What term is used to describe an oscillator that “runs down” and eventually stops?

FOR SCIENTISTS AND ENGINEERS

physics

a strategic approach

THIRD EDITION

randall d. knight

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Chapter 14 Lecture

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Chapter 14 Oscillations

Chapter Goal: To understand systems that oscillate with simple harmonic motion.

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Chapter 14 Preview

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Chapter 14 Preview

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Chapter 14 Preview

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Chapter 14 Preview

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Chapter 14 Preview

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Chapter 14 Preview

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Chapter 14 Reading Quiz

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A “sinusoidal” function of x can be represented by

Cos(x).
Sin(x).
Tan(x).
Either A or B.
Either A, B or C.
Reading Question 14.1

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Answer: B

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A “sinusoidal” function of x can be represented by

Cos(x).
Sin(x).
Tan(x).
Either A or B.
Either A, B or C.
Reading Question 14.1

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Answer: D

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What is the name of the quantity represented by the symbol ?

Angular momentum.
Angular frequency.
Phase constant.
Uniform circular motion.
Centripetal acceleration.
Reading Question 14.2

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Answer: B

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What is the name of the quantity represented by the symbol ?

Angular momentum.
Angular frequency.
Phase constant.
Uniform circular motion.
Centripetal acceleration.
Reading Question 14.2

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Answer: B

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What is the name of the quantity represented by the symbol 0?

Angular momentum.
Angular frequency.
Phase constant.
Uniform circular motion.
Centripetal acceleration.
Reading Question 14.3

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Answer: B

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What is the name of the quantity represented by the symbol 0?

Angular momentum.
Angular frequency.
Phase constant.
Uniform circular motion.
Centripetal acceleration.
Reading Question 14.3

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Answer: C

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Wavelength is

The time in which an oscillation repeats itself.
The distance in which an oscillation repeats itself.
The distance from one end of an oscillation to the other.
The maximum displacement of an oscillator.
Not discussed in Chapter 14.
Reading Question 14.4

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Answer: E

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Wavelength is

The time in which an oscillation repeats itself.
The distance in which an oscillation repeats itself.
The distance from one end of an oscillation to the other.
The maximum displacement of an oscillator.
Not discussed in Chapter 14.
Reading Question 14.4

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Answer: E

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What term is used to describe an oscillator that “runs down” and eventually stops?

Tired oscillator.
Out of shape oscillator.
Damped oscillator.
Resonant oscillator.
Driven oscillator.
Reading Question 14.5

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Answer: C

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What term is used to describe an oscillator that “runs down” and eventually stops?

Tired oscillator.
Out of shape oscillator.
Damped oscillator.
Resonant oscillator.
Driven oscillator.
Reading Question 14.5

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Answer: C

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Chapter 14 Content, Examples, and
QuickCheck Questions

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Oscillatory Motion

Objects that undergo a repetitive motion back and forth around an equilibrium position are called oscillators.
The time to complete one full cycle, or one oscillation, is called the period T.
The number of cycles per second is called the frequency f, measured in Hz:
1 Hz = 1 cycle per second = 1 s1

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Example 14.1 Frequency and Period of a Loudspeaker Cone

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Simple Harmonic Motion

A particular kind of oscillatory motion is simple harmonic motion.
In figure (a) an air-track glider is attached to a spring.
Figure (b) shows the glider’s position measured 20 times every second.
The object’s maximum displacement from equilibrium is called the amplitude A of the motion.
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QuickCheck 14.1

A and B but not C.
None are.
Which oscillation (or oscillations) is SHM?

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QuickCheck 14.1

A and B but not C.
None are.
Which oscillation (or oscillations) is SHM?

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Simple Harmonic Motion

Figure (a) shows position versus time for an object undergoing simple harmonic motion.
Figure (b) shows the velocity versus time graph for the
same object.
The velocity is zero at the times when x   A; these are the turning points of the motion.
The maximum speed vmax is reached at the times when x = 0.
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Simple Harmonic Motion

If the object is released from rest at time t = 0, we can model the motion with the cosine function:
Cosine is a sinusoidal function.
 is called the angular frequency, defined as
 = 2/T.
The units of  are rad/s.
 = 2f.
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Simple Harmonic Motion

The maximum speed is vmax  

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Example 14.2 A System in Simple Harmonic Motion

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Example 14.2 A System in Simple Harmonic Motion

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Example 14.2 A System in Simple Harmonic Motion

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Example 14.3 Finding the Time

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Simple Harmonic Motion and Circular Motion

Figure (a) shows a “shadow movie” of a ball made by projecting a light past the ball and onto a screen.
As the ball moves in uniform circular motion, the shadow moves with simple harmonic motion.
The block on a spring in figure (b) moves with the same motion.
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The Phase Constant

What if an object in SHM is
not initially at rest at x = A
when t = 0?
Then we may still use the
cosine function, but with a phase constant measured
in radians.
In this case, the two primary kinematic equations of SHM are:
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Oscillations described by the phase constants 0  /3 rad, /3 rad, and  rad.

The Phase Constant

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Example 14.4 Using the Initial Conditions

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Example 14.4 Using the Initial Conditions

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Example 14.4 Using the Initial Conditions

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Example 14.4 Using the Initial Conditions

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QuickCheck 14.2

 /2 rad.
0 rad.
 /2 rad.
 rad.
None of these.
This is the position graph of a mass oscillating on a horizontal spring. What is the phase constant 0?

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QuickCheck 14.2

 /2 rad.
0 rad.
 /2 rad.
 rad.
None of these.
This is the position graph of a mass oscillating on a horizontal spring. What is the phase constant 0?

Initial conditions:

x = 0

vx > 0

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QuickCheck 14.3

This is the position graph of a mass oscillating on a horizontal spring. What is the phase constant 0?

 /2 rad.
0 rad.
 /2 rad.
 rad.
None of these.
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QuickCheck 14.3

This is the position graph of a mass oscillating on a horizontal spring. What is the phase constant 0?

 /2 rad.
0 rad.
 /2 rad.
 rad.
None of these.
Initial conditions:

x = –A

vx = 0

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QuickCheck 14.4

The figure shows four oscillators at t = 0. For which is the phase constant 0   / 4?

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QuickCheck 14.4

The figure shows four oscillators at t = 0. For which is the phase constant 0   / 4?

Initial conditions:

x = 0.71A

vx > 0

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Energy in Simple Harmonic Motion

An object of mass m on a frictionless horizontal surface
is attached to one end of a spring of spring constant k.
The other end of the spring
is attached to a fixed wall.
As the object oscillates,
the energy is transformed between kinetic energy and potential energy, but the mechanical energy E  K  U doesn’t change.
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Energy in Simple Harmonic Motion

Energy is conserved in Simple Harmonic Motion:
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QuickCheck 14.5

A block oscillates on a very long horizontal spring. The graph shows the block’s kinetic energy as a function of position. What is the spring constant?

1 N/m.
2 N/m.
4 N/m.
8 N/m.
I have no idea.
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QuickCheck 14.5

A block oscillates on a very long horizontal spring. The graph shows the block’s kinetic energy as a function of position. What is the spring constant?

1 N/m
2 N/m.
4 N/m.
8 N/m.
I have no idea.
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Frequency of Simple Harmonic Motion

In SHM, when K is maximum, U  0, and when U is maximum, K  0.
K  U is constant, so Kmax  Umax:
Earlier, using kinematics, we found that:
So:
So:
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QuickCheck 14.6

A mass oscillates on a horizontal spring with period T  2.0 s. If the amplitude of the oscillation is doubled, the new period will be

1.0 s.
1.4 s.
2.0 s.
2.8 s.
4.0 s
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QuickCheck 14.6

A mass oscillates on a horizontal spring with period T  2.0 s. If the amplitude of the oscillation is doubled, the new period will be

1.0 s.
1.4 s.
2.0 s.
2.8 s.
4.0 s.
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QuickCheck 14.7

A block of mass m oscillates on a horizontal spring with period T  2.0 s. If a second identical block is glued to the top of the first block, the new period will be

1.0 s.
1.4 s.
2.0 s.
2.8 s.
4.0 s.
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QuickCheck 14.7

A block of mass m oscillates on a horizontal spring with period T  2.0 s. If a second identical block is glued to the top of the first block, the new period will be

1.0 s.
1.4 s.
2.0 s
2.8 s.
4.0 s.
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QuickCheck 14.8

Two identical blocks oscillate on different horizontal springs. Which spring has the larger spring constant?

The red spring.
The blue spring.
There’s not enough information to tell.
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QuickCheck 14.8

Two identical blocks oscillate on different horizontal springs. Which spring has the larger spring constant?

The red spring.
The blue spring.
There’s not enough information to tell.
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Example 14.5 Using Conservation of Energy

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Example 14.5 Using Conservation of Energy

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Example 14.5 Using Conservation of Energy

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Simple Harmonic Motion Motion Diagram

The top set of dots is a motion diagram for SHM going to the right.
The bottom set of dots is a motion diagram for SHM going to the left.
At x  0, the object’s speed is as large as possible, but it is not changing; hence acceleration is zero at x  0.
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Acceleration in Simple Harmonic Motion

Acceleration is the time-derivative of the velocity:

In SHM, the acceleration is proportional to the negative of the displacement.

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Dynamics of Simple Harmonic Motion

Consider a mass m oscillating on a horizontal spring with no friction.

The spring force is:

Since the spring force is the net force, Newton’s second law gives:

Since ax  2x, the angular frequency must be .

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QuickCheck 14.9

A mass oscillates on a horizontal spring. It’s velocity is vx and the spring exerts force Fx. At the time indicated by the arrow,

vx is  and Fx is .
vx is  and Fx is .
vx is  and Fx is 0.
vx is 0 and Fx is .
vx is 0 and Fx is .
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QuickCheck 14.9

A mass oscillates on a horizontal spring. It’s velocity is vx and the spring exerts force Fx. At the time indicated by the arrow,

vx is  and Fx is .
vx is  and Fx is .
vx is  and Fx is 0.
vx is 0 and Fx is .
vx is 0 and Fx is .
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QuickCheck 14.10

A mass oscillates on a horizontal spring. It’s velocity is vx and the spring exerts force Fx. At the time indicated by the arrow,

vx is  and Fx is .
vx is  and Fx is .
vx is  and Fx is 0.
vx is 0 and Fx is .
vx is 0 and Fx is .
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QuickCheck 14.10

A mass oscillates on a horizontal spring. It’s velocity is vx and the spring exerts force Fx. At the time indicated by the arrow,

vx is  and Fx is .
vx is  and Fx is .
vx is  and Fx is 0.
vx is 0 and Fx is .
vx is 0 and Fx is .
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Vertical Oscillations

Motion for a mass hanging from a spring is the same as for horizontal SHM,
but the equilibrium position is affected.

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Example 14.7 Bungee Oscillations

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VISUALIZE

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Example 14.7 Bungee Oscillations

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Example 14.7 Bungee Oscillations

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QuickCheck 14.11

Negative.
Zero.
Positive.
A block oscillates on a vertical spring. When the block is at the lowest point
of the oscillation, it’s acceleration ay is

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QuickCheck 14.11

Negative.
Zero.
Positive.
A block oscillates on a vertical spring. When the block is at the lowest point
of the oscillation, it’s acceleration ay is

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The Simple Pendulum

Consider a mass m attached to a string of length L which is free to swing back and forth.
If it is displaced from its lowest position by an angle , Newton’s second law for the tangential component of gravity, parallel
to the motion, is:
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The Simple Pendulum

If we restrict the pendulum’s oscillations to small angles ( 10), then we may use the small angle approximation sin   , where 
is measured in radians.

and the angular frequency of the motion is found to be:

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Example 14.8 The Maximum Angle of a Pendulum

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Example 14.8 The Maximum Angle of a Pendulum

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QuickCheck 14.12

1.0 s.
1.4 s.
2.0 s.
2.8 s.
4.0 s.
A ball on a massless, rigid rod oscillates as a simple pendulum with a period of 2.0 s. If the ball is replaced with another ball having twice the mass, the period will be

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QuickCheck 14.12

1.0 s.
1.4 s.
2.0 s.
2.8 s.
4.0 s.
A ball on a massless, rigid rod oscillates as a simple pendulum with a period of 2.0 s. If the ball is replaced with another ball having twice the mass, the period will be

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QuickCheck 14.13

1.0 s.
1.4 s.
2.0 s.
2.8 s.
4.0 s.
On Planet X, a ball on a massless, rigid rod oscillates as a simple pendulum with a period of 2.0 s. If the pendulum is taken to the moon of Planet X, where the free-fall acceleration g is half as big, the period will be

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QuickCheck 14.13

1.0 s.
1.4 s.
2.0 s.
2.8 s.
4.0 s.
On Planet X, a ball on a massless, rigid rod oscillates as a simple pendulum with a period of 2.0 s. If the pendulum is taken to the moon of Planet X, where the free-fall acceleration g is half as big, the period will be

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Tactics: Identifying and Analyzing Simple Harmonic Motion

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The Physical Pendulum

Any solid object that swings
back and forth under the
influence of gravity can be modeled as a physical pendulum.
The gravitational torque for
small angles (  10) is:
Plugging this into Newton’s second law for rotational motion,   I, we find the equation for SHM, with:
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Example 14.10 A Swinging Leg as a Pendulum

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Example 14.10 A Swinging Leg as a Pendulum

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QuickCheck 14.14

The solid disk.
The circular hoop.
Both have the same period.
There’s not enough information to tell.
A solid disk and a circular hoop have the same radius and the same mass. Each can swing back and forth as a pendulum from a pivot at one edge. Which has the larger period of oscillation?

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QuickCheck 14.14

The solid disk.
The circular hoop.
Both have the same period.
There’s not enough information to tell.
A solid disk and a circular hoop have the same radius and the same mass. Each can swing back and forth as a pendulum from a pivot at one edge. Which has the larger period of oscillation?

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Damped Oscillations

An oscillation that runs down
and stops is called a damped oscillation.
One possible reason for dissipation of energy is
the drag force due to air resistance.
The forces involved in dissipation are complex, but a simple linear drag model is:
The shock absorbers in cars and trucks are heavily damped springs. The vehicle’s vertical motion, after hitting a rock or a pothole, is a damped oscillation.

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Damped Oscillations

When a mass on a spring experiences the force of the spring as given
by Hooke’s Law, as well as
a linear drag force of
magnitude |D|  bv, the
solution is:

where the angular frequency is given by:

Here is the angular frequency of the undamped oscillator (b  0).

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Damped Oscillations

Position-versus-time graph for a damped oscillator.

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Damped Oscillations

A damped oscillator has position x  xmaxcos(t + 0), where:
This slowly changing function xmax provides a border to the rapid oscillations, and is called the envelope.
The figure shows several oscillation envelopes, corresponding to different values of the damping constant b.
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Mathematical Aside: Exponential Decay

Exponential decay occurs in a vast number of physical systems of importance in science and engineering.
Mechanical vibrations, electric circuits, and nuclear radioactivity all exhibit exponential decay.
The graph shows the function:
where
e  2.71828… is
Euler’s number.
exp is the
exponential function.
v0 is called the decay constant.
u  e /0   exp(/0)

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Energy in Damped Systems

Because of the drag force, the mechanical energy of a damped system is no longer conserved.
At any particular time we can compute the mechanical energy from:
Where the decay constant of this function is called the time constant , defined as:
The oscillator’s mechanical energy decays exponentially with time constant .
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Driven Oscillations and Resonance

Consider an oscillating system that, when left to itself, oscillates at a natural frequency f0.
Suppose that this system is subjected to a periodic external force of driving frequency fext.
The amplitude of oscillations is generally not very high if fext differs much from f0.
As fext gets closer and closer to f0, the amplitude of the oscillation rises dramatically.
A singer or musical instrument can shatter a crystal goblet by matching the goblet’s natural oscillation frequency.

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Driven Oscillations and Resonance

The response curve shows the amplitude of a driven oscillator at frequencies near its natural frequency of 2.0 Hz.

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Driven Oscillations and Resonance

The figure shows the
same oscillator with
three different values
of the damping constant.
The resonance amplitude becomes higher and narrower as the damping constant decreases.
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QuickCheck 14.15

The red oscillator.
The blue oscillator.
The green oscillator.
They all oscillate for
the same length of time.
The graph shows how three oscillators respond as the frequency of a driving force is varied. If each oscillator is started and then left alone, which will oscillate for the longest time?

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QuickCheck 14.15

The red oscillator.
The blue oscillator.
The green oscillator.
They all oscillate for
the same length of time.
The graph shows how three oscillators respond as the frequency of a driving force is varied. If each oscillator is started and then left alone, which will oscillate for the longest time?

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Chapter 14 Summary Slides

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General Principles

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General Principles

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Important Concepts

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Important Concepts

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