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Oscillations about equilibrium(Synonym:Vibration)
What means Oscillation?Oscillation is the periodic variation, typically in time, of some measure
as seen, for example, in a swinging pendulum.
Many things oscillate/vibrate: Periodic motion
(a motion that repeats itself over and over)Pulse Oscillations are the origin of thesensation of musical tone
.. in Aerospace: OrbitsElectrical/Computer: LRC resonance in circuits
Physics:Atomic Vibrations, String Theory, Electromagnetic Waves
Why does something vibrate/oscillate?
Whenever the system is displaced from equilibrium, a restoring force pulls it back,but it overshoots the equilibrium position.
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more examples
.heart beat, breathing, sleeping, taking shower, eating, chewing, blinking, drinking
.motion of planets, stars, motion of electrons, atoms
.wind (Tacoma Bridge)
.vocal cords, ear drums
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Parameter used to describe vibrations
Period T Time taken to complete one
cycle of the vibration. Units: s
Frequency f = 1/T
Number of vibration cycles per
second. Units: 1/s (Hz, Hertz)
Amplitude A Maximumdisplacement from equilibrium
position
One cycle is time take for a pendulum:
center right center left
center
Pendulum
T
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Special cases of periodic motion
Simple harmonic motion (SHM)
occurs when the restoring force
(the force directed toward a stable
equilibrium point) is proportional tothe displacement from equilibrium.
For instance when the restoringforce is F = - k x.(Hooks Law)
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A
Period T
Simple harmonic motionThe displacement from equilibrium can be describes as a
cosinusoidal function
0
0
2
2
2
2
2
2
2
=+
=+
=
xtx
xm
k
t
x
kxt
x
m
Equation of motion
for SHM
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Simple harmonic motion is the projection of circular
motion on the x-axis
Angular velocity
is NOT necessarily
the same as
Angular frequency
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Displacement, Velocity and Acceleration of SHM
( )
( )
( ) ( )txtAdt
tdvta
tAdttdxtv
tAtx
22 )cos()(
)sin()(
)cos(
=+==
+==
+=
A is the amplitude of the
motion, the maximum
displacement from
equilibrium, A=vmax, andA2 =amax.
Mass-Spring Java applet
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Mass-Spring-SystemA Spring always pushes or pulls mass back towards equilibrium
position. The time period can be calculated from Hookes Law:
k
mTor
m
k
tAktAm
kxmamaFkxF
2
)]cos([)]cos([
2
2
==
=
===
Independent from amplitude!
(Application: measure the mass of astronauts in
space) Heavier mass slower oscil lationsStiffer spring (greater k) rapid oscillations
The period of oscillation is
21==
fT
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Mass Spring-System in vertical setup
When a mass-spring system is oriented
vertically, it will exhibit SHM with the sameperiod and frequency as a horizontally
placed system.
Same formulae as for the horizontal setup
but the system oscillates around a newequilibrium position y0.
y0 = mg/k
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Energy: E = U + K
U: Potential Energy
K: Kinetic Energy
SpringEpot = U = k x
2
Ekin = K = m v2
Turning points:
E = Umax + 0
(Displacement and U at maximum)Minimum:
E = Kmax + 0 (Velocity and K at maximum)
Total energy of system
E = U + K = k (A cos(t))2
+ m (A sin(t))2
= k A2 cos2(t) + m A2 2 sin2(t)
E = k A2 cos2(t) + m A2 k/m sin2(t) = k A2 (cos2(t)+
sin2(t))
And therefore:
E = k A2
Potential Energy of simple harmonicmotion
Energy Conservation in Oscillatory Motion
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Exercise 1: The period of oscillation of an object in an ideal
mass-spring system is 0.50 sec and the amplitude is 5.0 cm.
What is the speed at the equilibrium point?
At equilibrium x=0:2222
2
1
2
1
2
1
2
1mvkAkxmvUKE ==+=+=
Since E=constant, at equilibrium (x = 0) the KE must be
a maximum. Here v = vmax = A.
( )( )cm/sec8.62rads/sec6.12cm5.0and
rads/sec6.12s50.0
22
===
===
Av
T
The amplitude A is given, but is not.
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Exercise 2: The diaphragm of a speaker has a mass of 50.0 g
and responds to a signal of 2.0 kHz by moving back and forthwith an amplitude of 1.810-4 m at that frequency.
(a) What is the maximum force acting on the diaphragm?
( ) ( ) 2222maxmax 42 mAffmAAmmaFF =====The value is Fmax=1400 N.
(b) What is the mechanical energy of the diaphragm?
Since mechanical energy is conserved, E = KEmax = Umax.
2
maxmax
2
max
21
2
1
mvKE
kAU
=
=The value of k is unknown so use KEmax.
( ) ( )
2222
maxmax2
2
1
2
1
2
1fmAAmmvKE ===
The value is KEmax= 0.13 J.
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Exercise 3: The displacement of an object in SHM is given
by:( ) ( ) ( )[ ]tty rads/sec57.1sincm00.8=
What is the frequency of the oscillations?
Comparing to y(t)= A sint givesA = 8.00 cm and = 1.57 rads/sec.The frequency is:
Hz250.02
rads/sec57.1
2===
f
( )( )
( )( )
222
max
max
max
cm/sec7.19rads/sec57.1cm00.8
cm/sec6.12rads/sec57.1cm00.8
cm00.8
===
===
==
Aa
Av
Ax
Other quantities can also be determined:
The period of the motion is sec00.4rads/sec57.1
22===
T
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A torsional pendulum is an oscillator for which
the restoring force is torsion. For example,suspending a barfrom a thin wire and winding it
by an angle , a torsional torque
is produced, where is a characteristic property of the
wire, known as the torsional constant. Therefore, the
equation of motion is
where I is the moment of inertia. But this is just a
simple harmonic oscillatorwith equation of
motion
where is the initial angle,
is the angular frequency, and is the phase constant.
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PendulumA mass, called a bob, suspended from a fixed point so that it can
swing in an arc determined by its momentum and the force of gravity.The length of a pendulum is the distance from the point of
suspension to the center of gravity of the bob. Chance observation of
a swinging church lamp led Galileo to find that a pendulum made
every swing in the same time, independent of the size of the arc. He
used this discovery in measuring time in his astronomical studies. His
experiments showed that the longer the pendulum, the longer is thetime of its swing.
If we assume the angle is small, for then we can approximate sin with (expressed in radian measure). (As an example, if = 5.00 =
0.0873 rad, then sin = 0.0872, a difference of only about 0.1%.)With that approximation and some rearranging, we then have
02
2
=+
I
mgL
t
Physical Pendulum,
Small amplitude
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UPendulum = mgh = mgL (1-cos)
For smaller displacements, the movement of
an ideal pendulum can be described
mathematically as simple harmonic motion
(like the mass-spring), as the change in
potential energy at the bottom of a circular arc
is nearly proportional to the square of the
displacement. Real pendulums do not have
infinitesimal displacements, so their behaviour
is actually of a non-linear kind.
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The Physical Pendulum
A "physical" pendulum hasextended size and is a
generalization of the bob pendulum.
An example would be a bar rotating
around a fixed axle. A simple
pendulum can be treated as a
special case of a physical pendulum
with moment of inertia I. ( I = miri2)
Period of a physical pendulum
(Note: l is now the length from thesuspension point to the center of
mass CM instead of L)
Simple Pendulum
All the mass of a simplependulum is concentrated in the
mass m of the particle-like bob,
which is at radius L from the
pivot point. Thus, we cansubstitute I = mL2 for the
rotational inertia of the
pendulum.
gLT 2=
for small amplitudes!!
Example:
Simple Pendulum: I = mL2Leg: I = 1/3 mL2
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Exercise 4:A clock has a pendulum that performs one full
swing every 1.0 sec. The object at the end of the stringweights 10.0 N. What is the length of the pendulum?
( )( ) m25.04s0.1m/s8.9
4L
2
2
22
2
2
===
=
gT
gLT
Solving for L:
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CM
Pivot
mg
Length: L
Mass: M
ICM= 1/12 ML2
Parallel-Axis TheoremIPivot= 1/12 ML
2 + M ( L)2 = 1/3 ML2
g
L
LgM
ML
gMl
IT
3
22
)
2
1(
3
1
22
2
===
The period is
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Ring
CM
r
IPivot= Mr2 + Mr2 = 2Mr2
g
rT
gMr
Mr
gMl
IT
22
222
2
=
==
Disc
CM
r
IPivot= Mr2 + Mr2 = 3/2 Mr2
g
rT
gMr
Mr
gMl
IT
2
32
2
3
22
2
=
==
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http://lectureonline.cl.msu.edu/%7Emmp/applist/damped/d.htm
http://physics.usask.ca/~pywell/p121/Images/tacoma.avi
http://www.walter-fendt.de/ph14e/resonance.htm