Chapter 12
Vibrations
and
Waves
Chapter 12 Objectives
•
•
•
•
•
•
•
•
Hooke’s Law
Simple Harmonic Motion
Elastic Potential Energy
Wave properties
– Frequency
– Amplitude
– Wavelength
Wave Types
Pendulum
Superposition
Wave Interference
Hooke’s Law
• The simplest type of vibrational motion is a mass
attached to a spring moving without any
frictional forces.
– No friction
– No air resistance
• The force provided by the spring is
– Fs = -kx
• k is spring constant
• x is displacement from rest position of spring
– x will be negative when the spring is stretched
– x will be positive when the spring is compressed
• (-) because the spring is always providing a force opposite
the motion of the mass
Simple Harmonic Motion
• Simple harmonic motion occurs when the net
force acting in the direction of motion follows
Hooke’s Law.
– That is no frictional forces present and…
– The force is proportional to the displacement
but opposite in direction
• Basically with simple harmonic motion, the
motion will repeat a cycle of back and forth
forever.
– Must be back and forth along same path.
• Also called periodic motion.
Wave Properties
• Amplitude
– maximum distance object travels away from rest point
•A
– units: meters
• Period
– time it takes to complete one full cycle of motion
•T
– units: seconds
• Frequency
– the number of cycles per unit of time
• number of waves past a given a point in one second
– 

Inverse of the Period


= T-1
units: hertz (Hz)
which is equivalent to an inverse second. (s-1)
Elastic Potential Energy
• Remember elastic potential energy can be found
– PEelastic = ½ kx2
• k is called the spring constant
– units: N/m
• x is the distance the spring is stretched or compressed away
from its resting point
• The energy is only stored in a spring when it is
•
either stretched or compressed.
The potential energy in a spring is always
positive.
– That is because x is squared.
How to Use Elastic Potential Energy
• Be sure to identify what types of energy are present at
each position of the problem.
v
v
v=0
E = KE
E = KE + PEelastic
E = PEelastic
v
E = KE + PEelastic
x=0
Wave Types
• A transverse wave
is a wave that its
particles move
perpendicular to the
overall motion of the
wave.
• A longitudinal
wave is a wave in
which its particles
move in the same
direction as the
overall motion of the
wave.
More Wave Properties
• Besides calculating the amplitude and frequency of a
•
wave, we can also calculate the wavelength.
The wavelength (λ) of a wave is the distance between
two successive points on the wave.
– Typically measured from crest-to-crest.
λ
A
λ
Velocity vs Frequency and Wavelength
• The velocity of a wave can be found very simply by
remember what velocity is measuring.
– distance over time
v=
v=
Δx
Δt
λ
T
v=λ
Remember that T-1 is
the same as .
Superposition Principle
• If two or more waves are moving through a medium, the
resultant wave is found by adding together the
displacements of the individual waves point by point.
+
Types of Interference
• Constructive
interference occurs
• Destructive
interference occurs
when two waves meet
that are in phase.
– Waves that are in phase
have crests and valleys that
line up exactly.
• This type will make a
bigger wave.
•
•
when two waves meet
that are out of phase.
This will typically make a
smaller wave.
If the two waves are
180o out of phase, then
the waves will cancel
each other out.
Pendulum
• A pendulum also exhibits simple
θ
harmonic motion under certain
conditions.
– The force must follow Hooke’s Law by
being proportional to the displacement
at all times
– The initial angle of displacement must
be less than 15 degrees
L
mg sin θ
• The restoring force to maintain
simple harmonic motion acts
tangential to the path of the swing.
– That force is the component of the
weight of the object that is tangent to
the circular path of the pendulum.
mg
Period of a Pendulum
• A pendulum exhibiting simple harmonic motion
will have a period that depends on:
– length of the pendulum, L
• longer the pendulum, longer the arc (larger amplitude)
– So it would take more time to complete one cycle
– acceleration due to gravity
• the faster gravity can pull the pendulum, the shorter the time
it takes to complete its cycle (period)
T = 2  (L/g)
Period of a Spring System
• The period of a mass oscillating in simple harmonic
motion depends on
– mass of the spring system
• more mass takes more time to accelerate
• and the longer the path (larger amplitude) it creates
– spring constant
• stiffer the spring, the less time it takes to accelerate
– It only depends on these two things because gravity remains constant
and those are the only two values that could vary in the system.
T = 2  (m/k)
Standing Waves
• A standing wave is a wave that can be sent down a medium
and returned in such a way that it appears to stand still.
– One end is free to oscillate and the other is attached to a fixed
point.
•
• Guitar strings
The position where the string is fixed, or appears to be
motionless, is called a node.
– The large part of the wave is called the antinode.