phy201_11

1
Physics 211
12 Oscillatory Motion

• Simple Harmonic Motion
• Energy of a Simple Harmonic Oscillator
• The Pendulum
• Comparing Simple Harmonic Motion and Uniform
  Circular Motion
• Damped Oscillations
• Forced Oscillations

2
Simple Harmonic Motion
Hookes Law F -kx F restoring force x
displacement from equilibrium position
3
x0
x-A
XA
4
2
d
x
(
)
(
)
(
)
F
t

-
kx
t

ma
t

m
dt
2
2
d
x
k
Û

-
x
dt
m
2
Differential Equation,
need to find solution
so that left hand side

right hand side
From looking at graph of position versus time
(
)
Guess


x
t

A
cos(
w
t
)


A
and
w
constants
.


5
(No Transcript)
6
What is the meaning of the constant A
at time
t

0
(
)
(
)
x
0

A
cos
0

A
which is the displacement from equilibrium at
time
t

0
p
at time
t

w
p
p
æ
æ
ö
ö
(
)
x

A
cos
w

A
cos
p

-
A
è
ø
è
ø
w
w
The max and min values of
x
are

A
Û
A
is the amplitude of the motion
the maximum displacement from equilibrium
position

7
cos(x)
1
?
??
cos(2n?) 1
-1
cos(2n1?) -1
8
Meaning of
w
2
n
p
for
t

Þ
x


A
w
(
)
2
n

1
p
for
t

Þ
x

-
A
w
motion repeats between

A
Harmonic Motion
If it repeats itself exactly
Simple Harmonic Motion
(SHM)
(
)
(
)
2
n

2
p
2
n
p
2
p
Time between repeats


-


(
angle changes
2
p
)
w
w
w
This is called the Period of the Oscillation,

T
Angular Frequency
Rate of change of angle with time

2
p
w


T

9
The number of times motion repeats in
1 second
1
w
is the frequency
f


T
2
p
rad




f

Hz
º
cps

w

º
s
-
1
s
2
p
m
T


2
p
w
k

When is the velocity the greatest/least When is
the acceleration the greatest/least
10
(
)
2
n
p
2
n

1
p
x


A
Û
t



t

w
w
(
)
(
)
at these times the velocity v
t

-
A
w
sin
w
t
Þ
2
n
p
2
n
p
æ
æ
ö
ö
(
)
v

-
A
sin
w

-
A
sin
2
n
p

0
è
ø
è
ø
w
w
(
)
(
)
2
n

1
p
æ
2
n

1
p
æ
ö
ö
(
)
(
)
v

-
A
sin
w

-
A
sin
2
n

1
p

0
è
ø
è
ø
w
w
so velocity is zero at maximum displacement
the acceleration on the other hand is a maximum
(
)
2
n
p
2
n

1
p
æ
æ
ö
ö
a

-
w
A



a

w
A
2
2
è
ø
è
ø
w
w
The acceleration is zero when
x

0
The velocity is the greatest when
x

0
v


A
w

max
11
Energy of a Simple Harmonic Oscillator
Mass experiences spring force,
thus its P
.
E is
1
U
(
x
)

kx
2
2
The spring force is a conservative force
The total energy of the mass is
1
1
1
1
(
)
2
(
)
2
E

K

U

m
v

kx

m
v
t

kx
t

constant
2
2
2
2
2
2
tot
\
The total energy when v

0
is equal to
1
E

kA
2
2
tot
which must be its value at ALL
TIMES
!
1
1
1
(
)
Þ
E
t

m
v

kx

kA
2
2
2
2
2
2
tot
1
1
k
Þ
when
x

0
,

K

m
v

kA
Þ
v

A
2
2
max
max
2
2
m

12
The Pendulum
?
l
T
s
mg cos?
mg sin?
Wmg
13
Restoring Force
-mg
when
q
F

-
mg
sin
q

q
is small
s
mg
\
F

-
mg
q

-
mg

-
s
l
l
º
Hookes Law for the Pendulum
2
d
s
,
This force provides the tangential acceleration
2
dt
and we obtain a similar differential equation to
before
.
Comparing to before we see
mg
k




x

s
l
(
)
(
)
s
t

A
cos
w
t
m
ml
l
T

2
p

2
p

2
p
k
mg
g

14
The Physical Pendulum
d
????d x W
?
Wmg
Note that the pivot point could be inside the
boundaries of the object
15
We formulate this for an object suspended so that
its center of gravity is at a distance d from the
pivot point, by using angular quantities

The restoring torque
due to gravity
t


-
mgd
sin
q

-
mgd
q

(small angle approx
.
)
using
t
I
a

2
d
q
mgd
mgd
2
Þ
a


-
q

-
w
q
,
where
w

2
dt
I
I
(
)
(
)
Þ
q
t

A
cos
w
t
I
Þ
T

2
p
mgd


16
The Torsion Pendulum
By suspending a mass at the end of a wire
supported tightly at the other end,
we make
a torsion pendulum.
By twisting the object
through a small angle we produce a restoring
torque
d
q
2
t

-
kq


I
a

I

dt
2
2
d
q
k
Þ


-
q
dt
I
2
I
T

2
p

k

17
Comparing Simple Harmonic Motion and Uniform
Circular Motion
(
)
(
)
x
t

A
cos
w
t
is precisely the time variation
of the x coordinate of a particle performing
uniform circular motion about a fixed point

at a fixed distance
A
.
p
æ
ö
(
)
(
)
y
t

A
sin
w
t

A
cos
w
t
-
is the time variation
è
ø
2
of the y coordinate,
which is a SHM variation
.

p
Here though we have a phase shift of
j

-
2
p
The argument
w
t
-
is called the phase of the motion
.
2

18
Damped Oscillations
If there are frictional forces present
D
E

W
lt
0
tot
nc
Thus the total energy decreases
and becomes
a non constant function of time,

E
(
t
)
¹
constant.
1
(
)
2
Þ
E
(
t
)

kA
t
2
(
)
Þ
A
t
decreases with time
The differential equation describing the position

of a particle undergoing damped SHM is of the
form
d
x
dx
2
m

-
kx
-
b
dt
dt

2
19
Forced Oscillations
If there is another external oscillating force
acting on the object
(in the direction of motion
)
one says that the motion of the oscillator is
forced by this external force.
The differential
equation describing such motion is
d
x
dx
2
(
)
m

-
kx

-
b

F
cos
w
t
dt
dt
0
2

20
solutions to this equation give amplitudes of the
form
F
0
m
A
(
t
)




(
)
(
)
b
w
2
2
w
-
w

2
2
m
0
w
is the frequency of the SHM
(i
.
e
.
no friction
0
and forcing oscillation
)
If b
(friction
) is small,
then if
w

w
the amplitude
0
becomes larger and larger
º
RESONANCE