Module 1
PERIODIC MOTION
Learning Goals for Chapter 14
Looking forward at …
• how to describe oscillations in terms of
amplitude, period, frequency, and angular
frequency.
• how to apply the ideas of simple harmonic
motion to different physical situations.
• how to analyze the motions of a pendulum.
• what determines how rapidly an oscillation
dies out.
• how a driving force applied to an oscillator
at a particular frequency can cause a very
large response, or resonance.
What would happen if you doubled a pendulum’s
mass?
• Why do dogs walk faster than humans?
• Does it have anything to do with the
characteristics of their legs?
• Many kinds of motion (such as a pendulum,
musical vibrations, and pistons in car
engines) repeat themselves. We call such
behavior periodic motion or oscillation.
What causes periodic motion?
• If a body attached to a spring is displaced
from its equilibrium position, the spring
exerts a restoring force on it, which tends
to restore the object to the equilibrium
position.
• This force causes oscillation of the system,
or periodic motion.
Characteristics of periodic motion?
Period, T
• Time for one cycle
• Unit is second (s)
Frequency, f
• Number of cycles per unit time
• Unit is Hz (1 Hz = 1 cycle/second = 1 s-1)
1
𝑓=
𝑇
1
𝑇=
𝑓
Characteristics of periodic motion?
Angular frequency, w
• 2p times the frequency
SIMPLE HARMONIC MOTION
When the restoring
force is directly
proportional to the
displacement from
equilibrium,
the
resulting motion is
called
simple
harmonic motion
(SHM).
SIMPLE HARMONIC MOTION
• In many systems the restoring force is
approximately
proportional
to
displacement if the displacement is
sufficiently small.
• That is, if the amplitude is small enough,
the oscillations are approximately simple
harmonic.
SIMPLE HARMONIC MOTION
Simple harmonic motion viewed as
a projection
Simple harmonic motion viewed as
a projection
• The circle in which the ball moves so
that its projection matches the motion of
the oscillating body is called the
reference circle.
Simple harmonic motion viewed as
a projection
• As point Q moves around the reference
circle with constant angular speed,
vector OQ rotates with the same angular
speed. Such a rotating vector is called a
phasor
Characteristics of SHM
For a body of mass m vibrating by an ideal
spring with a force constant k:
Characteristics of SHM
The greater the
mass m in a
tuning
fork’s
tines, the lower
the frequency of
oscillation, and
the lower the
pitch
of
the
sound that the
tuning
fork
produces.
Displacement as a function of time
in SHM
The displacement as a function of time for
SHM is:
Displacement as a function of time
in SHM
Phase angle 𝜙 – tells the point in the cycle
the motion was at 𝑡 = 0 (equivalent to
where around the circle the point 𝑄 was at
𝑡 = 0)
𝑥! = 𝐴𝑐𝑜𝑠𝜙
Displacement as a function of time
in SHM
Amplitude in SHM
𝐴=
"
𝑣
!#
"
𝑥! + "
𝜔
Displacement as a function of time
in SHM
• Increasing m with the same A and k
increases the period of the displacement
vs time graph.
Displacement as a function of time
in SHM
• Increasing k with the same A and m
decreases
the
period
of
the
displacement vs time graph.
Displacement as a function of time
in SHM
• Increasing A with the same m and k
does not change the period of the
displacement vs time graph
Displacement as a function of time
in SHM
• Increasing 𝜙 with the same 𝐴, 𝑚, and 𝑘
only shifts the displacement vs time
graph to the left
Graphs of displacement
velocity for SHM
and
Graphs of displacement
acceleration for SHM
and
EXAMPLE 1
In a physics lab, you attach a 0.200-kg airtrack glider to the end of an ideal spring of
negligible mass and start it oscillating. The
elapsed time from when the glider first
moves through the equilibrium point to the
second time it moves through that point is
2.60 s. Find the spring’s force constant.
EXAMPLE 2
An ultrasonic transducer used for medical
diagnosis oscillates at 6.7 x 106 Hz. How
long does each oscillation take, and what
is the angular frequency?
EXAMPLE 3
A spring is mounted horizontally, with its
left end fixed. A spring balance attached to
the free end and pulled toward the right
(Fig. 3a) indicates that the stretching force
is proportional to the displacement, and a
force of 6.0 N causes a displacement of
0.030 m. We replace the spring balance
with a 0.50-kg glider, pull it 0.020 m to the
right along a frictionless air track, and
release it from rest (Fig. 3b). (a) Find the
force constant k of the spring. (b) Find the
angular frequency, frequency ƒ, and period
T of the resulting oscillation.
EXAMPLE 3 (SHM)
EXAMPLE 4 (SHM)
We give the glider of Example 3 an initial
displacement xo = + 0.015 m and an initial
velocity vox = +0.40 m/s. (a) Find the
period, amplitude, and phase angle of the
resulting motion. (b) Write the equations for
the
displacement,
velocity,
and
acceleration as a function of time.
ENERGY in SHM
The total mechanical energy 𝐸 = 𝐾 + 𝑈 is
conserved in SHM:
1
1 " 1 "
"
𝐸 = 𝑚𝑣# + 𝑘𝑥 = 𝑘𝐴 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
2
2
2
Energy Diagrams in SHM
The potential energy 𝑈
and total
mechanical energy 𝐸 for a body in SHM as
a function of displacement 𝑥.
Energy Diagrams in SHM
The potential energy 𝑈, kinetic energy 𝐾,
and total mechanical energy 𝐸 for a body
in SHM as a function of displacement 𝑥.
EXAMPLE 1
A small block is attached to an ideal spring
and is moving in SHM on a horizontal,
frictionless surface. The amplitude of the
motion is 0.120 m. The maximum speed of
the block is 3.90 m/s. What is the
maximum magnitude of the acceleration of
the block?
EXAMPLE 2
A tuning fork labeled 392 Hz has the tip of
each of its two prongs vibrating with an
amplitude of 0.600 mm. (a) What is the
maximum speed of the tip of a prong? (b) A
housefly (Musca domestica) with mass
0.0270 g is holding onto the tip of one of
the prongs. As the prong vibrates, what is
the fly’s maximum kinetic energy? Assume
that the fly’s mass has a negligible effect
on the frequency of oscillation.
EXAMPLE 3
A small block is attached to an ideal spring
and is moving in SHM on a horizontal,
frictionless surface. The amplitude of the
motion is 0.250 m and the period is 3.20 s.
What are the speed and acceleration of the
block when x = 0.160 m?
Applications of SHM
Vertical SHM
• If a body oscillates vertically from a
spring, the restoring force has
magnitude 𝑘𝑥 . Therefore the vertical
motion is SHM.
Vertical SHM
• If the weight mg compresses the spring
a distance ∆𝑙, the force constant is 𝑘 =
𝑚𝑔/∆𝑙
Angular SHM
• A coil spring exerts a restoring torque
𝜏" = −𝜅𝜃, where 𝜅 is called the torsion
constant of the spring.
• The result is
angular simple
harmonic
motion.
Angular SHM
• A coil spring exerts a restoring torque
𝜏" = −𝜅𝜃, where 𝜅 is called the torsion
constant of the spring.
• The result is
angular simple
harmonic
motion.
The Simple Pendulum
• consists of a point mass (the bob)
suspended by a massless, unstretchable
string.
• If the pendulum
swings with a
small amplitude
with the vertical,
its motion is
simple harmonic
The Simple Pendulum
Angular frequency w of a
pendulum with small amplitude
simple
The corresponding frequency and period
relationships are
EXAMPLE 1
Find the period and frequency of a simple
pendulum 1.000 m long at a location where
𝑔 = 9.80 m/s2.
EXAMPLE 2
A building in San Francisco has light
fixtures consisting of small 2.35-kg bulbs
with shades hanging from the ceiling at the
end of light, thin cords 1.50 m long. If a
minor earthquake occurs, how many
swings per second will these fixtures
make?
EXAMPLE 3
A Pendulum on Mars. A certain simple
pendulum has a period on the earth of 1.60
s. What is its period on the surface of
Mars, where 𝑔 = 3.71 m/s2.
The Physical Pendulum
• A physical pendulum is any real
pendulum that uses an extended body
instead of a point-mass bob.
• For small amplitudes,
its motion is simple
harmonic.
𝐼
𝑇 = 2𝜋
𝑚𝑔𝑑
𝜔=
𝑚𝑔𝑑
𝐼
Recall on Moment of Inertia
• depends on how the body’s mass is
distributed in space
Recall on Parallel Axis
Theorem
EXAMPLE 1
We want to hang a thin hoop on a
horizontal nail and have the hoop make
one complete small-angle oscillation each
2.0 s. What must the hoop’s radius be?
EXAMPLE 2
A 1.80-kg connecting rod from a car engine is
pivoted about a horizontal knife edge as
shown in the figure. The center of gravity of
the rod was located by balancing and is 0.200
m from the pivot. When the rod is set into
small-amplitude oscillation
it makes 100 complete
swings in 120 s. Calculate
the moment of inertia of
the rod about the rotation
axis through the pivot.
EXAMPLE 3
A holiday ornament in the shape of a
hollow sphere with mass M = 0.015 kg and
radius R = 0.050 m is hung from a tree
limb by a small loop of wire attached to the
surface of the sphere. If the ornament is
displaced a small distance and released, it
swings back and forth as a physical
pendulum with negligible friction. Calculate
its period. (Hint: Use the parallel-axis
theorem to find the moment of inertia of the
sphere about the pivot at the tree limb.)
Damped Oscillations
• Damping – decrease in amplitude
caused by dissipative forces and the
corresponding motion is called damped
oscillations
𝐹! = −𝑏𝑣!
Force on a body due to friction
! 𝐹! = − 𝑘𝑥 − 𝑏𝑣!
Damped Oscillations
• Motion if the damping force is small
#
" $% &
'
𝑥 = 𝐴𝑒
cos(𝜔 𝑡 + 𝜙)
(Oscillator with little damping)
• The angular frequency of oscillations 𝜔#
𝜔' =
𝑘
𝑏$
−
𝑚 4𝑚$
(Oscillator with little damping)
Damped Oscillations
• Critical damping
$
𝑏
𝑘
−
=0
$
𝑚 4𝑚
or
𝑏 = 2 𝑘𝑚
The system no longer oscillates but returns to
its equilibrium position without oscillation when
it is displaced and released.
Damped Oscillations
• Overdamping
𝑏 > 2 𝑘𝑚
There is no oscillation, but the system returns
to equilibrium more slowly than with critical
damping.
Damped Oscillations
• Underdamping
𝑏 < 2 𝑘𝑚
The system oscillates
decreasing amplitude.
with
steadily
Damped Oscillations
𝑏 < 2 𝑘𝑚
𝑏 = 2 𝑘𝑚
𝑏 > 2 𝑘𝑚
Energy in Damped Oscillations
𝑑𝐸
$
= 𝑣! −𝑏𝑣! = −𝑏𝑣!
𝑑𝑡
rate at which the
damping force does
(negative) work on the
system (that is, the
damping power).
Forced Oscillations
• periodically varying driving force with
angular frequency 𝜔+ to a damped
harmonic oscillator
• driven oscillation
𝐴=
𝐹%(!
𝑘 − 𝑚𝜔)$ $ + 𝑏 $ 𝜔)$
(amplitude of a driven oscillator)
Damping and Force Oscillations
https://www.youtube.com/watch?v=NF75zls5PxE
Resonance
• phenomenon that occurs when an
object or system is subjected to an
external force or vibration that matches
its natural frequency
Resonance
Examples:
EXAMPLE
A 50.0-g hard-boiled egg moves on the
end of a spring with force constant 𝑘 =
25.0 N/m. Its initial displacement is 0.300
m. A damping force 𝐹, = −𝑏𝑣, acts on the
egg, and the amplitude of the motion
decreases to 0.100 m in 5.00 s. Calculate
the magnitude of the damping constant 𝑏.
EXAMPLE
A mass is vibrating at the end of a spring of
force constant 225 N/m. The figure shows
a graph of its position 𝑥 as a function of
time 𝑡. (a) At what times is the mass not
moving? (b) How much energy did this
system originally contain? (c) How much
energy did the
system lose
between 𝑡 = 1.0 s
and 𝑡 =
4.0 s. Where did
this energy go?