
The motion involves a restoring force that is
proportional to the displacement from equilibrium
The motion is back and forth over the same path
Earth’s Orbit
 Is
the motion of the Earth orbiting the sun
simple harmonic?



NO
Why not?
The Earth does not orbit about an equilibrium
position
p. 438 of your book

The spring is stretched away from the
equilibrium position

Since the spring is being stretched toward the
right, the spring’s restoring force pulls to the left
so the acceleration is also to the left
p. 438 of your book
 When
the spring is unstretched the force
and acceleration are zero, but the velocity
is maximum
p.438 of your book
 The
spring is stretched away from the
equilibrium position
 Since the spring is being stretched toward
the left, the spring’s restoring force pulls to
the right so the acceleration is also to the
right
Damping
 In
the real world, friction eventually causes
the mass-spring system to stop moving
 This
effect is called damping
Mass-Spring Demo
 http://phet.colorado.edu/simulations/sims.p
hp?sim=Masses_and_Springs
I
suggest you play around with this
demo…it might be really helpful!
Hooke’s Law
 The
spring force always pushes or pulls
the mass back toward its original
equilibrium position
 Measurements
show that the restoring
force is directly proportional to the
displacement of the mass
Hooke’s Law
Felastic  kx



Felastic= Spring force
k is the spring constant
x is the displacement from equilibrium

The negative sign shows that the direction of F
is always opposite the mass’ displacement
Flashback
 Anybody
remember where we’ve seen the
spring constant (k) before?
 PEelastic
A
= ½kx2
stretched or compressed spring has
elastic potential energy!!
Spring Constant
 The
value of the spring constant is a
measure of the stiffness of the spring
 The
bigger k is, the greater force needed
to stretch or compress the spring
 The
units of k are N/m (Newtons/meter)
Sample Problem p.441 #2
A
load of 45 N attached to a spring that is
hanging vertically stretches the spring 0.14
m. What is the spring constant?
Solving the Problem
Felastic  kx
F
 45 N
N
k

 321
x
 0.14m
m

Why do I make x
negative?

Because the
displacement is down
Follow Up Question

What is the elastic potential energy stored
in the spring when it is stretched 0.14 m?
1 2 1
N
2
PEelastic  kx   321.43 0.14m   3.15 J
2
2
m
The simple pendulum
 The
simple pendulum is a mass attached
to a string
 The
motion is simple harmonic
because the restoring force is proportional
to the displacement and because the
mass oscillates about an equilibrium
position
Simple Pendulum
 The
restoring force is a component of the
mass’ weight
 As
the displacement increases, the
gravitational potential energy increases
Simple Pendulum Activity
 http://phet.colorado.edu/simulations/sims.p
hp?sim=Pendulum_Lab
 You
should also play around with this
activity to help your understanding
Comparison between pendulum
and mass-spring system (p. 445)
Measuring Simple Harmonic
Motion (p. 447)
Amplitude of SHM
 Amplitude
is the maximum displacement
from equilibrium
 The
more energy the system has, the
higher the amplitude will be
Period of a pendulum

T = period

L= length of string

g= 9.81 m/s2
L
T  2
g
Period of the Pendulum
 The
period of a pendulum only depends
on the length of the string and the
acceleration due to gravity
 In
other words, changing the mass of the
pendulum has no effect on its period!!
Sample Problem p. 449 #2
 You
are designing a pendulum clock to
have a period of 1.0 s. How long should
the pendulum be?
Solving the Problem
L
T  2
g
2
T g
 T 
L
 *g 
2
4
 2 
m

1  9.81 2 
2
2
T g
s 
T 

L

 .25m
 *g 
2
2
4
(4 )
 2 
2
2
Period of a mass-spring system

T= period

m= mass

k = spring constant
m
T  2
k
Sample Problem p. 451 #2
 When
a mass of 25 g is attached to a
certain spring, it makes 20 complete
vibrations in 4.0 s. What is the spring
constant of the spring?
What information do we have?
 M=
.025 kg
 The
mass makes 20 complete vibrations in
4.0s



That means it makes 5 vibrations per second
So f= 5 Hz
T= 1/5 = 0.2 seconds
Solve the problem
m
T  2
k
2
2




2

4

4

N


.025kg   24.7
k 
 m   2 m  
2 
m
T 
T 
 0.20 
2
Day 2: Properties of Waves

A wave is the motion of a disturbance


Waves transfer energy by transferring the motion of
matter instead of transferring matter itself
A medium is the material through which a
disturbance travels



What are some examples of mediums?
Water
Air
Two kinds of Waves
 Mechanical
Waves require a material
medium

i.e. Sound waves
 Electromagnetic
Waves do not require a
material medium

i.e. x-rays, gamma rays, etc
Pulse Wave vs Periodic Wave
A
pulse wave is a single, non periodic
disturbance
A
periodic wave is produced by periodic
motion

Together, single pulses form a periodic wave
Transverse Waves

Transverse Wave: The particles move
perpendicular to the wave’s motion
Particles move in
y direction
Wave moves in
X direction
Longitudinal (Compressional)
Wave
 Longitudinal
(Compressional) Waves:
Particles move in same direction as wave
motion (Like a Slinky)
Longitudinal (Compressional)
Wave
Crests: Regions of High Density because
The coils are compressed
Troughs: Areas of Low Density because
The coils are stretched
Wave Speed

The speed of a wave
is the product of its
frequency times its
wavelength

f is frequency (Hz)

λ (lambda) Is
wavelength (m)
v  f
Sample Problem p.457 #4
A
tuning fork produces a sound with a
frequency of 256 Hz and a wavelength in
air of 1.35 m


a. What value does this give for the speed of
sound in air?
b. What would be the wavelength of the wave
produced b this tuning fork in water in which
sound travels at 1500 m/s?
Part a

Given:



f = 256 Hz
λ = 1.35 m
v=?
m
v  f  (256 Hz )(1.35m)  345.6
s
Part b

Given:



f = 256 Hz
v =1500 m/s
λ=?
m
1500
v
s
 
 5.86m
f 256 Hz
Wave Interference
 Since
waves are not matter, they can
occupy the same space at the same time
 The
combination of two overlapping waves
is called superposition
The Superposition Principle
 The
superposition principle: When two or
more waves occupy the same space at the
same time, the resultant wave is the vector
sum of the individual waves
Constructive Interference (p.460)
 When
two waves are traveling in the same
direction, constructive interference
occurs and the resultant wave is larger
than the original waves
Destructive Interference
 When
two waves are traveling on opposite
sides of equilibrium, destructive
interference occurs and the resultant
wave is smaller than the two original
waves
Reflection
 When
the motion of a wave reaches a
boundary, its motion is changed
 There


are two types of boundaries
Fixed Boundary
Free Boundary
Free Boundaries
A
free boundary is
able to move with the
wave’s motion
 At
a free boundary,
the wave is reflected
Fixed Boundaries

A fixed boundary
does not move with
the wave’s motion
(pp. 462 for more
explanation)

Consequently, the
wave is reflected and
inverted
Standing Waves
 When
two waves with the same properties
(amplitude, frequency, etc) travel in
opposite directions and interfere, they
create a standing wave.
Standing Waves

Standing waves have
nodes and antinodes
A

Nodes: The points
where the two waves
cancel
N
N
A
N
N


Antinodes: The
places where the
largest amplitude
occurs
There is always one
more node than
antinode
N
A
A
N
A
N
N
A
N
Sample Problem p.465 #2
A
string is rigidly attached to a post at one
end. Several pulses of amplitude 0.15 m
sent down the string are reflected at the
post and travel back down the string
without a loss of amplitude. What is the
amplitude at a point on the string where
the maximum displacement points of two
pulses cross? What type of interference is
this?
Solving the Problem
 What


type of boundary is involved here?
Fixed
So that means the pulse will be reflected and
inverted
 What
happens when two pulses meet and
one is inverted?


Destructive interference
The resultant amplitude is 0.0 m
Helpful Simulations

Mass-Spring system:
http://phet.colorado.edu/simulations/sims.php?sim=Mass
es_and_Springs

Pendulum:
http://phet.colorado.edu/simulations/sims.php?sim=Pend
ulum_Lab

Wave on a string system:
http://phet.colorado.edu/simulations/sims.php?sim=Wav
e_on_a_String
http://www.walter-fendt.de/ph14e/stwaverefl.htm