Chapter13_1-3_FA05.ppt

Periodic systems are all around us
• Rising of the sun
• Change of the seasons
• The tides
• Bird songs
• Rotation of a bicycle wheel
Periodic motion is any motion that repeats on a regular time basis.
Time
Position
Period, T

Periodic Motion
t, time
Amplitude Period, T
The motion PERIOD is the time, T, to return to same point.
The FREQUENCY, f, is the inverse of the period, 1/T.
The AMPLITUDE, A, is the maximum displacement.
A
Since the motion returns to the same point at t=T, it must be true that

 2

T so 
 2
1 
 T
f

Harmonic Motion: Key concepts
• Harmonic motion is an important and common
type of repetitive, or “oscillatory” motion
• Harmonic motion is “sinusoidal”
• Oscillatory motion results when an applied
force (1) depends on position AND (2) reverses
direction at some position.
• The two most common harmonic motions are
the pendulum and the spring-mass system.

Modern clocks
Mechanical, quartz, and atomic.

Clocks use Harmonic Motion
• Stonehenge, a clock based on the sun
• Sundials
• Water clocks
• Pendulum based clocks
• Geneva escapement mechanism
• Quartz crystal clocks
• Atomic clocks
Demonstrate some “clocks”

Our journey begins with uniform circular motion.
• Uniform Circular Motion is closely related to Harmonic Motion (oscillations)
• Objects in UCM have a constant centripetal acceleration (ac=V2/R).
Dq
1
V

2
V

R
R
Dq
V

D V

R
V
a
R
V
V
t
V
V
t
V
t
V
V
V
V
R
V
c
2
sin










D
D

D
D

D
D
D

D

D













q
q
q


Theme-park Physics
• What is the angular speed and linear speed needed to
have a rider feel “zero G” at the top of the ride?
• What is the net acceleration at the bottom of the ride?
R
TOP:
BOTTOM:
ac
ac
ac=g=V2/R=R2
NOTE: There is more to this than meets the eye! The force of the ride on the
rider is zero at the top of the ride, and is Mg at the bottom of the ride.
F=Mac=Mg
Mg
Mg
Fc+Fg=2Mg
If the ride is 9.8 meters in radius,
=sqrt(g/R)=1 rad/sec
V=9.8 m/sec (about 20 MPH)

Theme-park Physics:
Feeling weightless
• What is the angular speed and linear speed needed to
have a rider feel “zero G” at the top of the ride?
• What is the net acceleration at the bottom of the ride?
R
ac
g
g=ac
ac=g=V2/R=R2
Vzero-G = sqrt(Rg)
We don’t feel the acceleration of
gravity acting on our bodies, only
the force of gravity of the floor
pushing up against gravity.
Weightlessness is “zero g”
acceleration, normal gravity is
“one g”. Unconsciousness can
result at around 9 g without
special equipment.

Connection of rotation and
harmonic motion
• Physlet Illustration 16.1

Connection between rotational and oscillatory motion
Y motion
X motion q
q
sin
cos
R
y
R
x


q
t
R
y
t
R
x


sin
cos


t
t

Harmonic motion, velocity
q
V
Vx
V q
t
R
V
Vx


q
sin
sin





R
V 
So, now have…
t
R
V
t
R
x
x 


sin
cos




Harmonic motion, acceleration
q
ac
ac
ax
q
t
R
t
R
V
a
ax



q
cos
cos
cos
2
2







Harmonic motion-summary
t
R
a
t
R
v
t
R
x
x
x





cos
sin
cos
2





Look at what this says….
x
ax
2



Or x
m
ma
F x
x
2




So, we have a force that depends on position, and reverses direction at x=0.

Spring-time remembered….
We know that for an ideal
spring, the force is related to
the displacement by
kx
F 

But we just showed
that harmonic motion
has
x
m
F 2



So, we directly find out
that the “angular
frequency of motion”
of a mass-spring
system is
m
k
m
k



2

Harmonic motion: all together now.
t
A
a
t
A
v
t
A
x
x
x





cos
sin
cos
2





t, time
x
v
a

Properties of Mass-Spring System
• Physlet Exploration 16-1.
• How does the period of oscillation change
with amplitude?
Exploration 16-1

Application: Tuning Forks and Musical Instruments
• A tuning fork is basically a type of spring. The
same is true for the bars that make up a
xylophone. They have a very large spring
constant.
• Since the oscillation frequency does not
change with amplitude, the tone of the tuning
fork and xylophone note is independent of
loudness.

Simulation: Mass on Spring
• Physlet Illustration 16-4: forced & damped
motion of spring/mass system.
Physlet Illustration 16-4

The pendulum: keeping time harmonically.
Mg
T
q
Ftangent
q
q
Mg
Mg
Ft



 sin
L
s

q

Small angle approximation
0
sin


q
q
q
Theta must be calculated in
RADIANS! Generally the
approximation is used for
angles less than 30
degrees, or about ½ radian.

Compare pendulum and spring.
Pendulum
Spring
L
s
Mg
F 

Kx
F 

Force
Forces depend on position, reverse direction at some position.
Periodic motion:
t
x
x 
cos
0
 t
s
s 
sin
0

Angular frequency:
M
K


or t

q
q sin
0

L
g



Chapter13_1-3_FA05.ppt

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