This document summarizes oscillatory motion and simple harmonic motion. It describes how oscillatory motion involves periodic movement like that of a mass-spring system without friction. The motion of such a system can be modeled using Newton's second law, resulting in a differential equation of the form d2x/dt2 = -kx, where k is a constant. This has the form of a simple harmonic motion equation. Other examples of simple harmonic motion discussed include angular simple harmonic motion using a simple pendulum, as well as damped simple harmonic motion where a frictional force is present. Key characteristics of simple harmonic motion like amplitude, angular velocity, phase constant, and the sinusoidal nature of the position, velocity and acceleration functions are also
5. ∴𝛼 =
−𝑚𝑔𝐿 𝑠𝑖𝑛𝜃
𝐼
***General formula of simple harmonic motion
equation is 𝑎 = −𝑘𝑥 where k is a constant
Since
𝑚𝑔𝐿
𝐼
is a constant, this equation is
in a form of a simple harmonic motion
equation when 𝑠𝑖𝑛𝜃 ≈ 𝜃, or when is
very small (<10°)
Since 𝑎 𝑡 = −𝜔2
𝑥(𝑡),
𝛼 = −𝜔 2
𝜃
𝐼 = 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑖𝑡𝑖𝑎 𝑜𝑟 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝜔 =
𝑚𝑔𝐿
𝐼
𝛼 =
−𝑚𝑔𝐿𝜃
𝐼
𝐼 = 𝑚𝐿2
𝜔 =
𝑔
𝐿
𝜃 = 𝜃0 𝑠𝑖𝑛(𝜔𝑡 + 𝜑)
𝜃
ℎ
𝑚𝑔
𝜃
𝑚𝑔 𝑐𝑜𝑠𝜃𝑚𝑔 𝑠𝑖𝑛𝜃
Physical Pendulum
*** In our real life, string has mass
Think of it as one whole body without T
Since 𝐼 = 𝑚𝑘2
, k = radius of gyration
𝛼 =
−𝑚𝑔ℎ𝜃
𝐼
𝜔 =
𝑚𝑔ℎ
𝐼
𝜔 =
𝑔ℎ
𝑘2
𝜔 = 2𝜋𝑓 =
2𝜋
𝑇
𝑇 = 2𝜋
𝐿
𝑔
6. Torsional Pendulum
*** In our real life, string has mass
Think of it as one whole body without T
𝜏 = −𝜅𝜃
𝜅 = 𝐾𝑎𝑝𝑝𝑎
𝜏 = 𝐼𝛼
𝛼 = −𝜔2
𝜃
𝛼 = −
𝜅
𝐼
𝜃
𝝎 =
𝜿
𝑰
= 𝟐𝝅𝒇 =
𝟐𝝅
𝑻
Damped Simple Harmonic motion
*** In real life, there is a frictional force
(friction, air resistance)
Free-body diagram at 𝑥 𝑚𝑎𝑥
***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
2nd Newton law
𝐹 = 𝑚𝑎
−𝑘𝑥 − 𝑏𝑣 = 𝑚𝑎
−𝑘𝑥 − 𝑏
𝑑𝑥
𝑑𝑡
= 𝑚
𝑑2
𝑥
𝑑𝑡2
𝑚𝑥′′
+ 𝑏𝑥′
+ 𝑘𝑥 = 0
Change form,
𝑚𝐷2
+ 𝑏𝐷 + 𝑘 = 0
m 𝑥 = 0
m
−𝑘𝑥 − 𝑏𝑣
𝑚𝑔
𝜃
𝑠𝑝𝑟𝑖𝑛𝑔
7. 𝑚𝐷2
+ 𝑏𝐷 + 𝑘 = 0
𝐷 =
−𝑏 ± 𝑏2 − 4𝑚𝑘
2𝑚
𝐷 = −
𝑏
2𝑚
±
𝑏2
4𝑚2
−
𝑘
𝑚
There will be 3 cases
1. If there are 2 distinct roots, 𝐷1, 𝐷2
Over damped
𝑏2
4𝑚2
>
𝑘
𝑚
𝑥(𝑡) = 𝑐1 𝑒 𝑟1 𝑡
+ 𝑐2 𝑒 𝑟2 𝑡
2. If there are a repeated single root, (D)
Critically damped
𝑏2
4𝑚2
=
𝑘
𝑚
𝑥(𝑡) = 𝑐1 𝑒 𝑟𝑡
+ 𝑐2 𝑡𝑒 𝑟𝑡
3. If there are complex roots, (𝛼 ± 𝛽𝑗)
Under damped
𝑏2
4𝑚2
<
𝑘
𝑚
𝑥(𝑡) = 𝑒 𝛼𝑡
[𝑐1 cos 𝛽𝑡 + 𝑐2 𝑠𝑖𝑛 𝛽𝑡 ]
The x-t Graph
𝑡
𝑥
𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑑𝑎𝑚𝑝𝑒𝑑
𝑂𝑣𝑒𝑟 𝑑𝑎𝑚𝑝𝑒𝑑
𝑈𝑛𝑑𝑒𝑟 𝑑𝑎𝑚𝑝𝑒𝑑
In this level, we will learn only
underdamped;
𝑏2
4𝑚2 <
𝑘
𝑚
Suppose, 𝛾 =
𝑏
2𝑚
, 𝜔0 =
𝑘
𝑚
𝛾 = 𝑑𝑎𝑚𝑝𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜔0 = 𝑢𝑛𝑑𝑎𝑚𝑝𝑒𝑑 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
Since, 𝐷 = −
𝑏
2𝑚
±
𝑏2
4𝑚2 −
𝑘
𝑚
𝐷 = −𝛾 ± 𝑗 𝛾2 − 𝜔0
2
𝐷 = −𝛾 ± 𝑗 𝜔0
2 − 𝛾2
𝑥(𝑡) = 𝑒−𝛾𝑡[𝑐1 cos 𝑡 𝜔0
2 − 𝛾2 + 𝑐2 𝑠𝑖𝑛 𝑡 𝜔0
2 − 𝛾2 ]
𝑥(𝑡) = 𝑒−𝛾𝑡
[𝐴 sin 𝑡 𝜔0
2 − 𝛾2 + 𝜑 ]
Suppose 𝜔′
= 𝜔0
2 − 𝛾2
𝐴 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒
𝑒−𝛾𝑡
𝑤𝑖𝑙𝑙 𝑚𝑎𝑘𝑒 𝑡ℎ𝑒 𝐴 𝑑𝑖𝑒 𝑑𝑜𝑤𝑛
𝑥(𝑡) = 𝐴𝑒−𝛾𝑡
cos 𝜔′
𝑡 + 𝜑 ]