1. No category

Chapter #14 Oscillatory Motion and
Chapter #15 Wave Motion
• 
Usual Examples of Periodic Motion
•  Motion which repeats
themselves over and
over called periodic
motion or oscillation.
Physical Quantities for Periodic
Motion
•  Amplitude: the maximum
magnitude of displacement
from equilibrium.
•  Cycle: one complete round
trip
•  Period: the time for one
whole cycle.
•  Frequency: number of cycles
in a unit of time.
•  Angular velocity: ω=2πf
1
Units
•  Amplitude is the
maximum magnitude of
displacement from
equilibrium. Usual
units [m].
•  Frequency, f, is
number of cycles in a
unit of time. Unit [Hz].
1
[Hz],
T
1 Hz = 1 cycle /second
f =
•  The period, T, is the
time for one whole
cycle. Unit [s].
1 kHz = 10 3 Hz
•  Angular frequency,
ω, is 2π times the
frequency:
ω=
1 MHz = 10 6 Hz
1 GHz = 10 9 Hz
2π
1
= 2πf [ ]
T
s
€
€
A spring - example of simple
harmonic motion
•  A spring stretched or compressed
by the amount x from its
equilibrium length exerts a force
given by:
F = − k x, a = −
k
d2 x
k
x,
+
x =0
m
d t2
m
•  Periodic motion under the action of a restoring force
that is directly proportional to the displacement is
€
called simple harmonic motion.
Equations of simple harmonic motion
•  While the ball moves in uniform circular
motion the shadow moves in simple
harmonic motion
x = A sin (ω t + Φ) ,
v = + ω Acos (ω t + Φ)
a = − ω 2 Asin (ω t + Φ)
x = A cos (ω t + Φ) ,
€
v = − ω Asin (ω t + Φ)
a = − ω 2 Acos (ω t + Φ)
€
2
The period of a mass on a spring
ma = m
d 2x
=−kx
d t2
#2π &
x = A cos%
t ( = A cos(ω t )
$ T '
€
v = − A ω sin(ω t )
€
a = − Aω 2 cos(ω t ) = − ω 2 x
− mω 2 x = − k x ⇒ ω 2 =
€
k
m
⇒ T = 2π
m
k
€
€
What type of motion is this?
•  Each picture represents deflection from the equilibrium position.
Problems
•  You observe 74 heartbeats in a minute. What
are the period and frequency of your
heartbeat?
•  Show that the units of the quantity (k/m)1/2 is
seconds.
3
Simple Pendulum
F = − m g sin(θ )
mg
x, for small θ
L
mg L
g
=
m
L
F ≈ − mgθ = −
ω=
k
=
m
T = 2π
L
g
• 
€
Problem
•  Calculate period of
oscillation of
Foucault pendulum
of length 67 [m].
Foucault Pendulum
• 
4
Problem
•  Calculate period of
oscillation of torsional
pendulum, which
consists of solid disk of
mass M and radius R,
suspended by a wire.
When disk is twisted
through angle θ, a wire
exerts on an object a
restoring force:
τ = −κθ
€
Energy in harmonic motion
•  We can get it
directly from the
equation of motion.
1
1
E = K + U = mv2 + k x2
2
2
1
1
1
k A2 = m v 2 + k x 2
2
2
2
k
v =±
A2 − x 2
m
€
5
Damped Oscillations
d2 x
+ ω 02 x = 0, ω 0 =
d t2
€
k
, x = A cos(ω 0 t )
m
•  Each the real system has energy lost. That is why
energy of the system decreases and amplitude of
oscillation decreases as well. We called such
oscillation damped oscillation.
d2 x
dx
+ ω 02 x = − b
, x = Ae−bt / 2m cos(ω t )
d t2
dt
ω = ω 02 −
b2
4
€
Resonance
•  Resonance is a condition in which a vibrating
system responds with maximum amplitude to
an alternating driving force.
d2 x
dx
+ ω 02 x = − b
+ f 0 sin(ω t )
d t2
dt
x = A cos(ω t + Φ)
A=
f0
(ω
2
−ω
2 2
0
) + (bω /m)
2
€
• Tacoma
Bridge
6
Wave definition
•  A mechanical wave is a disturbance
that travels through some material
or substance called the medium for
the wave.
•  All mechanical waves require
a medium Transverse waves and Longitudinal
waves
•  Longitudinal wave
•  Transverse wave
Definitions
•  In a longitudinal wave, the displacement
of individual particle is in the same
direction as the direction of propagation
of the wave.
•  In a transverse wave, the displacement
of individual particle is at right angles to
the direction of propagation of the wave.
7
Three examples
• 
Types of waves
• 
• 
• 
• 
Periodic
Nonperiodic
Sinusoidal (harmonic)
Nonsinusoidal
How you can describe these
waves?
• 
8
Amplitude, Period, Frequency, and
Wavelength of periodic wave
•  The amplitude is the pick value of an
alternating physical quantity, which
represents the wave.
•  The period is the time interval required
for two identical wave points to pass by
a given point.
•  The frequency is number of crests (or
troughs) that pass a given point in a unit
of time interval.
•  The wavelength is the minimum
distance between two identical points
(crests or troughs).
Wavelength and period
•  The wavelength, λ, is the
minimum distance between two
identical points (crests or
troughs). Dimension - [m].
•  The period, T, is the time
interval required for two
identical wave points to pass by
a given point. Dimension - [s].
• 
Frequency
•  The frequency, f, of a periodic wave is
number of crests (or troughs) that pass
a given point in a unit of time interval.
f =
1
T
[s−1 ] = [Hz]
€
9
Wave Speed
•  Wave travels a distance of one wave
length in one period. That is why wave
speed is:
v=
λ
=λ f
T
[m /s]
€
The speed of a wave on a string
•  For a string with a
tension force F and a
mass per unit length
µ[kg/m] is:
v=
F
µ
[m /s]
€
Normal Modes of a String
λn =
2L
, ⇒
n
fn =
v n F
=
λn 2 L µ
€
• 
10
Wave Equation and its solutions
2
∂2y
2 ∂ y
=v
∂ t2
∂ x2
) #
y(x,t) = Asin +ω% t +
* $€
* $
y(x,t) = Asin ,ω& t −
+ %
x ')/
v (.
x &,
(.
v '-
€
• 
€
Mathematical Description of a Wave
* $
*
$
x 'x 'y(x,t) = Asin ,ω& t − )/ = Asin ,2π f & t − )/
% v (.
+ % v (.
+
+ % t x (.
y(x,t) = Asin -2π ' − *0
, & T λ )/
€
+ % x (.
+
y(x,0) = Asin -2π ' − *0 = − Asin -2π
,
, & λ )/
€
€
• 
x.
λ 0/
$ x'
y(0,t) = Asin &2π )
% λ(
€
Traveling Wave
* $
y(x,t) = Asin ,ω& t −
+ %
x ')/
v (.
€
• 
11
Two harmonic
waves traveling
in opposite
directions
) #
y(x,t) = Asin +ω% t +
* $
) #
x &,
(. + Asin +ω% t −
v '* $
# x&
2Asin (ωt ) ⋅cos% ω (
$ v'
x &,
(. =
v '-
• 
€
Standing Waves
λ
L = n , n =1, 2, 3, 4,…
2
• 
€
Example 15.3
•  What is the speed of a
transverse wave on the
rope?
•  What is the wavelength of
a transverse wave with
frequency 2 [Hz]?
12
• 
Latitude and longitude
•  Latitude - the angular distance of a place
north or south of the earth's equator. •  Longitude - the angular distance of a place
east or west of the meridian at Greenwich,
England.
13
Latitude
Time Measuring
•  The highest precision pendulum clock
have accuracy 0.01 [s/day]
•  Very stable quartz clock has accuracy
0.001 [s/day]
•  Atomic (cesium clock) has accuracy 1
part in 1013 . This is 1 sec per 2,000,000
years!
Longitude Prize
•  The Longitude Prize was a reward offered by the British
government for a simple and practical method for the precise
determination of a ship's longitude. The prize, established
through an Act of Parliament (the Longitude Act) in 1714, was
administered by the Board of Longitude.
•  £ 10,000 for a method that could determine longitude within 60
nautical miles (111 km)
•  £ 15,000 for a method that could determine longitude within 40
nautical miles (74 km)
•  £ 20,000 for a method that could determine longitude within 30
nautical miles (56 km)
14
Mechanical
Watch
Huygens,
Christiaan(16291695), Dutch
physicist,
mathematician,
and astronomer.
15