Physics 1321 Chapter 14 Lecture Slides.pdf

PowerPoint® Lectures for
University Physics, 14th Edition
– Hugh D. Young and Roger A. Freedman Lectures by Jason Harlow
Periodic Motion
Chapter 14
© 2016 Pearson Education Inc.

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.

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?
• 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.

What causes 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, is 2π times the frequency:
• The frequency and period are reciprocals of each other:
f = 1/T and T = 1/f.

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

Simple harmonic motion (SHM)
• 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 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.
• 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:

• Increasing m with the same A and k increases the period of
the displacement vs time graph.

• Increasing k with the same A and m decreases the period of
the displacement vs time graph.

• Increasing A with the same m and k does not change the
period of the displacement vs time graph.

• 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:
E = 1/2 mvx
2 + 1/2 kx2 = 1/2 kA2 = constant

Energy diagrams for SHM
• The potential energy U and total mechanical energy E for a
body in SHM as a function of displacement x.

Energy diagrams for SHM
• The potential energy U, kinetic energy K, and total
mechanical energy E for a body in SHM as a function of
displacement x.

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 Δl, the
force constant is k = mg/Δl .

Angular SHM
• A coil spring exerts a restoring torque 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
Fr = –(72U0/R0
2)x
• So k = 72U0/R0
2, 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 with the
vertical, its motion is simple
harmonic.

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.

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
• 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
• 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.

Physics 1321 Chapter 14 Lecture Slides.pdf

More Related Content

Similar to Physics 1321 Chapter 14 Lecture Slides.pdf

More from MarlonAliceLopezMagh

Recently uploaded

Physics 1321 Chapter 14 Lecture Slides.pdf