Electromagnetic wave Simple Harmonic Motion...
Transcript of Electromagnetic wave Simple Harmonic Motion...
Ch
ap. 1
4: O
scillatio
ns,
Perio
dic M
otio
n,
Sim
ple H
arm
on
ic Motio
n
11
Characterized by:
Period (T) and Frequency (f)
Preparation for: M
echanical wave m
otionE
lectromagnetic w
ave
Dynam
ics: F= m
a, �= I �
�Equation of M
otion: G
eneral solution:
LLook
ing B
ack
to C
hap. 7
•
To describe oscillations in terms of
amplitude,
period, frequency
and angular frequency
•To
do calculations
with sim
ple harm
onic motion
(SHM
); To
analyze sim
ple harmonic m
otion using energy
•To apply the ideas of sim
ple harmonic
motion to different physical situations
•
To analyze
the m
otion of
a sim
ple pendulum
, followed
by a
physical pendulum
•To explore how oscillations die out
•To learn how a driving force can cause resonance
2
LList o
f Previo
usly C
overed
Topics
•To describe oscillations in term
s of am
plitude, period,
frequency and
angular frequency
•To
do calculations
with sim
ple harm
onic motion
(SHM
); To
analyze sim
ple harmonic m
otion using energy
•To apply the ideas of sim
ple harmonic
motion to different physical situations
•
To analyze
the m
otion of
a sim
ple pendulum
, followed
by a
physical pendulum
3
Calculus Trig. functions F = m
a (Chap. 4) Circular m
otion (Chap. 5) Restoring force (i.e. spring
force) (Chao. 6) Energy conservation (Chap. 7) M
oment of inertia (Chap. 9)
� = I ��(Chap. 10)�
•To
explore how
oscillations die out •
To learn
how a
driving force
can cause resonance
Oscillations
WWh
at is S
.H.M
.?
5
Periodic Motion
Horizontal oscillation
Vertical oscillation Vibration
Common characteristics
�Sim
plified Model
Simple H
armonic M
otion
Oscillations
SS.H
.M.
Usefu
l Math
an
d P
hysics
Trig. functions:
sin(��+ ��2) = cos(�) sin(��+ ����) = �sin(�) cos(��� ��2) = sin(�) D
erivative and integral T
rig. functions A
pproximation:
sin� ~ ��
S.H.O
. 1) Spring plus block�
Horizontal
Vertical 2) Pendulum
Sim
ple pendulum
Physical pendulum F = m
a
� = I ��
6
Oscillations
SS.H
.M.
Usefu
l Sk
ill –– Sk
etch /
Visu
aliza
tion
7
Oscillations
Circular M
otion to SHM
x R
0 ��
Starts here
8
x = R0 cos ��
where � = t
Oscillations
Analyzing Spring+B
lock System
10
Oscillations
S.H.M
. : Spring+Block system
and Sim
ple Pendulum system
Spring+block System
Simple Pendulum
12
Oscillations
Physical Pendulum – S.H
.M.?
Physical Pendulum
Simple Pendulum
m
Axis of rotation
15
Oscillations
��
��
Im
gdl g
xm k
a
�
�
�
Math about a and ��
XC
td
Xd
2
2
�
Oscillations
Physical Pendulum
Physical Pendulum – –
S.H.M
.?S.H
.M.?
Physical PendulumPhysical Pendulum
Simple Pendulum
Simple Pendulum
Oscillations
S.H.M
. : Spring+Block system
and S.H
.M. : Spring+B
lock system and
Simple Pendulum
systemSim
ple Pendulum system
Spring+block SystemSpring+block System
Simple Pendulum
Simple Pendulum
18
Acceleration is proportional to displacem
ent.
Oscillations
S.H.M
. – Dynam
ics (Summ
ary)
��
xC
td
xd
xC
Dt
CA
C
Dt
CA
CC
td
xd
Dt
CA
Cdt dx
Dt
CA
tx
Ct
d dx
Ct
dx
d
)cos(
)cos(
)sin(
:then
),
cos()
(
If
or
2
2 2
2
2
2
2
2
�
�
��
��
��
��
�
�� ��� �
�
�
��
Math about C
Physics about C
Oscillations
)cos(
)(
φωt
At
x��
22
0
Oscillations
Circular M
otion to SHM
x
x = R0 cos ���
where � = t
R0
x(t) = R0 cos ( t)
��
Starts here
Kin. Equation of SHM
x(t) = R0 cos (�������������)
��
21
Oscillations
Acceleration is proportional to displacem
ent. S.H
.M.
�
� �� � �� �
�
�
��
position)
starting
the
on
(depending
2)
cos(
)(
onaccelerati
2
2
� �
�
T
tA
tX
XC
Ct
dX
d
��
�
224
XC
td
Xd
2
2
�
Oscillations
WWh
at is � ?
26
t
x(t) �/
� = ��/2�
x(t) = Asin(
�t ) = A
cos( �t )
= A cos(
��t ���/) )
= A cos(
�t � � )
Oscillations
S.H.M
.
3 independent variables: A
ngular frequency��A
mplitude
Phase angle
x(t) = A cos(
�t + �)
�(t) = A cos(
�t + �)
Period: T = 1 / f �
T = 2�/�
Note: A
ngular frequency () is N
OT the angular velocity.
Exam
ple of x(t), w
here ��= 0
Quick C
heck: H
ow do v(t) and a(t) look like?
28
Oscillations
Graphs
330
Oscillations
3 31
Oscillations
3 32
Oscillations
3 33
Oscillations
(1)Identify the force along
the m
otion and find the
equation of motion.
(2) D
etermine the values of three independent variables
Am
plitude
A
ngular frequency (and period)
Phase angle
(3) Sketch ( x-t, vv-t, a-t ) or ( �-t, -t, �-t ) graphs
and find the m
aximum
values of …
Tech
nica
l Step
s
Im
gdI
mgd
ge
�
�
�
�
.,
.
XC
td
Xd
2
2
�
Oscillations
S.H.M
.S.H
.M.
3 independent variables:A
ngular frequencyA
mplitude
Phase angle
x(t) = Acos(
�t + �)
�(t) = Acos(
�t + �)
Period:T = 1 /f �
T= 2�/
Period:T = 1 /f �
T= 2�/
Note:A
ngular frequency () is N
OT
the angular velocity.
Exam
ple of x(t), w
here ��= 0
Quick C
heck:Q
uick Check:
How
do v(t) and a(t) look like?
35
Oscillations
Rotational M
otion
2(e) A solid disk (m
ass M= 3.00 kg and radius R
= 20.0 cm) is
hung from the w
all by means of a m
etal pin through the hole, and used as a pendulum
. Calculate the m
oment of inertia of the
disk about the pin (= the axis of the rotation).
Rotational M
otion
2(c) A m
eter stick (mass M
= 0.500 kg and length L= 1.00 m
) ishung from
the wall by m
eans of a metal pin through the hole,
and used as a pendulum. E
xpress the mom
ent of inertia of thestick about the pin (= the axis of the rotation) in term
s of M, L,
and x.
Rotational M
otion
Torque due to G
ravity?T
orque due to Gravity?
)(
sin2
�� !
"# $�
%
�
�
m
g
xl
F
r
�
��
r �x
F �
lC
Mm
assm
?
%
F
r
�
���
ations
Rotatitttittttittittiiiona
oooooooool M
otion
2(e) A solid
hung fromk (m
ass M= 0.500 kg and length L
= 1.00 m) is
all by means of a m
etal pin through the hole,d di
m the iske w
= 3.00ans of
M=
mea
ss Mby m
k (mas
wall b
kg f a m
Rrodius n th
and radm
etal pin
Looking B
ack at I and ���
36
Oscillations
Con
cepts o
f S.H
.M..
Dynam
ics & Kinematics
Spring+block �
F = m
a & (x, v, a)
Pendulum
� a part of circular m
otion���
� = I ���&
(�, , �) Force:
Conservative force
Restoring force
Conservation:�
K + U
= constant
+ S.H.M
. (�as angular frequency)
37
Oscillations E
xercise 1: Find �
= [Find I and d] 40
Step
1: P
hysica
l Pen
du
lum
s Im
gdI
mgd
�
�
�
�
Oscillations
4 43
Exercise 1: Find
� = [Find I and d]
Exercise 2: Find ���
= [ Find I, d, and ���]
I d
I, d��
�
Oscillations
Step
1: P
hysica
l Pen
du
lum
s (II)
Exercise 2: Find ���
= [ Find I, d, and ���] 44
Oscillations
4 45
Want to find T
Need to know �
Then, find I
Step
1: P
hysica
l Pen
du
lum
s (III)
OOOOOOOOOOOsssssssssssccccccccccciiiiiiiiiiillllllllllllllllllllllaaaaaaaaaaatttttttttttiiiiiiiiiiiooooooooooonnnnnnnnnnnsssssssssss
�Find I
�Calculate
�Then, find T
Exercise 3: Find T
Oscillations
Cau
tion
: An
gula
r Velo
city?
446
Oscillations
PPra
ctice Pro
blem
1
47
Oscillations
Exercise 1+: Find
�
��= [Find I and d]
Top-Dow
n Steps: 1.W
hat is asked?����
�
(rigid body) � I and d
2. How
to find I and d for a system of tw
o rigid bodies?
I = I1 + I2
3. How
to find I1 (or I2 )?
W
here is the c.m. position? U
se parallel-axis theorem
4. How
to find d ?
W
here is the c.m. position of the system
?
Im
gdI
mgd
�
�
�
�
48
Pra
ctice Pro
blem
2
kg 0
2 cm 0
5kg
500 cm
10
2 1
.m
.r
.m
�
Oscillations
Oscillations
x= 2.00 cm
x=
2.00cm
50
Oscillations
Nam
e: _______________ Section: ______ UIN
: __________
Work on (a)
Oscillaaa
Repeat (a) without and w
ith a particle (mass m
)
51
Oscillations
Nam
e: _______________ Section: ______ UIN
: __________
Oscillaaa
Work on (a), (b), (c), and (d) 55
2
Oscillations
Problem 4: (25 points)
Determ
ine the net torque (magnitude
and direction) due to gravity on the system about the
pin, shown in the figure below
. A beam
has mass M
and length l; a big solid sphere has m
ass Mand radius R; a sm
all sphere has mass M
/2 and radius R/2. Assum
e l> Rand l> 2x.
Also determ
ine the mom
ent of inertia of the system about the pin. [H
int: use Parallel-axis theorem
.]
l
ations
But, this can also be a C
hap.14 problem,
If I ask you to find . (angular frequency) in S.H
.M. of a physical pendulum
.
Chap. 10
53
Oscillations
AAn
ato
my o
f S.H
.M. P
roblem
+ S.H.M
. (�as angular frequency) 54
Chap. 9
Chap. 10
Chap. 8
Calculus
Oscillations
Example 8: Physical Pendulum
s�2.00 cm 55
Oscillations
Equation of Motion:
d2x/dt 2 +
2 x = 0 or d2�/dt 2 +
2 ��= 0 G
eneral solution:
x(t) = A cos( �t + �)
�(t) = A cos(�t + �) �
Position (x or �) as a function of time.
Am
plitude
Angular Frequency
Oscillations
Periodic Motion
Simple H
armonic M
otion (S.H
.M.)
56
Oscillations
Periodic Motion
T = 2�� L cos� / g T 2 = (4�
2/GM
) s 3
57
Oscillations
Example 1: (a) F
T = ? (b) T = ?
mg sin��
Be C
ritcal Thinker
gL
T�
�cos
2
g lT
�2
FT =m
gcos��F
T =mg/cos��
58
Oscillations
Torsion Pendulum
59
Oscillations
S.H.M
. : T = 2�/�
660
Oscillations
Example 2(A
)
+
+
+
+
+ 0
T/4
T/2
3T/4
T
Equilibrium
Positions
A
61
Oscillations
Example 2(B
)
Ea = E
b = Ec = E
d
62
Oscillations
Example 4: M
omentum
conservation!�
What is the speed of the bullet?
Express the speed in term
s of m, M
, k, and d.
d
664
Oscillations
Example 5
You holdthe block
at x = A (= 0.030 m
) by applying 6.0 N
. T
hen, the block was
released.
The m
otion of the block undergoes SHM
. C
an you show that a = –(k/m
) x ? A
lso find: (a) k
(b)�
(c) T
(d) vm
ax (where?, w
hen?)
(e) x, v and a at t = 2 sec
k 0.50 kg
66
Oscillations
Example 9: V
ertical S.H.M
.�
67
Oscillations
Physical Pendulum (I)
����
Small oscillation
�� = sm
all �
L|| = L cos� ~ L
L||
68
Oscillations
Physical Pendulum (II)
+ x
����
+
mg
x0 x = 0
Fsp = k x
0
Equilibrium
(&�i = 0) �
mg(L/2) – kx
0 (L) = 0
69
Oscillations
Physical Pendulum (III)
+ x
����
+
mg
x0 x = 0
Fsp = k x
Equation of m
otion (&�i = I �) &�i = m
g(L/2) – kx(L) = kx0 (L) – kx(L)
–k(x–x
0 )(L) = Irod(P) d2�/dt 2
P
+ x
L� = x– x
0
70
Oscillations
889