[Mechanics II] Final Exam

2020-08-11 Sin-iu 發表迴響

Mechanics II (Course Number: Phys 2005), Department of Physics, National Taiwan University

Total: 100 points
Date: 17 June 2019
Duration: 3 hr
Instructor: Yih-Yuh Chen

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古典力學

Kepler Problem

2020-08-10 Sin-iu 發表迴響

1 Central Force and Central Potential

The central force on an object is a force directed along the line joining the object and the origin. We often describe such forces as

$\mathbf{F}(\mathbf{r})=F(\mathbf{r})\hat{\mathbf{r}},$ …(1)

where $\mathbf{r}$ is the position, or called radius vector, with $\hat{\mathbf{r}}$ being its unit vector (or called unit radial vector) and $r$ being its magnitude. $\mathbf{F}$ is the force as a vector-valued function of position. The (vector) field derived from a central force is called a central field.

Here we restrict our attention to conservative systems (i.e. without energy dissipation due to non-conservative forces, such as friction), then the magnitude $F(r)$ of a central force can always be expressed as the derivative of a time-independent potential energy function $U(r)$ :

$F(r)=-\dfrac{\text{d}U(r)}{\text{d}r}.$ …(2)

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台語

李臨秋、鄧雨賢－四季紅

2020-08-10 Sin-iu 發表迴響

台語流行歌曲〈四季紅〉是由知名作詞人李臨秋（1909-1979）和知名作曲家鄧雨賢合作，原唱是純純（1914-1943）與豔豔，1938年由台灣日東唱片發行。

〈四季紅〉的歌詞大意是敘述熱戀情侶的感情，不受季節的變遷所影響，而它的曲風也一反台語情歌哀怨的基調，屬於活潑且生動的作品。鄧泰超（鄧雨賢之孫）在開放博物館以中國傳統樂理分析它的曲調如下（有蟲膠曲盤可試聽）：

歌曲曲調為四拍子，分二段，前段以兩小節為一樂句，前兩句前均起於徵音，分別進行至角音、商音，屬徵調式；第三句宮音起，結束於徵音；第四句則角音起，結束於宮音，屬宮調式。後段可分為兩、兩、三小節句式。第一句起於徵音進行至商音，第二句起於宮音亦進行至宮音，第三句起於徵音亦結束於徵音，屬徵調式。因此此曲仍屬徵調式，只是其中含有轉至宮調式的性質。另本曲雖用七聲音階，但清角（第六級音）及變宮（第三級音）音的運用屬鄰音或經過音性質、因此仍具明顯的五聲音階性格。

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