Lecture 01
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Transcript of Lecture 01
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Physics 232 Lecture 01 1
Periodic Motion or
Oscillations
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Physics 232 Lecture 01 2
Periodic Motion Periodic Motion is motion that repeats about a point of stable equilibrium
A necessary requirement for periodic motion is a Restoring Force
Stable Equilibrium
Unstable Equilibrium
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Physics 232 Lecture 01 3
Characteristics
Amplitude - A The maximum displacement from the equilibrium position
Period - T The time to complete one cycle of motion, peak to peak or valley to valley
Frequency - f The number of cycles per unit time
Tf 1=
A
A
T
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Physics 232 Lecture 01 4
Simple Harmonic Motion Consider a mass m attached to a horizontal spring having a spring constant k with the spring unstretched
We now pull the mass to the right and then let the mass go
What is the subsequent motion of the mass?
The restoring force is given by Hookes Law xkmaF ==
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Physics 232 Lecture 01 5
Simple Harmonic Motion This is a second order differential equation
xkdt
xdm =22
Since this is a second order differential equation, there are two constants of integration and the general solution is
( ) += tAx cos
where A is the maximum displacement, is a phase angle, and is the angular velocity and is given by
mk
=
A and are determined from the initial, boundary, conditions
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Physics 232 Lecture 01 6
Phase Angle
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Physics 232 Lecture 01 7
Simple Harmonic Motion The period of the motion is related to by
kmT
22 ==
Note that the period of the motion is independent of the displacement!
The velocity of the particle is given by
( ) +== tAdtdxv sin
with the maximum velocity being A
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Physics 232 Lecture 01 8
Simple Harmonic Motion We have so far that
( ) += tAx cos and ( ) +== tAdtdxv sin
The constants A and are determined from the initial conditions, that is what are x and v at a specified time or some other suitable initial conditions The maximum displacement occurs when the velocity is zero, which occurs at the extreme of motion (two locations: x = A)
The maximum velocity occurs when the displacement from the equilibrium position is zero (two values: v = )
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Physics 232 Lecture 01 9
SHM Energy Considerations We assume that the system is isolated and frictionless
With the force being given by xkF =It can be shown that there is a potential energy in the system that is 2
21 xkU =
The total energy is then given by
2221
21 xkvm
UKEETotal
+=
+=
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Physics 232 Lecture 01 10
SHM Energy Considerations If we substitute for x and v we find that
221 AkETotal =
The total energy is constant
The energy shifts back and forth between the kinetic energy and the potential energy, but with the sum of the two being constant
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Physics 232 Lecture 01 11
SHM Energy Considerations The relationship between kinetic energy, potential energy, displacement, velocity and acceleration can be seen in the following diagram
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Physics 232 Lecture 01 12
Choice of Oscillatory Function Note that we used the cosine function in our development of Simple Harmonic Motion
We could have also used the sine function for our description of SHM
The only difference is in the phase angle
( ) += tAx cos
( )'sin += tAx
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Physics 232 Lecture 01 13
Simple Harmonic Motion Simple Harmonic Motion can be used to describe motion in many situations under appropriate approximations
Any potential energy function that under appropriate circumstances can be approximated by a parabolic function will exhibit SHM
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Physics 232 Lecture 01 14
Simple Pendulum Consider a mass m suspended from a massless, unstretchable string of length L The forces acting on the mass are as shown The restoring force is the one perpendicular to the string
sinmgFrestoring =But this is a nonlinear function of However for small angles sinWe then have that
mgFrestoring =
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Physics 232 Lecture 01 15
Simple Pendulum From before we have that mgFrestoring =
We also have that
Lmgk =
this is then becomes the equation of SHM
Lx=
This then yields that xL
mgFrestoring =
If we then let
The frequency of oscillation is given by Lgf
21
=
which is independent of the mass attached to the string
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Physics 232 Lecture 01 16
Damped Motion Real life situations have dissipative forces
The dissipative force is often related to the velocity that is in the motion
The fact that there is a dissipative force, leads to motion that is damped, that is the amplitude decreases with time
bvFdiss =
The total force is then the sum of the restoring force and this dissipative force
vbxkFnet =
The minus sign indicates that this force is in the opposite direction to the velocity
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Physics 232 Lecture 01 17
Damped Motion This net force leads to a slightly more complicated second order differential equation
022
=++ kxdtdxb
dtxdm
The exact solution to this equation depends upon the damping constant
There are three possible solutions
Underdamped: kmb 2
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Physics 232 Lecture 01 18
Damped Motion - Underdamped kmb 2
The general solution for this situation is given by ( ) ( )tttmb eAeAex 22 212/ +=
This solution looks like
with mk
mb
=4
2
2
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Physics 232 Lecture 01 21
Damped Motion The most efficient damping, as far getting back to zero amplitude, is the critically damped case
Note that the overdamped case may yield some interesting behavior depending on the relative values of the parameters
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Physics 232 Lecture 01 22
Forced Oscillation It is also possible to drive a system such as an oscillator with an external force that is also time varying
The general differential equation is of the form
tFbvxkdt
xdm dcosmax22
=++
where the term on the right hand side is the driving force
This solution to this equation involves two functions, a complementary function and a particular solution
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Physics 232 Lecture 01 23
Forced Oscillation The solution is given by
( )( )
+= t
bmk
Ftx ddd
cos)(2222
max
with ( )
= 2
1
/2/2tan
d
dmk
mb
represents the phase difference between the driving force and the resulting motion
There can be a delay between the action of the driving force and the response of the system
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Physics 232 Lecture 01 24
Resonance
( ) 2222max
dd bmk
FA +
=
The term in front of the cosine function represents the amplitude
If we vary the angular velocity of the driving force, we find that the system has its maximum amplitude when
mk
d
This phenomenon of the amplitude peaking at a driving frequency that is near the natural frequency of the system is known as resonance
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Physics 232 Lecture 01 25
Resonance The strength of the amplitude depends upon the magnitude of the damping coefficient
The smaller the value of b the more pronounced the peak
The larger the damping the peak becomes broader, less sharp, and shifts to lower frequencies
If mkb 2> the peak disappears completely
Periodic MotionorOscillationsPeriodic MotionCharacteristicsSimple Harmonic MotionSimple Harmonic MotionPhase AngleSimple Harmonic MotionSimple Harmonic MotionSHM Energy ConsiderationsSHM Energy ConsiderationsSHM Energy ConsiderationsChoice of Oscillatory FunctionSimple Harmonic MotionSimple PendulumSimple PendulumDamped MotionDamped MotionDamped Motion - UnderdampedDamped Motion- Critically DampedDamped Motion - OverdampedDamped MotionForced OscillationForced OscillationResonanceResonance