YF15e_CH14_ADA_PPT_LectureOutline (1).pptx

University Physics with Modern Physics
Fifteenth Edition, Global Edition, in SI Units
Chapter 14
Periodic Motion
Copyright © 2020 Pearson Education Ltd. All Rights Reserved

Learning Outcomes
In this chapter, you’ll learn…
• 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.

Introduction
• 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? (1 of 4)
• If an object 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
• The amplitude, A, is the maximum magnitude of
displacement from equilibrium.
• The period, T, is the time for one cycle.
• The frequency, f, is the number of cycles per unit
time.
• The angular frequency, ω, is 2π times the frequency:
2 .
f
 


• ω is a useful quantity. It represents the rate of change
of an angular quantity (not necessarily related to a
rotational motion) that is always measured in radians,
so its units are rad/s. since f is in cycle/s, we may
regard the number 2π as having units rad/cycle.
• The frequency and period are reciprocals of each
other:
 
1 1
and .
f T
T f

• Write equation which relates angular frequency with
period?

Example 14.1
• An ultrasonic transducer used for medical diagnosis
oscillates at 6.7 MHz. How long does each oscillation
take and what is the angular frequency?

Simple Harmonic Motion (SHM) (1 of 3)
• The simplest kind of
oscillation occurs when the
restoring force Fx is directly
proportional to the
displacement from
equilibrium x.

proportional to the
displacement from
equilibrium x. This happens
if the spring system is an
ideal one that obeys
Hooke’s law

proportional to the
displacement from
equilibrium x. This happens
if the spring system is an
ideal one that obeys
Hooke’s law
We are assuming that
there is no friction, so Eq. (14.3)
gives the net force on the body.

• When the restoring force
is directly proportional
to the displacement from
equilibrium, the resulting
motion is called simple
harmonic motion
(SHM).

• Derive the acceleration, ax of a body in simple
harmonic motion.

• Derive the acceleration, ax of a body in simple
harmonic motion.
- The minus sign means that, in SHM, the acceleration and displacement always
have opposite signs.
- This acceleration is not constant.
- A body that undergoes simple harmonic motion is called a harmonic oscillator.

• 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.

Why is simple harmonic motion
important?
 Not all periodic motions are simple harmonic; in periodic motion in general,
the restoring force depends on displacement in a more complicated way than in
Eq. (14.3).
 But 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 of such systems are approximately simple
harmonic and therefore approximately described by Eq. (14.4).
 Thus we can use SHM as an approximate model for many different periodic
motions:
 such as the vibration of a tuning fork,
 the electric current in an alternating-current circuit,
 the oscillations of atoms in molecules and solids.

Simple Harmonic Motion Viewed as a
Projection (1 of 2)

Simple Harmonic Motion Viewed as a
Projection (2 of 2)
• The circle in which the ball
moves so that its projection
matches the motion of the
oscillating object is called the
reference circle.
• 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.

• The x-component of the phasor at time t is
just the x-coordinate of the point Q:
• x = A cos θ
• This is also the x-coordinate of the shadow
P, which is the projection of Q onto the x-
axis.
• Hence the x-velocity of the shadow P along
the x-axis is equal to the x-component of
the velocity vector of point Q.

• The x-acceleration of P is equal to the
x-component of the acceleration vector
of Q.
• Since point Q is in uniform circular
motion, its acceleration vector
• aQ is always directed toward O.
• the magnitude of aQ is constant and
given by the angular speed squared
times the radius of the circle
• aQ = 2A

• Find:
• 1. The x-component of aQ.
• 2. Using equation, x = A cos θ and aQ
= 2A, find the acceration of point P.
•

Characteristics of SHM (1 of 2)
• For an object of mass m vibrating by an ideal spring with a
force constant k:
• Video Tutor Solution: Example 14.2

Example 14.2
• A spring is mounted horizontally, with its left end fixed. A spring balance
attached to the free end and pulled toward the right (Fig. a) 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. b).
• (a) Find the force constant k of the spring.
• (b) Find the angular frequency v, frequency f, and period T of the resulting
oscillation.

Characteristics of SHM (2 of 2)
• 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 (1 of 5)
• The displacement as a function of time for SHM is:

in SHM (2 of 5)
• Increasing m with the same A and k increases the
period of the displacement vs time graph.

in SHM (3 of 5)
• Increasing k with the same A and m decreases the
period of the displacement vs time graph.

in SHM (4 of 5)
• Increasing A with the same m and k does not change
the period of the displacement vs time graph.

in SHM (5 of 5)
• Increasing ϕ with the same A, m, and k only shifts the
displacement vs time graph to the left.

Graphs of Displacement and Velocity
for SHM

Graphs of Displacement and
Acceleration for SHM

Energy in SHM
• The total mechanical energy E = K + U is conserved in
SHM:
2 2 2
1 1 1
constant
2 2 2
x
E mv kx kA
   
• Video Tutor Solution: Example 14.4

Energy Diagrams for SHM (1 of 2)
• The potential energy U and total mechanical energy E
for an object in SHM as a function of displacement x.

Energy Diagrams for SHM (2 of 2)
• The potential energy U, kinetic energy K, and total
mechanical energy E for an object in SHM as a
function of displacement x.

Vertical SHM (1 of 2)
• If an object oscillates vertically from a spring, the restoring
force has magnitude kx. Therefore the vertical motion is
SHM.

Vertical SHM (2 of 2)
• If the weight mg compresses the spring a distance Δl,
the force constant is 

.
mg
k
l

Angular SHM
• A coil spring exerts a restoring torque , where
z k k
 
 
is called the torsion constant of the spring.
• The result is angular simple harmonic motion.

Potential Energy of a Two Atom
System

Vibrations of Molecules
• Shown are two atoms having centers a distance r apart,
with the equilibrium point at r = R0.
• If they are displaced a small distance x from equilibrium,
the restoring force is approximately
 
  
 
0
2
0
72
r
U
F x
R
• So  0
2
0
72
,
U
k
R
and the motion is SHM.

The Simple Pendulum
• A simple pendulum
consists of a point mass
(the bob) suspended by a
massless, unstretchable
string.
• If the pendulum swings
with a small amplitude θ
with the vertical, its motion
is simple harmonic.

The Physical Pendulum
• A physical pendulum is any
real pendulum that uses an
extended object instead of a
point-mass bob.
• For small amplitudes, its
motion is simple harmonic.
• Video Tutor Solution:
Example 14.9

Tyrannosaurus Rex and the Physical
Pendulum
• We can model the leg of Tyrannosaurus rex as a
physical pendulum.

Damped Oscillations
• Real-world systems
have some dissipative
forces that decrease
the amplitude.
• The decrease in
amplitude is called
damping and the
motion is called
damped oscillation.

Forced Oscillations and
Resonance (1 of 2)
• A damped oscillator left to itself will eventually stop moving.
• But we can maintain a constant-amplitude oscillation by
applying a force that varies with time in a periodic way.
• We call this additional force a driving force.
• If we apply a periodic driving force with angular frequency ωd
to a damped harmonic oscillator, the motion that results is
called a forced oscillation or a driven oscillation.

Forced Oscillations and
Resonance (2 of 2)
• This lady beetle flies by means of a forced oscillation.
• Unlike the wings of birds, this insect’s wings are
extensions of its exoskeleton.
• Muscles attached to the inside of the exoskeleton apply a
periodic driving force that deforms the exoskeleton
rhythmically, causing the attached wings to beat up and
down.
• The oscillation frequency of
the wings and exoskeleton is
the same as the frequency of
the driving force.

YF15e_CH14_ADA_PPT_LectureOutline (1).pptx

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