十一 27
![](http://www.qiusir.com/wp-content/gallery/net/Nobelnote.jpeg)
“世界会不会变好不重要,我们要变好才重要。”网上没查到是不是诺贝尔说的,不过这话对莫名焦虑的人来说也不是没有一点道理:)
上课带领学生推双星的公式,
![\frac{L^3}{T^2}=\frac{G(m_1+m_2)}{4\pi^2}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_6b220cebced2fb04c3ba5e6120f9e32b.gif)
![\frac{R^3}{T_1^2}=\frac{GM}{4\pi^2}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_4c1d48700d608367b368c9cd8b01927d.gif)
![T_1=2\pi \sqrt{\frac{R^3}{GM}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_aa9ccc2c870c93b561c9528f5f678bd2.gif)
![T_1=2\pi \sqrt{\frac{R}{g}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_08623f905b8d1659dd0cad23bb1b5cb3.gif)
类比单摆的公式,
![T=2\pi \sqrt{\frac{l}{g}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_5920822aff6dc914e8ebb9b67b331d95.gif)
![T=2\pi \sqrt{\frac{lcos\theta}{g}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_9c4e0316a6808a304521e5214c007c8f.gif)
![T=2\pi \sqrt{\frac{R}{g}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_c6d96c2fb33d14cf05eda216c1b343d3.gif)
![](http://www.qiusir.com/wp-content/gallery/physics/epqneycc.png)
和简谐振动一章推到单摆的周期很接近,设想地面上(可以设想是在很高建筑上)悬点o悬挂摆长l的单摆(摆角很小,一样能近似简振的),万有引力的切向分力提供回复力。
摆球在地表,所有有
![\frac{GMm}{R^2}=mg](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_49296849fc1c326e00943d8b5d706cd8.gif)
![F_t=mgsin(\alpha+\beta)](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_b28be8b3b0a107823bd8e10c90f3cd07.gif)
![sin\theta\approx\theta](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_780fe565c11c4eda55d2915f8b2221a1.gif)
![F_t=mg(\alpha+\beta)](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_dae5cd83f052875281956530374c9eef.gif)
![F_t=-mg(\frac{1}{R}+\frac{1}{l})x](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_5512894d951ce5a7ff38e12a485b8262.gif)
套用弹簧振子周期公式,
![T=2\pi\sqrt{\frac{m}{k}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_237cb1e42fcfd11922c5aba8e572bfe9.gif)
![T=2\pi\sqrt{\frac{Rl}{g(R+l)}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_d6a35eeda0517121ba145136d56d74a6.gif)
带入特殊值到上述公式,比如摆长远小于地球半径,
![T=2\pi \sqrt{\frac{l}{g}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_5920822aff6dc914e8ebb9b67b331d95.gif)
![T=2\pi \sqrt{\frac{R}{2g}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_7777d004528723cfea992ed49fdb8d6d.gif)
![T=2\pi \sqrt{\frac{R}{g}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_c6d96c2fb33d14cf05eda216c1b343d3.gif)
![](http://www.qiusir.com/wp-content/gallery/physics/eqbzh.jpg)
关于贯穿地球的地下铁通行时间的推算,也是用简谐振动的周期。由于球壳对球壳内物体的万有引力为零,所以物体穿行地球时所受的万有引力仅与此处内部球体有关,
![F=\frac{GMm}{r^2}=\frac{Gm\rho\frac{4}{3}\pi r^3}{r^2}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_2b544928845bf9b36c405a9839055b83.gif)
![F_x=-Gm\rho\frac{4}{3}\pi rsin\alpha=-kx](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_5ef3b1e6fddc2100c10004e7a9da24c9.gif)
继续套用弹簧振子周期公式,
![T=2\pi\sqrt{\frac{m}{k}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_237cb1e42fcfd11922c5aba8e572bfe9.gif)
![gR^2=GM=G\rho\frac{4}{3}\pi R^3](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_dee28b9ea45cb787bf1d21245ceb6091.gif)
![G\rho\frac{4}{3}\pi=\frac{g}{R}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_8387b88969888ed16737e7b423a3a282.gif)
![T=2\pi \sqrt{\frac{R}{g}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_c6d96c2fb33d14cf05eda216c1b343d3.gif)
![\theta](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_2554a2bb846cffd697389e5dc8912759.gif)
如果是横贯地心的地下铁,不仅时间和贴地卫星周期相同,更是大撒心心相映的狗粮(圆周运动的投影点)...
当然,这个周期也是和地球半径(大小)无关的,由Kepler's Law III,
![T=\sqrt{\frac{3\pi}{G\rho}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_1fb174bc7df52b8128f04b2022e47165.gif)
X上看到这种推导...
![](http://www.qiusir.com/wp-content/gallery/physics/fpninp.jpg)
物理作为高考理科的主力是要背很多的锅,不单是计算不准的数学锅,还有审题失误的语文锅,以及基础知识相关的地理锅、化学锅等等,甚至偶有符号乱用的英语锅...
数学L老师就曾提出过[?]上面梗图的困惑。地球自转的线速度
![v_0=\frac{2\pi R}{T_0}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_121344edbae060da136d436aa23889c6.gif)
On this day..
- 把握美丽 - 2008
- 太空,中国人来了实践单元设计初稿 - 2003
- 群星的闪耀就是天空 - 2003
11月 27th, 2023 at 18:42
“纯真而不欠闻达,善良而不失坚强,把生命高举在尘俗之上,又融化于社会中,这应当是我们这一代人的共同追求。”
11月 28th, 2023 at 22:01
今天构造了天球上的8字轨迹...