---
title: 普通物理學(一)
tags: 普通物理學(一), 共同筆記
---

# 普通物理學(一)
>原文書:University Physics with modern physics 15th ed
>
>>Author: Hugh D. Young , Roger A. Freedman
>>Publisher: 高立圖書/普林斯頓國際有限公司南方業務部
>>Phone: [phone]
>>Telephone: [phone]
>>Mail: [e-mail]
>>Line-ID: gan0932083861
## 教授相關資訊
- 教授的名字 : 田興龍
- Office_hour
Tue. 12:00~13:00
- Email
[e-mail]
- Office
理學教學大樓 6F 36652
## 考試日程及配分
- 考試日期
Mid. Exam I 2020-10-19
Mid. Exam II 2020-11-23
Final Exam 2021-01-04
- 配分
HW=10%

>late HW receive a 50% reduction in grade

MEI=20%
MEII=30%
FE=40%

## 教學內容:
1. $Mathematics$
1. $Derivatives$(微分)
(1) $Secant$(割線)
$\Delta x \equiv x_f-x$
$\Delta y \equiv y_f-y$
$\Delta: \;Difference$
$Geometry : slope :\frac{\Delta y}{\Delta x}$

-2020/9/7
(2) $Tangent$(切線)
$\Delta x_\downarrow\to0\Rightarrow dx$
$B \;approaches\; to\; A$
$dx$ : $infinitesimal$(無窮小)
$\Delta x\rightarrow0 : two\;seperated\;points$
$\quad\Rightarrow one\;point$
$\Rightarrow\;discontinuous\;\rightarrow\;continuous$

$\frac{dy}{dx}\equiv\lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}$
$\Rightarrow\frac{dy}{dx}:\;slope \;of \;point\;A$
$if \;x\rightarrow\;time\;t\Rightarrow concept\; of\;instantaneous\;value\;of\;time\;t$
$\frac{dy}{dx}=f(x)\Rightarrow dy = f(x)dx$

>實際上可把微分符號直接移項

(3) $Properties$(運算性質)
$\quad$(a) $Sum \frac{d}{dx}(f(x)+g(x))=\frac{df}{dx}+\frac{dg}{dx}$
$\quad \quad EX:3x^2+sin(x)\quad\frac{d}{dx}\Rightarrow6x+cos(x)$
$\quad$(b) $Product:\frac{d}{dx}[f(x)*g(x)]$
$\quad\quad=\frac{df}{dx}g(x)+\frac{dg}{dx}f(x)$
$\quad\quad EX:\frac{d}{dx}(x^2*cos(x))=2x*cos(x)+x^2*(-sin(x))$
$\quad$($c$)$Quotient:\frac{d}{dx}[\frac{f(x)}{g(x)}]$
$\quad\quad \frac{df}{dx}*\frac{1}{g}+f\;\frac{d}{dx}(\frac{1}{g})$
$\quad$(d) $chain\;rule$
$\quad\quad f=f(u),u=u(x)$
$\quad \frac{df}{dx}=\frac{df}{du}*\frac{du}{dx}$
$\quad\quad EX:\frac{d}{dt}(sin(wt))=cos(wt)*w$
$\quad\quad EX:\frac{d}{dx}(sin(x^2))=cos(x^2)*2x$
$\quad$(e) $2nd\;deriviative$
$\quad\quad\frac{d}{dx}(\frac{dy}{dx})=\frac{d^2y}{dx^2}\quad EX:y=ax^3\quad a=c^+\quad\frac{d^2y}{dx^2}=6ax$
2. $Integrals$(積分)
$\quad\quad Objective:\;Area\;between\;x_1\;and\;x_2$
$\quad\quad N\;divisions\;in\;[x_1,x_2]\;width\;\Delta x$
$\quad\quad \sum_{i=0}^n\;f(xi)\Delta x\approx Area\;between [x_1,x_2](\approx:discontinous)$
$\quad\quad Now\;N\rightarrow\infty ,\Delta x\rightarrow0\Rightarrow dx\Rightarrow continuous$
$\quad\quad then\;\sum\rightarrow\int ,f(xi)\rightarrow f(x)$
$\quad\quad \lim_{\Delta x \to 0} \sum f(xi)\Delta x\equiv\int_{x1}^{x2}f(x)dx=A(x)|^{x2}_{x1}=A(x_2)-A(x_1)$
>$Note:\;A(x) = \int f(x)dx$

-2020/9/9

3. $Exponentials$ & $Natural\;logarithm$
$\quad\quad Euler:e\equiv\frac{\lim}{n\rightarrow\infty}(1+\frac{1}{n})\sim2.718$
$\quad\quad Exponential\;funtion(指數):e^x\equiv exp(x)$
$\quad\quad \frac{d}{dx}(e^x)=e^x$
$\quad\quad Natural\;logarithm(對數):ln$
$\quad\quad ln(exp(x))=x$
$\quad\quad \frac{d}{dx}(lm\;x):\frac{1}{x}$
>$Note:\int\frac{1}{x}dx=ln(x)$
4. $Trangonometry$
$\quad\quad sin(\alpha\pm\beta)=sin(\alpha)cos(\beta)\pm sin(\beta)cos(\alpha)$
$\quad\quad cos(\alpha\pm\beta)=cos(\alpha)cos(\beta)\mp sin(\alpha)sin(\beta)$
$\quad\quad cos^2(\theta)=\frac{1+cos(2\theta)}{2}$
$\quad\quad sin^2(\theta)=\frac{1-cos(2\theta)}{2}$
5. $Vector\;algebra$
$\quad\quad$(1)$Dot:$
$\quad\quad meaning:Projection$

$\quad\quad\vec A*\vec B =|\vec A||\vec B|cos(\theta)$
$\quad\quad$(2)$Cross:$
$\quad\quad Right-hand\;screw\;rule(direction)$
$\quad\quad meaning:Area$
$\quad\quad constructed\;by\;\vec A$&$\vec B$
6. $Approximations$
$\quad\quad For|x|<<1$
$\quad\quad e^x \sim 1+x$
$\quad\quad sin(x)\sim x$
$\quad\quad ln(1+x)\sim x$
$\quad\quad(1+x)^p \sim 1+px$

#### Review
1. $why\;\vec V,\vec a,\vec p?\;enegry?\;work?$
$\vec F\frac{\rightarrow}{\Delta 無窮小} enegry$
2. $Newton's\;law$

3. $Work-enegry\;Theorem$
$\quad W_{net}=\int_i^f \vec{F} \cdot \mathrm{d} \vec{r}=\int_i^f F_{x} \mathrm{d} x+\int_i^f F_{y} \mathrm{d} y+\int_i^f F_{z} \mathrm{d} z$

$\quad\quad(\text In\;one-dimensional\; space)$
$\quad\quad W_{net}=\int_i^f F\cdot \mathrm{d} x=\int_i^f F\cdot \mathrm{d} x=$

$$
\begin{aligned}
&\int_i^f (ma)\cdot \mathrm{d} x \\
= &\int m\frac{\mathrm{d}v_x}{\mathrm{d}t}\mathrm{d}x \\
= &\int m\frac{\mathrm{d}x}{\mathrm{d}t}\mathrm{d}v_x \\
= &\int_{v_i}^{v_f} mv_x \;\mathrm{d}v_x \\
= &\frac{1}{2}m{v_x}^2\big|_{v_i}^{v_f} \\
= &\frac{1}{2}m{v_f}^2 - \frac{1}{2}m{v_i}^2
\end{aligned}
$$

4. $Potential\;Enegry$

$\quad\quad \because v=c^+ \therefore \Delta K=0=W_{net}$
$\quad\quad \because \Delta x \ne 0 \therefore \vec{F}_{net}=0$
$\quad\quad \vec{F}_{net}=\vec{F}_{ext}+\vec{F}_{g}=0$
$\quad\quad W_{net}=\int_i^f \vec{F}_{net} \cdot \mathrm{d \vec{r_y}}=0$
$\quad\quad W_{ext}=int_i^f \vec{f}_{ext} \cdot \mathrm{d} \vec{r_y}>0$

$\quad\quad Q:\text Where\;is\;this\;energy\;?$
$\quad\quad A:\text We\;say\;it\;stored\;as\;potential\;energy\;,U\;.$
$\quad \quad \quad \;\;W_{ext}=\int_i^f \vec{F}_{ext} \cdot \mathrm{d} \vec{r_y}\equiv U_f-U_i\equiv\Delta U$
$\quad\quad\quad \;\;\because W_{net}+W_g=0\;,\;\therefore W_g=-W_{ext}=-\Delta U$
$\quad\quad\quad\;\;\Longrightarrow \int_i^f \vec{F}_{ext} \cdot \mathrm{d} \vec{r}\;=\;-\Delta U"$
$\quad\quad\quad\;\; \text保守力作功之目的在於"減少""位能"$
$\quad\quad\quad\;\;(低能是穩定狀態)$
$\quad\quad\quad\;\;*\;\;\;\;\;\vec{F}_g\;:\text a\;kind\;of\;conservative\;force$
$\quad\quad\quad\;\;\;\;\;\;"-"\;:\text work\;done\;by\;con.force\;leads\;to\;a\;decrease\;in\;U$

5. $Conservative\;Force$

$\;\;Definition\;:\oint \vec{F}_{con.}\cdot \mathrm{d} \vec{r}=0$
$\;\;\oint \vec{F}_{con.} \cdot \mathrm{d} \vec{r}=\int_{A,1}^B \vec{F}_c\cdot\mathrm{d}\vec{r}+\int_{B,2}^A\vec{F}_c\cdot\mathrm{d}\vec{r}=\int_{A,1}^B\vec{F}_c\cdot\mathrm{d}\vec{r}-\int_{A,2}^B\vec{F}_c\cdot\mathrm{d}\vec{r}=0$
$\;\;\Longrightarrow\int_{A,1}^B\vec{F}_c\cdot\mathrm{d}\vec{r}=\int_{A,2}^B\vec{F}_c\cdot\mathrm{d}\vec{r}$
$\;\;\Longrightarrow\text 起/終點相同的情況下，不論路徑，保守力所作之功皆相同$
$\;\;\;\;\;\;\;\;\;\text Work\;done\;by\;the\;con.force\;is\;independent\;of\;the\;path.$
$\;\;\;\;\;\;\;\;\;\text 在保守力場中，力學能守恆$

6. $Mechanical\;enegry$
$\;\color{red}{if\;\vec{F}_{net}=\vec{F}_{con.}\;"only"\;!!}$
$\;\Delta U=-(\int\vec{F}_{con.}\cdot\mathrm{d}\vec{r})$
$\;\;\;\;\;\;\;\;=-(\int\vec{F}_{net}\cdot\mathrm{d}\vec{r})$
$\;\;\;\;\;\;\;\;=-(\Delta K)$
$\;\Longrightarrow\Delta U+\Delta K=0\;\;\equiv\;\;\Delta (U+K)=0$

$\;\;\;\;\;\;\;\;E=U+K\;,\color{red}{if\;\Delta E=0\;\;\land\;\;E=C^+},we\;call\;the\;mechanical\;energy\;conservative.$

7. $Potential\;Enegry \xrightarrow{?} \vec F_{con}$
$\int\vec{F}_{con.}\cdot\mathrm{d}\vec{r}=-\Delta U=-\int\mathrm{d} U$
$\;\Longrightarrow\vec{F}_{con.}\cdot\mathrm{d}\vec{r}=F_{con.\;r}\cdot\mathrm{d}r=-\mathrm{d}U$
$\;\Longrightarrow\color{red}{F_{con.\;r}={-\mathrm{d}U \over \;\mathrm{d}r}}$
$\;\;\;\;\color{green}{ex.}$
$\;\;\color{green}{|}\;\;\;\;Elastic\;P.E.$
$\;\;\color{green}{|}\;\;\;\;已知\;\;U(X)={1 \over 2}kx^2\;\;,求\;\;F(x)=\;\;?$
$\;\;\color{green}{|}\;\;\;\;\Longrightarrow\;\;F(x)={-\mathrm{d} U \over \;\mathrm{d}x}={{-\mathrm{d}({1 \over 2}kx^2)} \over \;\mathrm{d}x}=-kx$

8. $Enegry\;conservation\;with\;dissipative\;force(散逸力)$
$W_{net}=\color{red}{(}W_{con.}\color{red}{)}+W_{non-con.}=\Delta K$
$\quad\quad\;=\color{red}{(}-\Delta U\color{red}{)}+W_{non-con.}$
$\Longrightarrow W_{non-con.}=\Delta U+\Delta K=\Delta E \Longrightarrow\color{green}{cf.\;\;5.\;Conservative\;Force}$

> 園中花化為灰
> 夕陽一點已西墜
> 相思淚心已碎
> 空聞馬蹄歸
> 秋日殘紅螢火飛
> [name=秦少游]

[維基百科-秦少游](https://zh.wikipedia.org/zh-tw/%E7%A7%A6%E8%A7%82)

# CH8.5 Center of mass (CM)
$\;*Translated\;motion\;（平移運動）\;of\;CM\;is\;characteristic\\ \;\;\;of\;the\;system\;as\;a\;whole$
$1.Types:\\\;\;(1)Two\;particles:$

$\;\;\;\;\;\;\;\;X_{CM}={m_1x_1+m_2x_2 \over m_1+m_2}$
$\;\;\;(2)Many\;particles:\\\;\;\;\;\;\;\vec{r}_{CM}$

## 名言
>永遠不要相信在台上講話的人
>[name=田興龍]
>不能表達自己意見的,都是被宰割的一群
>[name=田興龍]

## 2020/9/7 Mon.
>Moodle:https://drive.google.com/file/d/1dIVi_UMTRI58FlImSACMmUyfHm6B72GR/view?usp=sharing
## 2020/9/9 Wed.
>Moodle:https://drive.google.com/file/d/12zyrPj9vytJyr9oRbyKE-0BzTOIXWnmF/view?usp=sharing
## 2020/9/14 Mon.
>Moodle:https://drive.google.com/file/d/1M8kk85BpXPx5tyci_fcnzHebNqCcE36w/view?usp=sharing