6). oscillatory motion (finished)

1. VI
Oscillatory Motion

2. Oscillatory Motion
; periodic movement

Simple Harmonic motion
E.g.; mass-spring system (no friction)





















2
nd
Newton law

� =��
At (a); � =−�
�����??????�=��
Since � =
??????
2
??????
??????�
2
,
−�
�����??????�=�
�
2
�
��
2

�
2
�
��
2
=−
�
�����??????
�
�

Let
�
�??????�??????????????????
�
be ??????
2
,

�
2
�
��
2
=−??????
2
�
m
m
m
(�)
(�)
(�)
�
-�
�=0
�
�
�
�
Math
Differential equation

��
��
=�
2
�
��
�
=�
2
��

��
�
= �
2
��
���=
�
3
3
+�

Homogeneous Second Order Linear
Differential Equations (2
nd
ODE)

���
′′
+���
′
+���=�(�)

With constant coefficient
��
′′
+��
′
+��=0

Change the form
��
2
+��+�=0

1. If there are 2 distinct roots, �
1,�
2
�=�
1�
�1??????
+�
2�
�2??????


2. If there are a repeated single root, (�)
�=�
1�
�??????
+�
2��
�??????


3. If there are complex roots, (�±��)
�=�
????????????
[�
1cos��+�
2�����]

Euler’s identity

�
�??????
=���??????+����??????
***  is in radian

R-Formulae
� ���??????+� ���??????=�cos(??????−??????)
� ���??????+� ���??????=�sin(??????+??????)
�=�
2
+�
2
,??????=���
−1
�
�

�
�=��������� �����

3. Graph plotted when ??????=??????























�
2
�
��
2
=−??????
2
�
�
′′
+0�
′
+??????
2
�=0

Change form,
�
2
+??????
2
=0
�
2
=−??????
2

�=�??????
�=±�??????

The roots are complex (�=0±�??????)
��= �
1cos??????�+�
2���??????�

Using R-Formulae,
��=Acos (??????�+??????)
Or
??????�=??????????????????� (??????�+??????)
??????�=
????????????
??????�
=??????????????????�?????? (??????�+??????)
??????�=
????????????
??????�
=−??????
??????
??????????????????� (??????�+??????)

***When ??????≠0, the graph is shifted to
the left or right

??????=??????�������� �
??????=������� ��������
���
�

??????=�ℎ���� �������� [���]

There are Sinusoidal functions as they
can be shifted to the left or right

??????
�????????????=??????
??????
�????????????=????????????
??????
�????????????=−??????
??????
??????

??????(�)
??????(�)
??????(�)
�
�
�
??????
−??????
????????????
−????????????
−??????
2
??????
??????
2
??????
� =−�
�����??????�=��
�=−
�
�
�,�=−??????
2
�
***every motion with this form of
equation is Simple Harmonic motion











�=�������� �� ������ ��ℎ���� ������
�=−
�
�
�
�=−??????
2
�
??????=
2??????
�
=2??????�=
�
�

4. Since ��=??????sin??????�+??????
�??????�??????�+??????=
??????(�)
??????
,










Since ��=????????????cos (??????�+??????)




*** ± because there are 2 direction at 1
position

Since ��=−??????
2
??????sin (??????�+??????)

??????�=−??????
??????
??????(�)

Energy of Simple Harmonic motion
*** energy is conserved
�
�����=��+���

Kinetic Energy (KE)
��(�)=
1
2
��
2
=
1
2
�??????
2
(??????
2
−�(�)
2
)
Since ??????=
�
�
,
��(�)=
1
2
�(??????
2
−�(�)
2
)

Elastic Potential Energy (PEP)
���=
1
2
��(�)
2


∴�
�����=
1
2
�??????
2
−��
2
+
1
2
��(�)
2

�
�����=
1
2
�??????
2

�
�����=
1
2
�??????
2

??????�+??????
�(�)
??????
??????
??????
−??????(�)
??????

??????�= ±????????????
??????
−??????(�)
??????

????????????
,
??????????????????

�
????????????
??????????????????
0 � �
2

????????????,??????????????????
−?????? 0 ??????
�
Angular Simple Harmonic motion
Simple Pendulum













??????=??????�
??????=������=��=−�� ���??????�
−�� ���??????�=??????�

∴�=
−�???????????? ���??????
??????

??????
�
��
??????
�� ���?????? �� ���??????
�

5. ∴�=
−�???????????? ���??????
??????


***General formula of simple harmonic motion
equation is �= −�� where k is a constant

Since
�????????????
??????
is a constant, this equation is
in a form of a simple harmonic motion
equation when ���??????≈??????, or when  is
very small (<10°)




Since &#3627408462;&#3627408481;=−??????
2
&#3627408485;(&#3627408481;),

&#3627409148;=−??????
2
??????





?????? = &#3627408474;&#3627408476;&#3627408474;&#3627408466;&#3627408475;&#3627408481; &#3627408476;&#3627408467; &#3627408470;&#3627408475;&#3627408470;&#3627408481;&#3627408470;&#3627408462; &#3627408476;&#3627408479; &#3627408479;&#3627408476;&#3627408481;&#3627408462;&#3627408481;&#3627408470;&#3627408476;&#3627408475;&#3627408462;&#3627408473; &#3627408470;&#3627408475;&#3627408470;&#3627408481;&#3627408470;&#3627408462;&#3627408473;









??????=
&#3627408474;&#3627408468;&#3627408447;
??????

&#3627409148;=
−&#3627408474;&#3627408468;&#3627408447;??????
??????

??????=&#3627408474;&#3627408447;
2

??????=
&#3627408468;
&#3627408447;

??????=??????
0&#3627408480;&#3627408470;&#3627408475;(??????&#3627408481;+??????)
??????
ℎ
&#3627408474;&#3627408468;
??????
&#3627408474;&#3627408468; &#3627408464;&#3627408476;&#3627408480;?????? &#3627408474;&#3627408468; &#3627408480;&#3627408470;&#3627408475;??????
Physical Pendulum
*** In our real life, string has mass
Think of it as one whole body without T






















Since ??????=&#3627408474;&#3627408472;
2
, k = radius of gyration

&#3627409148;=
−&#3627408474;&#3627408468;ℎ??????
??????

??????=
&#3627408474;&#3627408468;ℎ
??????

??????=
&#3627408468;ℎ
&#3627408472;
2

??????=2??????&#3627408467;=
2??????
&#3627408455;

&#3627408455;=2??????
&#3627408447;
&#3627408468;

6. Torsional Pendulum
*** In our real life, string has mass
Think of it as one whole body without T


??????=−????????????
??????=&#3627408446;&#3627408462;&#3627408477;&#3627408477;&#3627408462;
??????=??????&#3627409148;
&#3627409148;=−??????
2
??????
&#3627409148;=−
??????
??????
??????
??????=
??????
??????
=??????????????????=
????????????
??????

Damped Simple Harmonic motion
*** In real life, there is a frictional force
(friction, air resistance)













Free-body diagram at &#3627408485;
&#3627408474;&#3627408462;??????









***However, when we calculate the
resultant force, mg is canceled as at
x=0, mg is already canceled by –kx
***-bv is the fluid resistance or drag
force

2
nd
Newton law

&#3627408441; =&#3627408474;&#3627408462;
−&#3627408472;&#3627408485;−&#3627408463;&#3627408483;=&#3627408474;&#3627408462;
−&#3627408472;&#3627408485;−&#3627408463;
&#3627408465;&#3627408485;
&#3627408465;&#3627408481;
=&#3627408474;
&#3627408465;
2
&#3627408485;
&#3627408465;&#3627408481;
2

&#3627408474;&#3627408485;
′′
+&#3627408463;&#3627408485;
′
+&#3627408472;&#3627408485;=0

Change form,
&#3627408474;&#3627408439;
2
+&#3627408463;&#3627408439;+&#3627408472;=0
m &#3627408485;=0
m
−&#3627408472;&#3627408485;−&#3627408463;&#3627408483;
&#3627408474;&#3627408468;
??????
&#3627408480;&#3627408477;&#3627408479;&#3627408470;&#3627408475;&#3627408468;

7. &#3627408474;&#3627408439;
2
+&#3627408463;&#3627408439;+&#3627408472;=0

&#3627408439;=
−&#3627408463;±&#3627408463;
2
−4&#3627408474;&#3627408472;
2&#3627408474;

&#3627408439;=−
&#3627408463;
2&#3627408474;
±
&#3627408463;
2
4&#3627408474;
2
−
&#3627408472;
&#3627408474;


There will be 3 cases
1. If there are 2 distinct roots, &#3627408439;
1,&#3627408439;
2
Over damped
&#3627408463;
2
4&#3627408474;
2
>
&#3627408472;
&#3627408474;


&#3627408485;(&#3627408481;)=&#3627408464;
1&#3627408466;
&#3627408479;1&#3627408481;
+&#3627408464;
2&#3627408466;
&#3627408479;2&#3627408481;


2. If there are a repeated single root, (D)
Critically damped
&#3627408463;
2
4&#3627408474;
2
=
&#3627408472;
&#3627408474;


&#3627408485;(&#3627408481;)=&#3627408464;
1&#3627408466;
&#3627408479;&#3627408481;
+&#3627408464;
2&#3627408481;&#3627408466;
&#3627408479;&#3627408481;


3. If there are complex roots, (&#3627409148;±&#3627409149;&#3627408471;)
Under damped
&#3627408463;
2
4&#3627408474;
2
<
&#3627408472;
&#3627408474;


&#3627408485;(&#3627408481;)=&#3627408466;
??????&#3627408481;
[&#3627408464;
1cos&#3627409149;&#3627408481;+&#3627408464;
2&#3627408480;&#3627408470;&#3627408475;&#3627409149;&#3627408481;]

The x-t Graph
&#3627408481;
&#3627408485;
&#3627408438;&#3627408479;&#3627408470;&#3627408481;&#3627408470;&#3627408464;&#3627408462;&#3627408473;&#3627408473;&#3627408486; &#3627408465;&#3627408462;&#3627408474;&#3627408477;&#3627408466;&#3627408465;
&#3627408450;&#3627408483;&#3627408466;&#3627408479; &#3627408465;&#3627408462;&#3627408474;&#3627408477;&#3627408466;&#3627408465;
&#3627408456;&#3627408475;&#3627408465;&#3627408466;&#3627408479; &#3627408465;&#3627408462;&#3627408474;&#3627408477;&#3627408466;&#3627408465;
In this level, we will learn only
underdamped;
&#3627408463;
2
4&#3627408474;
2
<
&#3627408472;
&#3627408474;


Suppose, &#3627409150;=
&#3627408463;
2&#3627408474;
, ??????
0=
&#3627408472;
&#3627408474;

&#3627409150;=&#3627408465;&#3627408462;&#3627408474;&#3627408477;&#3627408470;&#3627408475;&#3627408468; &#3627408464;&#3627408476;&#3627408475;&#3627408480;&#3627408481;&#3627408462;&#3627408475;&#3627408481;
??????
0=&#3627408482;&#3627408475;&#3627408465;&#3627408462;&#3627408474;&#3627408477;&#3627408466;&#3627408465; &#3627408475;&#3627408462;&#3627408481;&#3627408482;&#3627408479;&#3627408462;&#3627408473; &#3627408467;&#3627408479;&#3627408466;&#3627408478;&#3627408482;&#3627408466;&#3627408475;&#3627408464;&#3627408486;

Since, &#3627408439;=−
&#3627408463;
2&#3627408474;
±
&#3627408463;
2
4&#3627408474;
2
−
&#3627408472;
&#3627408474;


&#3627408439;=−&#3627409150;±&#3627408471;&#3627409150;
2
−??????
0
2

&#3627408439;=−&#3627409150;±&#3627408471;??????
0
2
−&#3627409150;
2


&#3627408485;(&#3627408481;)=&#3627408466;
−??????&#3627408481;
[&#3627408464;
1cos&#3627408481;??????
0
2
−&#3627409150;
2
+&#3627408464;
2&#3627408480;&#3627408470;&#3627408475;&#3627408481;??????
0
2
−&#3627409150;
2
]
&#3627408485;(&#3627408481;)=&#3627408466;
−??????&#3627408481;
[??????sin&#3627408481;??????
0
2
−&#3627409150;
2
+??????]

Suppose ??????
′
=??????
0
2
−&#3627409150;
2





??????=&#3627408470;&#3627408475;&#3627408470;&#3627408481;&#3627408470;&#3627408462;&#3627408473; &#3627408462;&#3627408474;&#3627408477;&#3627408473;&#3627408470;&#3627408481;&#3627408482;&#3627408465;&#3627408466;
&#3627408466;
−??????&#3627408481;
&#3627408484;&#3627408470;&#3627408473;&#3627408473; &#3627408474;&#3627408462;&#3627408472;&#3627408466; &#3627408481;ℎ&#3627408466; ?????? &#3627408465;&#3627408470;&#3627408466; &#3627408465;&#3627408476;&#3627408484;&#3627408475;
&#3627408485;(&#3627408481;)=??????&#3627408466;
−??????&#3627408481;
cos??????
′
&#3627408481;+??????]