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Periodic MotionPeriodic Motion 11
Chapter 15Chapter 15
Oscillatory MotionOscillatory Motion
April 17April 17thth, 2006, 2006
Periodic MotionPeriodic Motion 22
The last steps …The last steps …
If you need to, file your taxes TODAY!If you need to, file your taxes TODAY!– Due at midnight.Due at midnight.
This weekThis week– Monday & Wednesday – OscillationsMonday & Wednesday – Oscillations– Friday – Review problems from earlier in Friday – Review problems from earlier in
the semesterthe semester Next Week Next Week
– Monday – Complete review.Monday – Complete review.
Periodic MotionPeriodic Motion 33
The The FINAL EXAMFINAL EXAM
Will contain 8-10 problems. One will Will contain 8-10 problems. One will probably be a collection of multiple probably be a collection of multiple choice questions.choice questions.
Problems will be similar to WebAssign Problems will be similar to WebAssign problems but only some of the actual problems but only some of the actual WebAssign problems will be on the WebAssign problems will be on the exam.exam.
You have 3 hours for the examination.You have 3 hours for the examination. SCHEDULE: MONDAY, MAY 1 @ 10:00 AMSCHEDULE: MONDAY, MAY 1 @ 10:00 AM
Periodic MotionPeriodic Motion 44
Things that Bounce Things that Bounce AroundAround
Periodic Motion 5
The Simple PendulumThe Simple Pendulum
0
1
)sin(
2
2
2
22
L
g
dt
d
dt
dmLLmg
I
Periodic Motion 6
The SpringThe Spring
02
2
2
2
xm
k
dt
xd
dt
xdmkx
maF
Periodic Motion 7
Periodic Motion From our observations, the motion of these
objects regularly repeats The objects seem t0 return to a given position
after a fixed time interval A special kind of periodic motion occurs in
mechanical systems when the force acting on the object is proportional to the position of the object relative to some equilibrium position If the force is always directed toward the
equilibrium position, the motion is called simple harmonic motion
Periodic Motion 8
The Spring … for a moment Let’s consider its motion at each
point. What is it doing?
Position Velocity Acceleration
Periodic Motion 9
Motion of a Spring-Mass System A block of mass m is
attached to a spring, the block is free to move on a frictionless horizontal surface
When the spring is neither stretched nor compressed, the block is at the equilibrium position
x = 0
Periodic Motion 10
More About Restoring Force The block is
displaced to the right of x = 0 The position is
positive The restoring
force is directed to the left
Periodic Motion 11
More About Restoring Force, 2 The block is at the
equilibrium position x = 0
The spring is neither stretched nor compressed
The force is 0
Periodic Motion 12
More About Restoring Force, 3 The block is
displaced to the left of x = 0 The position is
negative The restoring
force is directed to the right
Periodic Motion 13
Acceleration, cont. The acceleration is proportional to the
displacement of the block The direction of the acceleration is opposite
the direction of the displacement from equilibrium
An object moves with simple harmonic motion whenever its acceleration is proportional to its position and is oppositely directed to the displacement from equilibrium
Periodic Motion 14
Acceleration, final The acceleration is not constant
Therefore, the kinematic equations cannot be applied
If the block is released from some position x = A, then the initial acceleration is –kA/m
When the block passes through the equilibrium position, a = 0
The block continues to x = -A where its acceleration is +kA/m
Periodic Motion 15
Motion of the Block The block continues to oscillate
between –A and +A These are turning points of the
motion The force is conservative In the absence of friction, the
motion will continue forever Real systems are generally subject to
friction, so they do not actually oscillate forever
Periodic Motion 16
The Motion
Periodic Motion 17
Vertical Spring
Equilibrium Point
Periodic Motion 18
Ye Olde Math
02
2
xm
k
dt
xd0
2
2
L
g
dt
d
022
2
qdt
qd
Periodic Motion 19
022
2
qdt
qd
)cos(
:
0 tqq
Solution
q is either the displacement of the spring
(x) or the angle from equilibrium (). q is MAXIMUM at t=0 q is PERIODIC, always returning to its
starting position after some time T called the PERIOD.
Periodic Motion 20
Example – the Spring
m
kf
k
mT
tm
kTt
m
k
m
k
tm
kxx
xm
k
dt
xd
2
1
2
2)()(
so same, thestaysfunction T,tWhen t
sin
0
2
0
2
2
Periodic Motion 21
Example – the Spring
L
gf
g
LT
tL
gTt
L
g
L
g
tL
g
L
g
dt
d
2
1
2
2)()(
so same, thestaysfunction T,tWhen t
sin
0
2
0
2
2
Periodic Motion 22
Simple Harmonic Motion – Graphical Representation A solution is x(t)
= A cos (t + A, are all
constants A cosine curve
can be used to give physical significance to these constants
Periodic Motion 23
Simple Harmonic Motion – Definitions A is the amplitude of the motion
This is the maximum position of the particle in either the positive or negative direction
is called the angular frequency Units are rad/s
is the phase constant or the initial phase angle
Periodic Motion 24
Motion Equations for Simple Harmonic Motion
Remember, simple harmonic motion is not uniformly accelerated motion
22
2
( ) cos ( )
sin ( t )
cos( t )
x t A t
dxv A
dt
d xa A
dt
Periodic Motion 25
Maximum Values of v and a Because the sine and cosine
functions oscillate between 1, we can easily find the maximum values of velocity and acceleration for an object in SHM
max
2max
kv A A
mk
a A Am
Periodic Motion 26
Graphs The graphs show:
(a) displacement as a function of time
(b) velocity as a function of time
(c ) acceleration as a function of time
The velocity is 90o out of phase with the displacement and the acceleration is 180o out of phase with the displacement
Periodic Motion 27
SHM Example 1
Initial conditions at t = 0 are x (0)= A v (0) = 0
This means = 0 The acceleration
reaches extremes of 2A
The velocity reaches extremes of A
Periodic Motion 28
SHM Example 2 Initial conditions at
t = 0 are x (0)=0 v (0) = vi
This means = /2 The graph is shifted
one-quarter cycle to the right compared to the graph of x (0) = A
Periodic Motion 29
Energy of the SHM Oscillator Assume a spring-mass system is moving
on a frictionless surface This tells us the total energy is constant The kinetic energy can be found by
K = ½ mv 2 = ½ m2 A2 sin2 (t + ) The elastic potential energy can be found
by U = ½ kx 2 = ½ kA2 cos2 (t + )
The total energy is K + U = ½ kA 2
Periodic Motion 30
Energy of the SHM Oscillator, cont The total mechanical
energy is constant The total mechanical
energy is proportional to the square of the amplitude
Energy is continuously being transferred between potential energy stored in the spring and the kinetic energy of the block
Periodic Motion 31
As the motion continues, the exchange of energy also continues
Energy can be used to find the velocity
Energy of the SHM Oscillator, cont
2 2
2 2 2
kv A x
m
A x
Periodic Motion 32
Energy in SHM, summary
Periodic Motion 33
SHM and Circular Motion This is an overhead
view of a device that shows the relationship between SHM and circular motion
As the ball rotates with constant angular velocity, its shadow moves back and forth in simple harmonic motion
Periodic Motion 34
SHM and Circular Motion, 2 The circle is
called a reference circle
Line OP makes an angle with the x axis at t = 0
Take P at t = 0 as the reference position
Periodic Motion 35
SHM and Circular Motion, 3 The particle moves
along the circle with constant angular velocity
OP makes an angle with the x axis
At some time, the angle between OP and the x axis will be t +
Periodic Motion 36
SHM and Circular Motion, 4 The points P and Q always have the
same x coordinate x (t) = A cos (t + ) This shows that point Q moves with
simple harmonic motion along the x axis
Point Q moves between the limits A
Periodic Motion 37
SHM and Circular Motion, 5 The x component
of the velocity of P equals the velocity of Q
These velocities are v = -A sin (t +
)
Periodic Motion 38
SHM and Circular Motion, 6 The acceleration of
point P on the reference circle is directed radially inward
P ’s acceleration is a = 2A
The x component is –2 A cos (t + )
This is also the acceleration of point Q along the x axis
Periodic Motion 39
SHM and Circular Motion, Summary Simple Harmonic Motion along a straight
line can be represented by the projection of uniform circular motion along the diameter of a reference circle
Uniform circular motion can be considered a combination of two simple harmonic motions One along the x-axis The other along the y-axis The two differ in phase by 90o
Periodic Motion 40
Simple Pendulum, Summary The period and frequency of a
simple pendulum depend only on the length of the string and the acceleration due to gravity
The period is independent of the mass
All simple pendula that are of equal length and are at the same location oscillate with the same period
Periodic Motion 41
Damped Oscillations 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
Periodic Motion 42
Damped Oscillations, cont A graph for a
damped oscillation The amplitude
decreases with time
The blue dashed lines represent the envelope of the motion
Periodic Motion 43
Damped Oscillation, Example 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 b is called the damping
coefficient
Periodic Motion 44
Damping Oscillation, Example Part 2 The restoring force is – kx From Newton’s Second Law
Fx = -k x – bvx = max
When the retarding force is small compared to the maximum restoring force we can determine the expression for x This occurs when b is small
Periodic Motion 45
Damping Oscillation, Example, Part 3 The position can be described by
The angular frequency will be
2 cos( )bt
mx Ae t
2
2
k b
m m
Periodic Motion 46
Damping Oscillation, Example Summary When the retarding force is small, the
oscillatory character of the motion is preserved, but the amplitude decreases exponentially with time
The motion ultimately ceases Another form for the angular frequency
where 0 is the angular frequency in the absence of the retarding force
220 2
b
m
Periodic Motion 47
Types of Damping
is also called the natural frequency of the
system If Rmax = bvmax < kA, the system is said to
be underdamped When b reaches a critical value bc such that
bc / 2 m = 0 , the system will not oscillate The system is said to be critically damped
If Rmax = bvmax > kA and b/2m > 0, the system is said to be overdamped
0
k
m
Periodic Motion 48
Types of Damping, cont Graphs of position
versus time for (a) an underdamped
oscillator (b) a critically
damped oscillator (c) an overdamped
oscillator For critically damped
and overdamped there is no angular frequency
Periodic Motion 49
Forced Oscillations 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
Periodic Motion 50
Forced Oscillations, 2 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
Periodic Motion 51
Forced Oscillations, 3 The amplitude of a driven oscillation
is
0 is the natural frequency of the undamped oscillator
0
222 2
0
FmA
bm
Periodic Motion 52
Resonance When the frequency of the driving force
is near 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
Periodic Motion 53
Resonance At resonance, the applied force is in
phase with the velocity and the power transferred to the oscillator is a maximum The applied force and v are both
proportional to sin (t + ) The power delivered is F . v
This is a maximum when F and v are in phase
Periodic Motion 54
Resonance Resonance (maximum
peak) occurs when driving frequency equals the natural frequency
The amplitude increases with decreased damping
The curve broadens as the damping increases
The shape of the resonance curve depends on b
Periodic Motion 55
WE ARE DONE!!!