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Simple Harmonic
Motion
Periodic Motion
• defined: motion that
repeats at a constant rate
• equilibrium position: forces
are balanced
Periodic Motion
• For the spring example,
the mass is pulled down to
y = -A and then released.
• Two forces are working on
the mass: gravity (weight)
and the spring.
Periodic Motion
• for the spring:
ΣF = Fw + Fs
ΣFy = mgy + (-kΔy)
Periodic Motion
• Damping: the effect of
friction opposing the
restoring force in
oscillating systems
Periodic Motion
• Restoring force (Fr): the
net force on a mass that
always tends to restore the
mass to its equilibrium
position
Simple Harmonic
Motion
• defined: periodic motion
controlled by a restoring
force proportional to the
system displacement from
its equilibrium position
Simple Harmonic
Motion
• The restoring force in SHM
is described by:
Fr x = -kΔx
• Δx = displacement from
equilibrium position
Simple Harmonic
Motion
• Table 12-1 describes
relationships throughout
one oscillation
Simple Harmonic
Motion
• Amplitude: maximum
displacement in SHM
• Cycle: one complete set of
motions
Simple Harmonic
Motion
• Period: the time taken to
complete one cycle
• Frequency: cycles per unit
of time
• 1 Hz = 1 cycle/s = s-1
Simple Harmonic
Motion
• Frequency (f) and period
(T) are reciprocal
quantities.
1
1
f=
T=
T
f
Reference Circle
• Circular motion has many
similarities to SHM.
• Their motions can be
synchronized and similarly
described.
Reference Circle
• The period (T) for the
spring-mass system can be
derived using equations of
circular motion:
m
T = 2π
k
Reference Circle
• This equation is used for
Example 12-1.
• The reciprocal of T gives
the frequency.
m
T = 2π
k
Periodic Motion
and the
Pendulum
Overview
• Galileo was among the first
to scientifically study
pendulums.
Overview
• The periods of both
pendulums and springmass systems in SHM are
independent of the
amplitudes of their initial
displacements.
Pendulum Motion
• An ideal pendulum has a
mass suspended from an
ideal spring or massless
rod called the pendulum
arm.
• The mass is said to reside
at a single point.
Pendulum Motion
• l = distance from the
pendulum’s pivot point and
its center of mass
• center of mass travels in a
circular arc with radius l.
Pendulum Motion
• forces on a pendulum at
rest:
• weight (mg)
• tension in pendulum arm
(T p )
• at equilibrium when at rest
Restoring Force
• When the pendulum is not
at its equilibrium position,
the sum of the weight and
tension force vectors
moves it back toward the
equilibrium position.
Fr = Tp + mg
Restoring Force
• Centripetal force adds to
the tension (Tp):
Tp = Tw΄+ Fc , where:
Tw΄ = Tw = |mg|cos θ
Fc = mvt²/r
Restoring Force
• Total acceleration (atotal) is
the sum of the tangential
acceleration vector (at) and
the centripetal acceleration.
• The restoring forces causes
this atotal.
Restoring Force
• A pendulum’s motion does
not exactly conform to
SHM, especially when the
amplitude is large (larger
than π/8 radians, or 22.5°).
Small Amplitude
• defined as a displacement
angle of less than π/8
radians from vertical
• SHM is approximated
Small Amplitude
• For small initial
displacement angles:
T = 2π
l
|g|
Small Amplitude
• Longer pendulum arms
produce longer periods of
swing.
T = 2π
l
|g|
Small Amplitude
• The mass of the pendulum
does not affect the period
of the swing.
T = 2π
l
|g|
Small Amplitude
• This formula can even be
used to approximate g (see
Example 12-2).
T = 2π
l
|g|
Physical Pendulums
• mass is distributed to some
extent along the length of
the pendulum arm
• can be an object swinging
from a pivot
• common in real-world
motion
Physical Pendulums
• The moment of inertia of an
object quantifies the
distribution of its mass
around its rotational center.
• Abbreviation: I
• A table is found in
Appendix F of your book.
Physical Pendulums
• period of a physical
pendulum:
T = 2π
I
|mg|l
Oscillations in
the Real World
Damped Oscillations
• Resistance within a
spring and the drag of
air on the mass will
slow the motion of the
oscillating mass.
Damped Oscillations
• Damped harmonic
oscillators experience
forces that slow and
eventually stop their
oscillations.
Damped Oscillations
• The magnitude of the
force is approximately
proportional to the
velocity of the system:
fx = -βvx
β is a friction proportionality
constant
Damped Oscillations
• The amplitude of a
damped oscillator
gradually diminishes
until motion stops.
Damped Oscillations
• An overdamped
oscillator moves back
to the equilibrium
position and no further.
Damped Oscillations
• A critically damped
oscillator barely
overshoots the
equilibrium position
before it comes to a
stop.
Driven Oscillations
• To most efficiently
continue, or drive, an
oscillation, force should
be added at the maximum
displacement from the
equilibrium position.
Driven Oscillations
• The frequency at which
the force is most effective
in increasing the
amplitude is called the
natural oscillation
frequency (f0).
Driven Oscillations
• The natural oscillation
frequency (f0) is the
characteristic frequency
at which an object
vibrates.
• also called the resonant
frequency
Driven Oscillations
• terminology:
• in phase
• pulses
• driven oscillations
• resonance
Driven Oscillations
• A driven oscillator has
three forces acting on it:
• restoring force
• damping resistance
• pulsed force applied in
same direction as Fr
Driven Oscillations
• The Tacoma Narrows
Bridge demonstrated the
catastrophic potential of
uncontrolled oscillation in
1940.
Waves
Waves
• defined: oscillations of
extended bodies
• medium: the material
through which a wave
travels
Waves
• disturbance: an oscillation
in the medium
• It is the disturbance that
travels; the medium does
not move very far.
Graphs of Waves
Waveform
graphs
Vibration
graphs
Types of Waves
• longitudinal wave:
disturbance that displaces
the medium along its line of
travel
• example: spring
Types of Waves
• transverse wave:
disturbance that displaces
the medium perpendicular
to its line of travel
• example: the wave along a
snapped string
Longitudinal Waves
• Any physical medium can
carry a longitudinal wave.
• Compression
Rarefaction zone:
zone:
molecules are spread
pushedapart
and
haveand
lower
density
and
together
have
higher
pressure
density and pressure
Longitudinal Waves
• travel faster in solids than
gases
• water waves have both
longitudinal and transverse
components—a
“combination” wave
Periodic Waves
• carry information and
energy from one place to
another
Periodic Waves
• amplitude (A): the greatest
distance a wave displaces a
particle from its average
position
A = ½(ypeak - ytrough)
A = ½(xmax - xmin)
Periodic Waves
• wavelength (λ): the
distance from one peak (or
compression zone) to the
next, or from one trough (or
rarefaction zone) to the
next
Periodic Waves
• A wave completes one
cycle as it moves through
one wavelength.
• A wave’s frequency (f) is
the number of cycles
completed per unit of time
Periodic Waves
• wave speed (v): the speed
of the disturbance
• for periodic waves:
λf = v
Sound Waves
• longitudinal pressure
waves that come from a
vibrating body and are
detected by the ears
• cannot travel through a
vacuum; must pass
through a physical medium
Sound Waves
• travel faster through solids
than liquids, and faster
through liquids than gases
• have three characteristics:
Loudness
• the interpretation your
hearing gives to the
intensity of the wave
• intensity (Is): amount of
power transported by the
wave per unit area
• measured in W/m²
Loudness
• a sound must be ten times
as intense to be perceived
as twice as loud
• sound is measured in
decibels (dB)
Pitch
• related to the frequency
• high frequency is
interpreted as a high pitch
• low frequency is interpreted
as a low pitch
• 20 Hz to 20,000 Hz
Quality
• results from combinations
of waves of several
frequencies
• fundamental and harmonics
• why a trumpet sounds
different than an oboe
Sound Waves
• All three characteristics
affect the way sound is
perceived.
Doppler Effect
• related to the relative
velocity of the observer and
the sound source
approaching
object
has
• an
actual
object
sound
moving
emitted
away
by
has
the
pitch
than
there
a higher
object
lowerdoes
pitch
not
than
change
ififthere
were no relative velocity
Mach Speed
• measurement is dependent
on the composition and
density of the atmosphere
• speed of sound changes
with altitude