By Liew Sau Poh
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19. Oscillations
Objectives 19.1 Characteristics of simple harmonic motion 19.2 Kinematics of simple harmonic motion 19.3 Energy in simple harmonic motion 19.4 Systems in simple harmonic motion 19.5 Damped oscillations 19.6 Forced oscillations and Resonance
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Outcomes a) define simple harmonic motion by means of the
equation a = 2x b) Show that x = xo sin t as a solution of a = 2x c) derive and use the formula v = (A2 x2) d) describe, with graphical illustrations, the variation in
displacement, velocity and acceleration with time e) describe, with graphical illustrations, the variation in
velocity and acceleration with displacement f) derive and use the expressions for kinetic energy and
potential energy g) describe, with graphical illustrations, the variation in
kinetic energy and potential energy with time and displacement
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Outcomes h) derive and use expressions for the periods of
oscillations for spring-mass and simple pendulum systems
i) describe the changes in amplitude and energy for a damped oscillating system
j) distinguish between under damping, critical damping and over damping
k) distinguish between free oscillations and forced oscillations
l) state the conditions for resonance to occur
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19.1 Characteristics of SHM This type of motion is the most pervasive motion in the universe.
All atoms oscillate under harmonic motion.
We can model this motion with a linear restoring force.
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Periodic Motion Motion that repeats in a regular pattern over and over again is called periodic motion.
Simple harmonic motion is a specific type of periodic motion that has a simple sine or cosine wave shape.
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Position VS. Time graph What is the simple mathematical form of SHM motion? The displacement of the oscillating mass varies sinusoidally as a function of time.
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Periodic Motion Simple Harmonic Motion
Hearbeat Oscillating mass on a Spring
The restoring force of an ideal spring is given by: F = -kx where k is the spring constant and x is the displacement of the spring from its unstrained length. The minus sign indicates that the restoring force always points in opposite direction to the displacement of the spring. 9
Simple Harmonic Motion When there is a restoring force, F = -kx, simple harmonic motion occurs.
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19.2 Kinematics of SHM Simple Harmonic Motion (SHM) occurs when the force acting on a body is proportional to the displacement of the body from some equilibrium position (eg. a spring or a pendulum).
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x=0
Fs = -kx
x Fs = 0
Fs = +kx
-x
19.2 Kinematics of SHM
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x=0
Fs = -kx
x Fs = 0
Fs = +kx
-x
When the block attached to the spring (left) is displaced a small distance x from equilibrium, the spring exerts a restoring force which is proportional to the displacement:
19.2 Kinematics of SHM
xdt
xdSo
SHMx
tAdt
xda
tAdtdxv
22
2
2
22
2
)(
cos
sin
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a = - (k/m) x If we try x=A cos(wt+f) as a solution to this equation, we obtain:
Equation for Simple Harmonic Motion
19.3 Energy in SHM Total Energy = Kinetic Energy + Potential Energy E = K + U
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x k U 2 2
1 v m K 2 2
1
19.3 Energy in SHM
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txktU
txktU
m
m22
21
22
1
cos
cos22
1 sin txmtK m
txktK m22
21 sin
212 mE k x constant
k m 2
19.3 Energy in SHM Motion
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turning point
turning point
range of
motion
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1 xkUconstant2
21
mxkE
KE and PE Conversion
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222100 kAUKExavAx
02100 2 UkAKEaAvx
222100 kAUKEAavAx
x=0
Fs = -kx
x
Fs = 0
Fs = +kx
-x
Amplitude
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Amplitude is the magnitude of the maximum displacement.
tAAx coscos
Period, T
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For any object in simple harmonic motion, the time required to complete one cycle is the period T.
Frequency, f
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The frequency f of the simple harmonic motion is the number of cycles of the motion per second.
Tf 1
Energy of the Simple Harmonic Oscillator
21 2
222
222
21
cossin.21
cos21
21
kAE
ttAmkmE
PEKEE
tkAkxPE
total
total
total
tAmmvKE 2222 sin21
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Thus, total energy is proportional to amplitude2.
For a displacement x = A cos (wt+f), we can say that kinetic energy, KE is: Potential energy (elastic) PE is:
Energy transfer
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KE, U kA2/2
0 t t = 0 corresponds to the stretched spring.
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KE, U kA2/2
+A -A x
x = 0 corresponds to equilibrium position of spring.
Angular Frequency
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Since force restoring kx ) ( x m 2
m k T f or
k m dt
x d m
2 1 / 1
/
2
2
19.4 Systems in SHM
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1. Pendulums 1. The Simple Pendulum 2. The Physical Pendulum 3.
2. SHM & Uniform Circular Motion 3. Damped SHM 4. Forced Oscillations & Resonance
Gravitational Pendulum
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Simple Pendulum: a bob of mass m hung on an unstretchable massless string of length L.
Simple Pendulum
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a t x t
T
2
2
T Lg
2SHM for small
2LmI
L F L FI
mg LI
g gsin
acceleration ~ - displacement SHM
The Simple Pendulum
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m
T
mg
L
x
g L T
L g
L g
dt d
L x But dt
x d m mg
)
2
in rad
(sin
sin
2
2
2
2
2
Comparing a =
A pendulum leaving a trail of ink:
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Physical Pendulum
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Pivot
Center of Mass
quick method to measure g
A rigid body pivoted about a point other than its center of mass (com). SHM for small
acceleration ~ - displacement SHM
a t x t
T
2
2
T Img h
2
I
m g h I
h F h F g g sin
The Torsional Pendulum
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Torsion Pendulum:
2 IT
m Ik
2
2
dI Idt
Spring:
Simple Harmonic Motion
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Any Oscillating System:
2 mTk
T 2 inertiaspringiness
T Lg
2 2 IT
T Img h
2
SHM & Uniform Circular Motion
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The projection of a point moving in uniform circular motion on a diameter of the circle in which the motion occurs executes SHM.
The execution of uniform circular motion describes SHM. http://positron.ps.uci.edu/~dkirkby/music/html/demos/SimpleHarmonicMotion/Circula
r.html
SHM & Uniform Circular Motion
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radius = xm
The reference point
of radius xm. The projection of xm on a diameter of the circle executes SHM.
ta n g l eUC Irvine Physics of Music Simple Harmonic Motion Applet Demonstrations
txtx m c o s
SHM & Uniform Circular Motion
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xm. The projection of xm on a diameter of the circle executes SHM.
x(t) v(t) a(t)
radius = xm mxv mxa 2
txtx m c o stxtv m s in
txta m cos2
SHM & Uniform Circular Motion
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The projection of a point moving in uniform circular motion on a diameter of the circle in which the motion occurs executes SHM.
Measurements of the angle between Callisto and Jupiter: Galileo (1610)
earth
planet
Equations of Motion (SHM)
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a = - 2x [the definition]
x = A cos t
v = - A sin t
a = - 2A cos t
v = ± (A2 - x2 )0.5
Displacement-Time Graph
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x
t 0
x = A cos t A
-A
Velocity-Time Graph
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v
t 0
v = A sin t A
A
Acceleration-Time Graph
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a
t 0
a = A cos t A
A
Velocity-Displacement Graph
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v v = ± A x )0.5 A
A
A -A t 0
Acceleration-Displacement Graph
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a a = x [the definition]
A
A
A -A x 0
Phase Relationship
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0
x
v a
t
Free oscillations When a system oscillates without external forces acting on it, the system is in free oscillation. The amplitude of oscillation is constant, which will not drop.
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0
Displacement, x
Time
-x0
x0
19.5 Damped Oscillations
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In many real systems, nonconservative forces are present
This is no longer an ideal system (the type we have dealt with so far) Friction is a common nonconservative force
In this case, the mechanical energy of the system diminishes in time, the motion is said to be damped
19.5 Damped Oscillations
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Damped harmonic motion is harmonic motion with a frictional or drag force. If the damping is
modifies the undamped oscillation.
Damped SHM
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SHM in which each oscillation is reduced by an external force.
F k x
Damping Force In opposite direction to velocity Does negative work Reduces the mechanical energy
F bvD
Restoring Force SHM
Damped SHM
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netF m a
k x b v m a
differential equation
2
2
dx d xk x b mdt dt
19.5 Damped Oscillations
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A graph for a damped oscillation The amplitude decreases with time The blue dashed lines represent the envelope of the motion
19.5 Damped Oscillation
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One example of damped motion occurs when an object is attached to a spring and submerged in a viscous liquid The retarding force can be expressed as R = - b v where b is a constant and is called the damping coefficient
19.5 Damped Oscillation
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However, if the damping is large, it no longer resembles SHM at all.
A: underdamping: there are a few small oscillations before the oscillator comes to rest.
19.5 Damped Oscillation
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B: critical damping: this is the fastest way to get to equilibrium.
C: overdamping: the
system is slowed so much that it takes a long time to get to equilibrium.
19.5 Damped Oscillation
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There are systems where damping is unwanted, such as clocks and watches. Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings.
19.5 Damped Oscillation
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m d xdt
k x b dxdt
2
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2nd Order Homogeneous Linear Differential Equation: Solution of Differential Equation:
x t x e tm
bm
t
( ) cos2
where: km
bm
2
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b = 0 SHM
Damped Oscillations
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x t x e tm
bm
t
( ) cos2
km
2
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bm
1 small damping2
bm
bm
critically damped2
1 0 " "
21 0 " "2
b overdam pedm
the natural frequency
Exponential solution to the DE
Auto Shock Absorbers
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Typical automobile shock absorbers are designed to produce slightly under-damped motion
19.6 Forced Oscillations & Resonance
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Forced oscillations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency of the system.
If the frequency is the same as the natural frequency, the amplitude becomes quite large. This is called resonance.
19.6 Forced Oscillations & Resonance
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It is possible to compensate for the loss of energy in a damped system by applying an external force The amplitude of the motion remains constant if the energy input per cycle exactly equals the decrease in mechanical energy in each cycle that results from resistive forces
19.6 Forced Oscillations & Resonance
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After a driving force on an initially stationary object begins to act, the amplitude of the oscillation will increase After a sufficiently long period of time,
Edriving = Elost to internal Then a steady-state condition is reached The oscillations will proceed with constant amplitude
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19.6 Forced Oscillations & Resonance The sharpness of the resonant peak depends on the damping. If the damping is small (A), it can be quite sharp; if the damping is larger (B), it is less sharp.
Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it.
External frequency f
19.6 Forced Oscillations & Resonance
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When the frequency of the driving force is near to the natural frequency ( » ) an increase in amplitude occurs This dramatic increase in the amplitude is called resonance The natural frequency is also called the resonance frequency of the system
19.6 Forced Oscillations & Resonance
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Each oscillation is driven by an external force to maintain motion in the presence of damping:
F td0 cos
wd = driving frequency
19.6 Forced Oscillations & Resonance
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Each oscillation is driven by an external force to maintain motion in the presence of damping.
2nd Order Inhomogeneous Linear Differential Equation:
md xdt
k x m dxdt
F td
2
22
0 cos
km
19.6 Forced Oscillations & Resonance
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w = natural frequency wd = driving frequency
2nd Order Homogeneous Linear Differential Equation:
Steady-State Solution of Differential Equation:
x t x tm( ) coswhere: x F
m b
bm
m
d d
d
d
0
2 2 2 2 2 2
2 2tan
m d xdt
k x m dxdt
F td
2
22
0 cos
km
19.6 Forced Oscillations & Resonance
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w = natural frequency wd = driving frequency When w = wd resonance occurs!
km
The natural frequency, w, is the frequency of oscillation when there is no external driving force or damping.
less damping
more damping
x F
m bm
d d
0
2 2 2 2 2 2
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Stop the SHM caused by winds on a high-rise building
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The weight is forced to oscillate at the same frequency as the building but 1190 degrees out of phase.
400 ton weight mounted on a spring on a high floor of the Citicorp building in New York.
Summary
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OSCILLATION
Free Damped Forced Oscillations and Resonance
0
Displacement, x
Time -x0
x0
less damping
more damping