Download Simple Harmonic Motion

Lightening Review
Torque & Static Equilibrium
Which mass is heavier?
Cut at balance point
Balance
Point
1. The hammer portion.
2. The handle portion.
3. They have the same
mass.
Locate the Center of Mass
X=0
10N
1. 4 Meters
2. 5 Meters
3. 8 Meters
M
X=10M
40N
Periodic Motion
Chapter 13
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.
Position
Period, T
Time
Periodic Motion
Period, T
Amplitude
A
t, time
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.
Since the motion returns to the same point at t=T, it must be true that
T  2
so
f  1 T   2
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.
A history of time (keeping)
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).

V2

V1
Dq

DV

V
Dq
R
R


V  R
 

DV  V sin Dq  VDq

DV  Dq 
V
 V
Dt
Dt



V 
DV
 V  
Dt
R
V2
ac 
R
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?
TOP:
Mg
ac=g=V2/R=R2
ac
F=Mac=Mg
R
BOTTOM:
Fc+Fg=2Mg
If the ride is 9.8 meters in radius,
ac
=sqrt(g/R)=1 rad/sec
V=9.8 m/sec (about 20 MPH)
Mg
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.
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?
ac=g=V2/R=R2
g=ac
R
ac
g
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
q
t
x  R cos q
y  R sin q
X motion
x  R cos t
y  R sin t
t
Harmonic motion, velocity
Vx
V
V
q
q
V  R
Vx  V sin q
  R sin t
So, now have…
x  R cos t
Vx   R sin t
Harmonic motion, acceleration
ac
q
ax
q
ac
ax  a cos q
V2

cos t
R
  R 2 cos t
Harmonic motion-summary
x  R cos t
v x   R sin t
a x   R 2 cos t
Look at what this says….
a x   x
2
Or
Fx  max   m 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
But we just showed
that harmonic motion
has
So, we directly find out
that the “angular
frequency of motion”
of a mass-spring
system is
F  kx
F  m x
2
k  m

2
k
m
Harmonic motion: all together now.
x  A cos t
v x   A sin t
a x   A 2 cos t
x
a
v
t, time
Mass on a spring: x, v and 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.
T
Ftangent
q
Mg
Ft   Mg sin q
  Mgq
s
q
L
Small angle approximation
sin q  q
q 0
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.
Spring
Force
F  Kx
Pendulum
s
F   Mg
L
Forces depend on position, reverse direction at some position.
Periodic motion:
x  x0 cos t
s  s0 sin t
or q  q 0 sin t
Angular frequency:

K
M

g
L