一根长度为 1 、与墙夹角 30° 的木棒，下滑直至水平，求木棒滑落过程中扫过的面积

dodonaomikiki

请看图片

dodonaomikiki

期间疑问：木棒是否与包络线相切？切点瞬时速度必沿切线，也就是木棒的瞬时方向？
我们需要计算的是不是包络线包络的面积？

dodonaomikiki

一个不完备的分析

luyuanhong

王守恩

\(S(\theta)=\frac{3(2\pi+\sin(4\theta)-4\theta)}{64}\)

王守恩

\(S(\theta)=\frac{3(2\pi+\sin(4\theta)-4\theta)}{64}\)

\(有个有趣的现象。若\ a+b=90^\circ\ 则\ S(a)+S(b)=\frac{3\pi}{32}\)

王守恩

\(S(\theta)=\frac{3(2\pi+\sin(4\theta)-4\theta)}{64}\)

\(有个有趣的现象。若\ a+b=90^\circ\ 则\ S(a)+S(b)=\frac{3\pi}{32}\)

譬如

\(S(00^\circ)=\frac{3\pi}{32}\)

\(S(15^\circ)=\frac{3}{64}\big(\frac{\sqrt{3}}{2} + \frac{5\pi}{3}\big)\)

\(S(30^\circ)=\frac{3}{64}\big(\frac{\sqrt{3}}{2} + \frac{4\pi}{3}\big)\)

\(S(45^\circ)=\frac{3\pi}{64}\)

\(S(60^\circ)=\frac{3}{64}\big(-\frac{\sqrt{3}}{2} + \frac{2\pi}{3}\big)\)

\(S(75^\circ)=\frac{3}{64}\big(-\frac{\sqrt{3}}{2} + \frac{\pi}{3}\big)\)

\(S(90^\circ)=0\)


(3 \[Pi])/32,
3/64 ((89 \[Pi])/45 + Sin[\[Pi]/45]),
3/64 ((88 \[Pi])/45 + Sin[(2 \[Pi])/45]),
3/64 ((29 \[Pi])/15 + Sin[\[Pi]/15]),
3/64 ((86 \[Pi])/45 + Sin[(4 \[Pi])/45]),
3/64 ((17 \[Pi])/9 + Sin[\[Pi]/9]),
3/64 ((28 \[Pi])/15 + Sin[(2 \[Pi])/15]),
3/64 ((83 \[Pi])/45 + Sin[(7 \[Pi])/45]),
3/64 ((82 \[Pi])/45 + Sin[(8 \[Pi])/45]),
3/64 (Sqrt[5/8 - Sqrt[5]/8] + (9 \[Pi])/5),
3/64 ((16 \[Pi])/9 + Sin[(2 \[Pi])/9]),
3/64 ((79 \[Pi])/45 + Sin[(11 \[Pi])/45]),
3/64 ((26 \[Pi])/15 + Cos[(7 \[Pi])/30]),
3/64 ((77 \[Pi])/45 + Cos[(19 \[Pi])/90]),
3/64 ((76 \[Pi])/45 + Cos[(17 \[Pi])/90]),
3/64 (Sqrt[3]/2 + (5 \[Pi])/3),
3/64 ((74 \[Pi])/45 + Cos[(13 \[Pi])/90]),
3/64 ((73 \[Pi])/45 + Cos[(11 \[Pi])/90]),
3/64 (Sqrt[5/8 + Sqrt[5]/8] + (8 \[Pi])/5),
3/64 ((71 \[Pi])/45 + Cos[(7 \[Pi])/90]),
3/64 ((14 \[Pi])/9 + Cos[\[Pi]/18]),
3/64 ((23 \[Pi])/15 + Cos[\[Pi]/30]),
3/64 ((68 \[Pi])/45 + Cos[\[Pi]/90]),
3/64 ((67 \[Pi])/45 + Cos[\[Pi]/90]),
3/64 ((22 \[Pi])/15 + Cos[\[Pi]/30]),
3/64 ((13 \[Pi])/9 + Cos[\[Pi]/18]),
3/64 ((64 \[Pi])/45 + Cos[(7 \[Pi])/90]),
3/64 (Sqrt[5/8 + Sqrt[5]/8] + (7 \[Pi])/5),
3/64 ((62 \[Pi])/45 + Cos[(11 \[Pi])/90]),
3/64 ((61 \[Pi])/45 + Cos[(13 \[Pi])/90]),
3/64 (Sqrt[3]/2 + (4 \[Pi])/3),
3/64 ((59 \[Pi])/45 + Cos[(17 \[Pi])/90]),
3/64 ((58 \[Pi])/45 + Cos[(19 \[Pi])/90]),
3/64 ((19 \[Pi])/15 + Cos[(7 \[Pi])/30]),
3/64 ((56 \[Pi])/45 + Sin[(11 \[Pi])/45]),
3/64 ((11 \[Pi])/9 + Sin[(2 \[Pi])/9]),
3/64 (Sqrt[5/8 - Sqrt[5]/8] + (6 \[Pi])/5),
3/64 ((53 \[Pi])/45 + Sin[(8 \[Pi])/45]),
3/64 ((52 \[Pi])/45 + Sin[(7 \[Pi])/45]),
3/64 ((17 \[Pi])/15 + Sin[(2 \[Pi])/15]),
3/64 ((10 \[Pi])/9 + Sin[\[Pi]/9]),
3/64 ((49 \[Pi])/45 + Sin[(4 \[Pi])/45]),
3/64 ((16 \[Pi])/15 + Sin[\[Pi]/15]),
3/64 ((47 \[Pi])/45 + Sin[(2 \[Pi])/45]),
3/64 ((46 \[Pi])/45 + Sin[\[Pi]/45]),
(3 \[Pi])/64,
3/64 ((44 \[Pi])/45 - Sin[\[Pi]/45]),
3/64 ((43 \[Pi])/45 - Sin[(2 \[Pi])/45]),
3/64 ((14 \[Pi])/15 - Sin[\[Pi]/15]),
3/64 ((41 \[Pi])/45 - Sin[(4 \[Pi])/45]),
3/64 ((8 \[Pi])/9 - Sin[\[Pi]/9]),
3/64 ((13 \[Pi])/15 - Sin[(2 \[Pi])/15]),
3/64 ((38 \[Pi])/45 - Sin[(7 \[Pi])/45]),
3/64 ((37 \[Pi])/45 - Sin[(8 \[Pi])/45]),
3/64 (-Sqrt[5/8 - Sqrt[5]/8] + (4 \[Pi])/5),
3/64 ((7 \[Pi])/9 - Sin[(2 \[Pi])/9]),
3/64 ((34 \[Pi])/45 - Sin[(11 \[Pi])/45]),
3/64 ((11 \[Pi])/15 - Cos[(7 \[Pi])/30]),
3/64 ((32 \[Pi])/45 - Cos[(19 \[Pi])/90]),
3/64 ((31 \[Pi])/45 - Cos[(17 \[Pi])/90]),
3/64 (-(Sqrt[3]/2) + (2 \[Pi])/3),
3/64 ((29 \[Pi])/45 - Cos[(13 \[Pi])/90]),
3/64 ((28 \[Pi])/45 - Cos[(11 \[Pi])/90]),
3/64 (-Sqrt[5/8 + Sqrt[5]/8] + (3 \[Pi])/5),
3/64 ((26 \[Pi])/45 - Cos[(7 \[Pi])/90]),
3/64 ((5 \[Pi])/9 - Cos[\[Pi]/18]),
3/64 ((8 \[Pi])/15 - Cos[\[Pi]/30]),
3/64 ((23 \[Pi])/45 - Cos[\[Pi]/90]),
3/64 ((22 \[Pi])/45 - Cos[\[Pi]/90]),
3/64 ((7 \[Pi])/15 - Cos[\[Pi]/30]),
3/64 ((4 \[Pi])/9 - Cos[\[Pi]/18]),
3/64 ((19 \[Pi])/45 - Cos[(7 \[Pi])/90]),
3/64 (-Sqrt[5/8 + Sqrt[5]/8] + (2 \[Pi])/5),
3/64 ((17 \[Pi])/45 - Cos[(11 \[Pi])/90]),
3/64 ((16 \[Pi])/45 - Cos[(13 \[Pi])/90]),
3/64 (-(Sqrt[3]/2) + \[Pi]/3),
3/64 ((14 \[Pi])/45 - Cos[(17 \[Pi])/90]),
3/64 ((13 \[Pi])/45 - Cos[(19 \[Pi])/90]),
3/64 ((4 \[Pi])/15 - Cos[(7 \[Pi])/30]),
3/64 ((11 \[Pi])/45 - Sin[(11 \[Pi])/45]),
3/64 ((2 \[Pi])/9 - Sin[(2 \[Pi])/9]),
3/64 (-Sqrt[5/8 - Sqrt[5]/8] + \[Pi]/5),
3/64 ((8 \[Pi])/45 - Sin[(8 \[Pi])/45]),
3/64 ((7 \[Pi])/45 - Sin[(7 \[Pi])/45]),
3/64 ((2 \[Pi])/15 - Sin[(2 \[Pi])/15]),
3/64 (\[Pi]/9 - Sin[\[Pi]/9]),
3/64 ((4 \[Pi])/45 - Sin[(4 \[Pi])/45]),
3/64 (\[Pi]/15 - Sin[\[Pi]/15]),
3/64 ((2 \[Pi])/45 - Sin[(2 \[Pi])/45]),
3/64 (\[Pi]/45 - Sin[\[Pi]/45]),
0}

Table[3*((360 - 4 a) Pi/180 + Sin[4 a Pi/180])/64, {a, 0, 90}]

dodonaomikiki

请教陆老师，这个方程哪里得来的？甚感迷惑！诚心请教~~~~~~

其他基本能够理解啦！

dodonaomikiki

DING顶一顶

dodonaomikiki

不晓得下面这个方程怎么来的？
\(
    \displaystyle{  \boldsymbol      {    \frac{x}{sint}  +\frac{y}{cost}=1                       }}


...\)