Chapter 12
Vibrations and Waves
Section 12-1
Simple Harmonic Motion
 Force is maximum
at maximum
displacement.
 Force is zero at
equilibrium
Spring
Simple Harmonic Motion
con’t
• Simple Harmonic Motion–vibration about
an equilibrium position in which a restoring
force is proportional to the displacement from
equilibrium.
• Hooke’s Law–Discovered in 1678 by
Robert Hooke. The relationship between the
force and displacement in a mass-spring
system.
Felastic  -kx
Some problems
1.
A 76 N crate is attached to a spring (k = 450
N/m). How much displacement is
caused by the
weight of the crate?
Given:
F = 76 N
k = 450 N/m
Felastic  -kx
76 N  -(450 N /m)x
x  -0.17m
Some Problems con’t
2.
A spring of k = 1962 N/m loses its
elasticity if stretched more than 50.0
cm. What is the mass of the heaviest
object the spring can support without
being damaged?
Given:
k = 1962 N/m
x = 50.0 cm = 0.50 m
Felastic  -kx
Felastic  -1962N/ m0.50m
Felastic  -981N
F  ma

kgm
981N  m9.81 2 

s 
m  100 kg
Simple Pendulum
• A simple pendulum
consists of a mass
called a bob, which
is attached to a
fixed string.
Pendulum
Foucault Pendulum
•
This type of pendulum was first
used by the French physicist
Jean Foucault to verify the
Earth’s rotation experimentally.
As the pendulum swings, the
vertical plane in which it
oscillates appears to rotate as
the bob successively knocks
over the indicators arranged in
a circle on the floor.
© Bob Emott, Photographer
Section 12-2
Measuring Simple Harmonic Motion
• Amplitude–maximum displacement from
equilibrium.
• Period–the time it takes to execute a
complete cycle of motion.
• Frequency–number of cycles or vibrations
per unit of time.
Period of Pendulum and
Spring
L
m
T  2
T  2
g
k
A Couple of Problems
1.
What is the period of a 3.98 m long pendulum?
Given:
L  3.98 m
m
g  9.81 2
s
L
T  2
g
3.98m
T  2
2
9.81m / s
T  4.00 s
Problems con’t
2.
What is the free-fall acceleration at a
location where a 6.00 m long pendulum swings
through exactly 100 cycles in
492 s?
Given:
L  6.00m
100 cyclesin 492s
100cycles
f
492s
f  0.203 Hz
L
T  2
g
m
g  9.79 2
s
T
1
1

 4.92s
f 0.203 Hz
6.00m
4.92 s  2
g
Spring Problem
3.
A 1.0 kg mass attached to one end of a spring
completes one oscillation every 2.0 s. Find the spring
constant.
Given:
m  1.0 kg
T  2.0 s
m
T  2
k
1.0 kg
2.0s  2
k
N
k  9.9
m
Pendulum Problem
4.
A man needs to know the height of a tower, but darkness
obscures the ceiling. He knows, however,
that a long pendulum
extends from the ceiling
almost to the floor and that its
period is 12.0 s. How tall is the tower?
Given:
T  12.0s
m
g  9.81 2
s
L
T  2
g
L
12.0 s  2
9.81m / s2
L  35.8 m
Section 12-3
Properties of Waves
• Medium–material through which a
disturbance travels.
Types of Waves
• Mechanical Wave–a wave whose
propagation requires the existence of a
medium.
• Electromagnetic Waves–a wave consisting of
oscillating electric and magnetic fields at right
angles to each other, no medium is required.
• Matter Waves–Electrons show wave-like
behavior.
Kinds of Waves
• Transverse wave–A wave in which the vibration is at
right angles (perpendicular) to the direction in which
the wave is traveling.
• Longitudinal Wave–A wave in which the vibration is
in the same direction (parallel) as that in which the
wave is traveling.
• Surface Wave–Surface disturbance with
characteristics of both transverse and longitudinal
waves.
Transverse and
Longitudinal Waves
Surface Wave
Single or Multiple
• Wave Pulse–A single disturbance
traveling through a medium.
• Periodic Wave–A wave whose source is
some form of periodic motion.
Parts of Waves
• Crests–One of the places in a wave where the wave
is highest or the disturbance is greatest.
• Troughs–One of the places in a wave where the
wave is lowest or the disturbance is greatest in the
opposite direction from a crest.
• Wavelength–The distance from the top of crest of a
wave to the top of the following crest, or equivalently,
the distance between successive identical parts of
the wave.
Parts of Waves 2
Wave Speed
x
v
t
v

T
1
f
T
Velocity
x  tT
For waves




v   f
T
Speed of a Wave
v  f
Section 12-4
Wave Interactions
• Principle of Superposition–The displacement
of a medium caused by two or more waves is
the algebraic sum of the displacements
caused by the individual waves.
• Interference–Combining of two waves
arriving at the same point at the same time.
Constructive Interference
• Constructive Interference–The superposition of two
equal wave pulses resulting in a combined wave of
larger amplitude than its components.
Destructive Interference
• Destructive Interference–The superposition
of two wave pulses with equal but opposite
amplitude.
Reflection
Standing Waves
• Standing Waves–Wave
•
•
pattern that results when two
waves of the same frequency,
wavelength and amplitude
travel in opposite directions and
interfere.
Node–point in a standing wave
that always undergoes
complete destructive
interference.
Antinode–point in a standing
wave, halfway between two
nodes, at which the largest
amplitude occurs
More Standing Waves
Last Standing Wave