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14  OSCILLATIONS
Page
1
14.1
Periodic
and
Oscillatory
motion
Motion
of
a
system
at
regular
interval
of
time on
a
definite
path
about
a
definite
point
is
known
as
a
periodic
motion,
e.g.,
uniform
circular
motion
of
a
particle.
To
and
fro
motion
of
a
system
on
a
linear
path
is
called
an
oscillatory
motion,
e.g.,
motion
of
the
bob
of
a
simple
pendulum.
14.2
Simple
harmonic
motion
This
is
the
simplest
type
of
periodic
motion
which
can
be
understood
by
considering
the
following
example.
Suppose
a
body
of
mass
m
is
suspended
at
the
lower
end
of
a
massless
elastic
spring
obeying
Hooke’s
law
which
is
fixed
to
a
rigid
support
in
the vertical
position.
The
spring
elongates
by
length
Δ
l
and
attains
equilibrium
as
shown
in
Fig. ( b )
Here
two
forces
act
on
the
body.
( 1 )
Its
weight,
mg,
downwards
and
( 2 )
the
restoring
force
developed
in
the
spring,
k
Δ
l
,
upwards,
where
k
=
force
constant
of
the
spring.
For
equilibrium,
mg
=
k
Δ
l
…
( 1 )
The
spring
is
constrained
to
move
in
the
vertical
direction
only.
Now,
suppose
the
body
is
given
some
energy in
its
equilibrium
condition
and
it
undergoes
displacement
y
in
the
upward
direction
as
shown
in
Fig.
( c ).
Two
forces
act
on
the
body
in
this
displaced
condition
also.
( 1 )
Its
weight,
mg,
downwards
and
( 2 )
the
restoring
force
developed
in
the
spring,
k (
Δ
l

y ),
upwards.
The
resultant
force
acting
on
the
body
in
this
condition
is
given
by
F
=
 mg
+
k (
Δ
l

y )
…
…
( 2 )
From
equations
( 1 )
and
( 2 ),
F
=
 ky
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Displacement:
The
distance
of
the
body
at
any
instant
from
the
equilibrium
point
is
known
as
its
displacement
at
that
instant.
The
displacements
along
the
positive Yaxis
are
taken
as
positive
and
those
on
the
negative
Yaxis
are
taken
as
negative.
In
the
equation,
F
=
 ky,
F
is
negative
when
y
is
positive
and
vice
versa.
Thus,
the
resultant
force
acting
on
the
body
is
proportional
to
the
displacement
and
is
directed
opposite
to
the
displacement,
i.e., towards
the
equilibrium
point.
Differential
equation
of
simple
harmonic
motion
( SHM )
According
to
Newton’s
second
law
of
motion,
F
=
ma
=
m
dt
dv
=
m
2
2
dt
y
d
=
 ky
( for
springtype
oscillator
as
above )
∴
2
2
dt
y
d
=

m
k
y
=

ω
0
2
y
( taking
m
k
=
ω
0
2
)
∴
2
2
dt
y
d
+
ω
0
2
y
=
0
This
is
the
differential
equation
of
SHM.
To
obtain
the
solution
of
the
above
differential
equation
is
to
obtain
y
as
a
function
of
t
such
that
on
twice
differentiating
y
w.r.t.
t,
we
get
back
the
same
function
y
with
a
negative
sign.
Both
the
sine
and
the
cosine
functions
possess
such
a
property.
Hence,
taking
y
=
A
1
sin
ω
0
t
+
A
2
cos
ω
0
t
as
a
possible
solution
and
differentiating
twice
w.r.t.
t,
dt
dy
=
A
1
ω
0
cos
ω
0
t

A
2
ω
0
sin
ω
0
t
and
2
2
dt
y
d
=

A
1
ω
0
2
sin
ω
0
t

A
2
ω
0
2
cos
ω
0
t
=

ω
0
2
( A
1
sin
ω
0
t
+
A
2
cos
ω
0
t )
=

ω
0
2
y
Thus,
y
t
=
A
1
sin
ω
0
t
+
A
2
cos
ω
0
t
is
the
solution
of
the
differential
equation
and
is
known
as
its
general
solution,
where
y
t
is
the
displacement
of
simple
harmonic
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This note was uploaded on 11/28/2011 for the course PHYSICS 300 taught by Professor Smith during the Spring '06 term at ITT Tech Flint.
 Spring '06
 Smith
 Circular Motion, Simple Harmonic Motion

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