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Chapter 14 - Oscillations
I.
Introduction.
a. We discuss oscillatory motion in this chapter.
b. The kinematics of motion with constant acceleration is presented in Chapter 2 and
Chapter 3.
i. In this chapter, the kinematics and dynamics of motion with acceleration that is
proportional to displacement from equilibrium is presented.
ii. The word “oscillate” means to swing back and forth.
iii. Oscillation occurs when a system is disturbed from a position of stable
equilibrium.
iv. Many familiar examples exist: surfers bob up and down waiting for the right
wave, clock pendulums swing back and forth, and the strings and reeds of
musical instruments vibrate.
v. Other, less familiar examples are the oscillations of air molecules in a sound
wave and the oscillations of electric currents in radios, television sets, and
metal detectors.
vi. In addition, many other devices rely on oscillatory motion to function.
c. In this chapter, we deal mostly with the most fundamental type of oscillatory motion—
simple harmonic motion. We also consider both damped and driven oscillations.
II.
Simple Harmonic Motion.
a. A common, very important, and very basic kind of oscillatory motion is simple
harmonic motion such as the motion of a solid object attached to a spring.
b.
c.
d.
e.
f.
g.
h.
i.
In equilibrium, the spring exerts no force on the object.
When the object is displaced an amount x from its equilibrium position, the spring exerts
a force −kx, as given by Hooke’s law:
Combining with Newton’s Second Law:
The acceleration is proportional to the displacement and the minus sign indicates that
the acceleration and the displacement are oppositely directed.
This relation is the defining characteristic of simple harmonic motion and can be used to
identify systems that will exhibit it:
The time it takes for a displaced object to execute a complete cycle of oscillatory
motion—from one extreme to the other extreme and back—is called the period T.
The reciprocal of the period is the frequency f, which is the number of cycles per unit of
time:
The unit of frequency is the cycle per second (cy/s), which is called a hertz (Hz).
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Chapter 14 - Oscillations
Position in SHM
A,, and  are constants
j.
k.
The maximum displacement
from equilibrium is called the amplitude A.
The argument of the cosine function,
, is called the phase of the motion, and
the constant δ is called the phase constant, which equals the phase at t = 0.
i.
iv.
Note that
thus, whether the
equation is expressed as a cosine function or a sine function simply depends on
the phase of the oscillation at t = 0.
If we have just one oscillating system, we can always choose t = 0 so that δ =
0.
If we have two systems oscillating with the same frequency but with different
phases, we can choose δ = 0 for one of them.
The equations for the two systems are then
v.
If the phase difference δ is 0 or an integer times 2
ii.
iii.
, then the systems are said
to be in phase. If the phase difference δ is or an odd integer times
the systems are said to be 180° out of phase.
l.
, then
Velocity and Acceleration.
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Chapter 14 - Oscillations
m. Taking it Further:
A problem from our past:
A particle’s position varies with time according to the expression y = (4 m) sin (t). Draw a motion
diagram making sure to indicate the points where the position, velocity, and acceleration are zero and at
their extreme values.
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Chapter 14 - Oscillations
n.
PROBLEM-SOLVING STRATEGY
i. Solving Simple Harmonic Motion Problems
ii. PICTURE Choose the origin of the x axis at the equilibrium position. For a
spring, choose the +x direction so that x is positive if the spring is extended.
iii. SOLVE Do not use the kinematic equations for constant acceleration. Instead,
use the equations developed for simple-harmonic motion.
iv. CHECK Make sure your calculator is in the appropriate mode (degrees or
radians) when evaluating trigonometric functions and their arguments.
o.
Example 14 – 1: Riding the Waves - You are sitting on a surfboard that is riding up and
down on some swells. The board’s vertical displacement y is given by
(a) Find the amplitude, angular frequency, phase constant, frequency, and period of the
motion.
(b) Where is the surfboard at t = 1.0 s?
(c) Find the velocity and acceleration as functions of time t
(d) Find the initial values of the position, velocity, and acceleration of the surfboard.
p.
Frequency and Amplitude.
i.
ii.
The fact that the frequency in simple harmonic motion is independent
of the amplitude has important consequences in many fields.
In music, for example, it means that when a note is struck on the
piano, the pitch (which corresponds to the frequency) does not depend
on how loudly the note is played (which corresponds to the amplitude).
If changes in amplitude had a large effect on the frequency, then
musical instruments would be unplayable.
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Chapter 14 - Oscillations
q.
Example 14 – 2: An Oscillating Object - An object oscillates with angular frequency ω =
8.0 rad/s. At t = 0, the object is at x = 4.0 cm with an initial velocity
= 25 cm/s. (a)
Find the amplitude and phase constant for the motion. (b) Write x as a function of time.
r.
Example 14 – 3: A Block on a Spring - A 2.00-kg block is attached to a spring as in
Figure 14-1. The force constant of the spring is k = 196 N/m. The block is held a
distance 5.00 cm from the equilibrium position and is released at t = 0. (a) Find the
angular frequency ω, the frequency f and the period T. (b) Write x as a function of time.
s.
Example 14 – 4: Speed and Acceleration of an Object on a Spring - Consider an object
on a spring whose position is given by
. (a) What is
the maximum speed of the object? (b) When does this maximum speed first occur after
t = 0? (c) What is the maximum of the acceleration of the object? (d) When does the
maximum of the magnitude of the acceleration first occur after t = 0?
t.
Practice Problem 14 – 1: A 0.80-kg object is attached to a spring that has a force
constant k = 400 N/m. (a) Find the frequency and period of motion of the object when it
is displaced from equilibrium and then released. (b) Repeat Part (a) except with a 1.6-kg
object attached to the spring in place of the 0.80-kg object. Hint: Review Example 14 4 first.
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Chapter 14 - Oscillations
u.
Simple Harmonic Motion and Circular Motion.
i. A relation exists between simple harmonic motion and circular motion with
constant speed.
ii.
iii.
iv.
Its x component of position describes simple harmonic motion.
Its x component of velocity describes the velocity of the simple harmonic
motion.
Equations:
v.
vi.
The speed of a particle moving in a circle is rω, where r is the radius.
1. For the particle in Figure 14-6b, r = A, so its speed is Aω.
2. The projection of the velocity vector onto the x axis gives
.
3. Substituting for v and
III.
gives
Energy in Simple Harmonic Motion.
a. When an object on a spring undergoes simple harmonic motion, the system’s potential
energy and kinetic energy vary with time.
b. Their sum, the total mechanical energy E = K + U, is constant.
c. Consider an object a distance x from equilibrium, acted on by a restoring force −kx.
i. The system’s potential energy is
ii.
For simple harmonic motion,
. Substituting gives
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Chapter 14 - Oscillations
d.
The kinetic energy of the system is
where m is the object’s mass and v is its speed.
e.
f.
For simple harmonic motion,
Substituting gives
g.
Then using
h.
The total mechanical energy E is the sum of the potential and kinetic energies:
i.
Because
j.
This equation reveals an important general property of simple harmonic motion:
i.
ii.
iii.
iv.
v.
.
,
For an object at its maximum
displacement, the total energy is all
potential energy.
As the object moves toward its
equilibrium position, the kinetic energy
of the system increases and its
potential energy decreases.
As the object moves through its
equilibrium position, the kinetic energy
of the object is maximum, the
potential energy of the system is zero,
and the total energy is kinetic.
As the object moves past the
equilibrium point, its kinetic energy
begins to decrease, and the potential
energy of the system increases until
the object again stops momentarily at
its maximum displacement (now in the
other direction).
At all times, the sum of the potential
and kinetic energies is constant.
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Chapter 14 - Oscillations
k.
More from our past:
i. The potential energy of an object constrained to the x axis is given by U(x) =
2
3
3x – 2x , where U is in joules and x is in meters. (a) Determine the force Fx
associated with this potential-energy function. (b) Assuming no other forces act
on the object, at what positions is this object in equilibrium? (c) Which of these
equilibrium positions are stable and which are unstable?
ii.
The potential energy of an object constrained to the x axis is given by U(x) =
2
4
8x – x , where U is in joules and x is in meters. (a) Determine the force Fx
associated with this potential—energy function. (b) Assuming no other forces
act on the object, at what positions is this object in equilibrium? (c) Which of
these equilibrium positions are stable and which are unstable?
iii.
The net force acting on an object constrained to the x axis is given by Fx (x) =
3
x – 4x (The force is in newtons and x in meters.) Locate the positions of
unstable and stable equilibrium. Show that each position is stable or unstable
by calculating the force one millimeter on either side of the locations.
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Chapter 14 - Oscillations
l.
In Figure 14-8, the potential energy U is
graphed as a function of x.
i. The total energy E is constant and
is therefore plotted as a horizontal
line.
ii. This line intersects the potentialenergy curve at x = A and x =
−A. At these two points, called the
turning points, oscillating
objects reverse direction and head
back toward the equilibrium
position.
iii. Because U ≤ E, the motion is
restricted to −A ≤ x ≤ +A.
m. Example 14 – 5: Energy and Speed of an Oscillating Object - A 3.0-kg object attached to
a spring oscillates with an amplitude of 4.0 cm and a period of 2.0 s. (a) What is the
total energy? (b) What is the maximum speed of the object? (c) At what position x1 is
the speed equal to half its maximum value?
n.
Practice Problem 14 – 2: Calculate ω for this example and find
from
= ωA.
o.
Practice Problem 14 – 3: An object of mass 2.00 kg is attached to a spring that has a
force constant 40.0 N/m. The object is moving at 25.0 cm/s when it is at its equilibrium
position. (a) What is the total energy of the object? (b) What is the amplitude of the
motion?
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Chapter 14 - Oscillations
IV.
Some Oscillating Systems.
a. The Vertical Spring.
i. When a spring-mass system is hung vertically, two things happen.
1. Gravity is acting on the mass. Therefore, the total force acting on the
mass is F = -kx + mg.
2.
a. This seems to destroy the SHM condition (F = -kx).
b. The gravity force is not proportional to the displacement.
The equilibrium position of the mass shifts downward. Through careful
examination, gravity can be eliminated from the equation to re-satisfy
the SHM condition.
ii.
The spring has a period T = 2 √(k/m).
iii.
Example 14 – 6: Paper Springs - You are showing your nieces how to make
paper party decorations using paper springs. One niece makes a paper spring.
The spring is stretched 8 cm and has a single sheet of colored paper suspended
from it. You want the decorations to bounce at approximately 1.0 cy/s. How
many sheets of colored paper should be used for the decoration on that spring
if it is to bounce at 1.0 cy/s?
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Chapter 14 - Oscillations
b.
iv.
PRACTICE PROBLEM 14-4 How much is the paper spring stretched when a
decoration made from three sheets of paper is suspended from it and the paper
is in equilibrium?
v.
Example 14 – 7: A Bead on a Block - A block
securely attached to a spring oscillates vertically
with a frequency of 4.00 Hz and an amplitude of
7.00 cm. A tiny bead is placed on top of the
oscillating block just as it reaches its lowest
point. Assume that the bead’s mass is so small
that its effect on the motion of the block is
negligible. At what displacement from the
equilibrium position does the bead lose contact
with the block?
The Simple Pendulum.
i. The motion of a simple pendulum is a
common repetitive motion consisting of a
point mass m attached to a pivot by a light
spring of length L.
ii. When the mass (bob) is released from its
angle with the vertical, it swings back and
forth with some period T = 2√(L/g).
iii. The motion must meet the SHM criterion:
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Chapter 14 - Oscillations
iv.
Practice Problem 14 – 5: Find the period of a simple pendulum of length 1.00 m
undergoing small oscillations.
c.
The Torsional Oscillator.
i. A system that undergoes rotational
oscillations in a variation of simpleharmonic motion is called a torsional
oscillator.
ii. The motion must meet the SHM criterion:
d.
The Physical Pendulum.
i. A rigid object free to rotate about a horizontal
axis that is not through its center of mass will
oscillate when displaced from equilibrium.
Such a system is called a physical
pendulum.
ii. The period of a physical pendulum depends on
the distribution of the mass, but not on the
total mass M. The moment of inertia I is
proportional to M, so the ratio I/M is
independent of M.
iii. Analyzing the SHM:
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Chapter 14 - Oscillations
iv.
Example 14 – 10: A Comfortable Pace - You claim that the
pace of a comfortable walk can be predicted if we model
each leg as a physical pendulum. Your teacher is skeptical
about this claim and asks you to back it up. Is your claim
correct?
v.
Example 14 – 11: A Swinging Rod - A uniform rod of
mass M and length L is free to swing about a horizontal
axis perpendicular to the rod and a distance x from the
rod’s center. Find the period of oscillation for small
angular displacements of the rod.
vi.
PRACTICE PROBLEM 14-7 Show that the step-3 expression for the period
gives the same period for x = L/6 as for x = L/2.
vii.
Example 14 – 12: The Swinging Pendulum Revisited - Find the value of x in
Example 14-11 for which the period is a minimum.
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Chapter 14 - Oscillations
V.
Damped Oscillations.
a. Left to itself, a spring or a pendulum eventually stops oscillating because the mechanical
energy is dissipated by frictional forces.
b. Such motion is said to be damped.
i. If the damping is large enough, as, for example, a pendulum submerged in
molasses, the oscillator fails to complete even one cycle of oscillation.
1. Instead, it just moves toward the equilibrium position with a speed
that approaches zero as the object approaches the equilibrium
position.
2. This type of motion is referred to as overdamped.
ii. If the damping is small enough that the system oscillates with an amplitude
that decreases slowly with time—like a child on a playground swing when a
parent stops providing a push each cycle—the motion is said to be
underdamped.
iii. Motion with the minimum damping for nonoscillatory motion is said to be
critically damped. (With any less damping, the motion would be
underdamped.)
c. Undamped motion.
i.
ii.
iii.
iv.
The damping force exerted on an oscillator such as the one shown in
Figure 14-21a can be represented by the empirical expression
where b is a constant. THIS IS SIMILAR TO TERMINAL VELOCITY.
Such a system is said to be linearly damped.
The discussion here is for linearly damped motion. Because the damping force
is opposite to the direction of motion, it does negative work and causes the
mechanical energy of the system to decrease. This energy is proportional to the
square of the amplitude, and the square of the amplitude decreases
exponentially with increasing time. That is,
−1
The time constant is the time for the energy to change by a factor of e .
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Chapter 14 - Oscillations
d.
v.
The motion of a damped system can be obtained from Newton’s second law:
vi.
Example 14 – 13: Sprung Mass of a Passenger Car - The sprung mass of an
automobile is the mass that is supported by the springs. (It does not include
the mass of the wheels, axles, brakes, and so on.) A passenger car has a
sprung mass of 1100 kg and an unsprung mass of 250 kg. If the four shock
absorbers are removed, the car bounces up and down on its springs with a
frequency of 1.0 Hz. What is the damping constant provided by the four shocks
if the car, with shocks, is to return to equilibrium as quickly as possible without
passing it after hitting a speed bump?
Because the energy of an oscillator is proportional to the square of its amplitude, the
energy of an underdamped oscillator (averaged over a cycle) also decreases
exponentially with time:
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Chapter 14 - Oscillations
e.
A damped oscillator is often described by its Q factor (for quality factor),
f.
The Q factor is dimensionless. (Because ω0 has dimensions of reciprocal time,
without dimension.) We can relate Q to the fractional energy loss per cycle.
g.
If the damping is weak so that the energy loss per cycle is a small fraction of the energy
E, we can replace dE by ΔE and dt by the period T. Then |ΔE|/E in one cycle (one
period) is given by
h.
Example 14 – 14: Making Music - When middle C on a piano (frequency 262 Hz) is
is
struck, it loses half its energy after 4.00 s. (a) What is the decay time ? (b) What is
the Q factor for this piano wire? (c) What is the fractional energy loss per cycle?
i.
You can estimate
and Q for various oscillating systems.
i. Tap a crystal wine glass and see how long it rings.
ii.
j.
The longer it rings, the greater the value of
and Q and the lower the
damping.
iii. Glass beakers from the laboratory may also have a high Q. Try tapping a plastic
cup.
In terms of Q, the exact frequency of an underdamped oscillator is
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Chapter 14 - Oscillations
VI.
k.
The power dissipated by the damping force equals the instantaneous rate of change of
the total mechanical energy
l.
For a weakly damped oscillator with linear damping, the total mechanical energy
decreases slowly with time. The average kinetic energy per cycle equals half the total
energy
Driven Oscillations and Resonance.
a. To keep a damped system going indefinitely, mechanical energy must be put into the
system. When this is done, the oscillator is said to be driven or forced.
i. If the driving mechanism puts energy into the system at a greater rate than it
is dissipated, the system’s mechanical energy increases with time, and the
amplitude increases.
ii. If the driving mechanism puts energy in at the same rate it is being dissipated,
the amplitude remains constant over time.
iii. The motion of the oscillator is then said to be steady-state motion.
iv. In the steady state, the energy put into the system per cycle by the driving
force equals the energy dissipated per cycle due to the damping.
b. The amplitude, and therefore the energy, of a system in the steady state depends not
only on the amplitude of the driving force, but also on its frequency.
i. The natural frequency of an oscillator, ω0,
is its frequency when no driving or damping
forces are present.
ii. If the driving frequency is sufficiently close
to the natural frequency of the system, the
system will oscillate with a relatively large
amplitude.
iii. This phenomenon is called resonance.
iv. When the driving frequency equals the
natural frequency of the oscillator, the
energy per cycle transferred to the oscillator
is maximum.
v. The natural frequency of the system is thus
called the resonance frequency.
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Chapter 14 - Oscillations
c.
There are many familiar examples of resonance.
i. When you sit on a swing, you learn intuitively to pump with the same frequency
as the natural frequency of the swing.
ii. Many machines vibrate because they have rotating parts that are not in perfect
balance. (Observe a washing machine in the spin cycle, for example.)
1. If such a machine is attached to a structure that can vibrate, the
structure becomes a driven oscillatory system that is set in motion by
the machine.
2. Engineers pay great attention to balancing the rotary parts of such
machines, damping their vibrations, and isolating them from building
supports.
iii. A crystal goblet with weak damping can be broken by an intense sound wave at
a frequency equal to or very nearly equal to the natural frequency of vibration
of the goblet. The breaking of the goblet is often done in physics
demonstrations using an audio oscillator, a loudspeaker and an amplifier.
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