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Chapter
.
Oscillations
Oscillatory
motion :
is
a
repetitive
motion
back
and
Amplitude
(
Xm or
A) : the
object
's maximum
forth
around an
equilibrium
position
.
des
placement
from
equilibrium
.
Amplitude
(A)
Example
of SHM : ✗ it) =
Xmcoslwt
)
>
a mass
oscillating
on a
spring
.
wiangular frequency
)= 217T
T
pendulum
w=
f=¥,
B.
The
velocity
is Zero when ✗ =IXm
.
Any
system
with a
linear
restoring
*•vma×
when 0
Vxlt)=d±=
-2¥
Sin
(2-1,1+10)
force
dt
=2ñffA Sin
(2211-1-1-+0)
=
wxmsinlwt
-10)
Vmax =
WXM
velocity
amplitude
act
)=doY¥=
dfwxmsinlwt
=
-10)
am
=
WZXLT
acceleration
amplitude
Objects
that
undergo
a
repetitive
V=±wFÉx
motion back
and
forth around an
equilibrium
position
.
••
In SHM
,
the acceleration a
is
proportional
to
Period ( T)
: The time to
complete
one
full
cycle
or one the
displacement
✗ but
opposite
in
sign
,
and
oscillation
.
the
two
quantities
are related
by
the
Square
of
the
angular
frequency
w
.
frequency
(f) : The number of
cycles
per
second
measured in Hz.
The Force
Law
For
Simple
Harmonic
motion :
f-=
¥
or
1-=
I
F = ma = m 1-
WH
) = - LMWYX
f
,
,
'
(
9B¥
/ T
i
,
The minus
sign
means that the direction of
it
..
!
the force on the
particle
is
opposite
the
1
HZ
/
Cycle per
Second
= 151
\
,
direction of
the displacement
of the
particle
In SHM
,
the force is a
restoring
force in
the
sense that it
fights against
the
displacement
attempting
to restore the
particle
to the Center
point
at ✗=o.
simple
harmonic motion is
the motion
of a
particle
when
the
force
acting
on
it
is
proportional
to the
particle
's
displacement
but in the
opposite
direction
spring
system
is
called
a linear
simple
harmonic oscillator
F- =
,
f-
=
mut )
✗
-1k *
=
( mut) *
T=
21T
÷
W
Kmt
=
MWZ
Energy
in
simple
harmonic
motion
Pendulum
Energy
is conserved in
simple
harmonic motion
a
simple
pendulum
e : a bots of mass m
suspended
from
an uinstrechable
,
massless
string
.
acting
on
the
bob are :
force
F
from the
string
.
the
gravitational
force
Ég
resolve Ig
Fg
Sin
Q
Fg
Cos
OV
tangent
to the
path
taken
by
the bob
i
Produces a
restoring
torque
about the
pendulum
's
pivot
Point
because
the
component always
acts
opposite
the displacement
of the
bob so as
to
bring
the bob back toward its central
location -
qsaq
LB
>
minus
sign
toraque
acts to reduce
Potential
energ
of linear oscillator
The
kinetic
energy
I = -
L
(
Fg
8in -0)
>
L : the moment arm of
Fg
Sino
about the pivot point
.
I
Its value
depends
on
how much Its value
depends
on how
the
spring
is stretched or compressed
fast
the
block
is
moving
> I : the Pendulum 's rotational inertia
( (
Mg
Sino
)
=
I
✗
✗ :
angular
acceleration
that is
,
on vctl
.
UCH
=IzkÑ=£kXmco5(
wt
kltl-1-zmvEI-zkxmsintwt.to/)V=+wfFx
=
MFL
.
equation
of
angular
I
equivalent
.
"
The mechanic
le
energy
k
== B*
-9*8^87 The
angular
acceleration
✗ of
the
E = U
K
Pendulume
is
proportional to the
angular
displacement
but
opposite
=
.
.
.
=
tzkxm
[
coszcwt
-10 +8in
lwt -101]
Angular
amplitude
0m
of the
motion must
be
small
cost
+ Sink =L
"
E
= Ut
k
=
§
Kim
Angular frequency
: w =
MIL
I
Period :
21T
I
mgl
The mechanical
energy
of
a linear
oscillator is • (
for
simple pendulum ,
small
amplitude
I=mL
indeed constant and
independent
of
time
.
-1=2-
Iggy
= 21T
HE
nxgk
An
Angular
simple
harmonic oscillator
:
1-
= 21T
kg
elasticity
from
a
twisting
wire
.
"
"
angular
simple
harmonic motion.
I
It is called the torsion
constant
to
depends
on the
length ,
diameter
and material of the suspension
wire.
-11T¥
-098^ Damped
energy
Resonance
approximately
the
condition at
which
the
displacement amplitude
Xm
of
the
oscillations
is
greatest
.
Ect)
=
§
kxzm
e-
'
btlm
The
displacement
amplitude
of an
oscillator
the
mechanic
le
energy
depends on
the
angular
frequency
wa
of
decreases
exponentially
the
driving
force .
with time
.
•• Time
required
for
mechanical
energy
depends
less
damPi^
Upon
ratio blm
and
independent
in 1 €
IMO
gives
taller
and
narrower
smaller ratio
slow
hey
energy
dissipate
need more
time
A
larger
ratio
faster
energy
dissapaie
. need
less
time
yegONÑ&
peak
.
- forced oscillations and resonance
→
Two
angular
frequecies
are
associated
with
a
system undergoing
driven
oscillations :
the
natural
angular
frequency
w
of
the
system
,
which
is
the
angular
frequency
at
which
it would
oscillate
if it were
suddenly
disturbed and then
left to oscillate
freely
the
angular frequency
wa
of
the
external
driving
force
causing
the
driven oscillator.
✗It
) = Xm
Cos (
Wat
+0)
How
large
the
displacement amplitude
✗
m
is
depends
on
a
complicated
function of
iwd
and
w
The
velocity
amplitude
Vm
of the oscillations
is
easier
to describe :
it is
greatest
when
Wd =
W
