来投个文章试试||傅科摆问题的两种方法分析

//现在是2021.1.7，闲着无聊，并发现B站专栏文章居然支持LaTeX公式系统。

//所以随便写点之前的东西来玩玩。

//感谢Mathpix的公式识别！我真的不会想用TeX再把那些公式打一遍...

//这是去年物理研讨课程中我的报告内容。由于老师鼓励用报告用英文写，这里是直接将之前的报告原文搬运过来的，所以是全英文...请原谅英语渣的语法错误...

Abstract

This article demonstrates two different methods to solve the problem with the example of Foucault pendulum. We will find the exact motion of the pendulum by using both algebraic and geometric method and evaluate the advantages and disadvantages of the two methods. And we will discuss about how we choose our method to solve a new problem, and how our geometrical intuition help us to make a breakthrough.

1. Introduction

A Foucault Pendulum is a large mass suspended from a long line, which is often used as an example for the effect of Coriolis force. When it swings in a vertical plain, the earth rotates beneath it, creating a relative motion between them so that we can see the perpendicular plain of swing rotates slowly. In this article we will also discuss general Foucault Pendulum, where we put the pendulum on a rotational reference frame with any angular velocity.

We will find out the angular frequency 𝜔′ of the perpendicular plain’s rotation.

2. Algebraic Method

In this part we analyze the tragectory of the pendulum by solving the differential eqation of the pendulum's motion. (原谅画渣的全损画质示意图)

For a simple pendulum with mass 𝑚 and length of the string 𝑙, we know that the net force on the mass is

                                                                                                                (1)

Where  is the original angular frequency of the pendulum.  is the horizontal positon vector.

We assume that the Earth rotates with angular frequency  . Consider the effect of inertial centrifugal force, the inertial force is

                                                                       (2)

where  because we neglect the vertical motion (that is z direction).

And we can find that

                                                                          (3)

                                                                       (4)

where  is the latitude of the lab.

Then we set the equations of horizontal motion up by using Newton’s second law: 

                                                                     (5)

                                                                      (6)

This is a linear system. And if we let complex position , and let , we will find that the two equations can be written as

                                                  (7)

and there's only one variable in this complex-variable differential equation. (That's quite funny as we try to use "complex" to "simplify" our question)

The equation has a general solution:

                                                                                                      (8)

Where  are complex const determined by initial motion of the pendulum, and

                                                                                                 (9)

are the two roots of characteristic equation

                                                                (10)

or the solution can be written as

                                                                                                  (11)

where

                                                                                                         (12)

                                                                                                         (13)

so that

                                                                              (14)

which means that in a rotational reference frame with angular frequency , the pendulum looks just like a normal pendulum with angular frequency . (写到这里突然发现这里字数统计是按照字母算的...难怪这么快就2k字了)

So the angular frequency we’ve been trying to find is

                                                                                                                    (15)

3. Intuitionistic Geometrical Method

(公式轰炸终于结束了...如果有人看到这里了就给点耐心看下去吧，后面是比较轻松愉快的几何分析...现在其实我自己也觉得前面公式放多了...)

Let’s first consider a Foucault pendulum placed at the Arctic pole, and observe it from the space. The Earth rotates but there’s no force to turn the pendulum together. Obviously the angular frequency of the rotation is .

And if we put it somewhere else, it's just like the earth remains still but we're moving along a parallel line continuously. Obviously we need to turn to keep us on a certain latitude, and that's why the perpendicular plain seems to rotate. So we need to find how fast we turn when we move a circle along a parallel everyday. We only need to find out the total angle 𝛼 we turned this day.

For a polygon on the surface of a sphere with solid angle Ω and exterior angles , its solid angle writes(事实上我并不知道这个定理怎么证)

                                                                                                          (16)

And   is actually the total angle we turn when we walk along this polygon. That will also be true when we move along the parallel, and the solid angle determined by this parallel where we stand is (这里真的是最后的计算了)

                                                   (17)

so the angle we turn everyday will be

                                                                                         (18)

But we can find a much easier way.

Suppose a cone is tangent to the sphere and the tangent line is the parallel where we put the pendulum. And moving on the parallel of the sphere is equal to moving on the cone. 

And we can easily find that .

                                                                                                       (19)

That's how we use a completely geometrical method to solve the problem.

4. Conclusion

Both methods can solve the question correctly, but geometrical method is obviously easier. (写到这里因为后面都是文本了就换了手机继续写...结果发现手机端好多格式都乱了...)

Nowadays we can easily solve differential equations with the help of computers, so geometrical method seems less useful than before. However the geometrical method brings a new way of thinking, to find another view of the problem from space. Here we break the boundary of 2-dimensional space, and find a better solution of this problem. And if we break the boundary between space and time, we'll make a great breakthrough, theory of relativity. And that's why we need different views of a problem.

References

[1]辛国君,刘树新,舒幼生.傅科摆的进动与轨迹的周期性[J].大学物理, 2013,32(04):5-7.

Acknowledgement

感谢我的室友们帮助审稿及讨论，感谢李教授的指导与同学们的宝贵意见。

现在再去看又还是觉得有不少地方写得不太好...毕竟半学期真的能学很多东西。现在会了Mathematica之类的一些新东西之后再看前面做的过程，其实很多困难是可以轻松处理的。不管怎样这篇文章只是留作纪念，将来还有很长的路...

(当初准备这篇报告的时候挺有意思的就是...我找了一篇舒幼生教授的文章参考，并发现这篇文章里面的计算过程丢了一项离心力的变化量，导致计算结果的特征频率与我的结果出现了一些偏差。在地球的低角速度下，以本题近似级别这个偏差是一个高阶小量，但是如果在一个角速度无法看作小量的转动系这一偏差会有点大，所以我坚持我的结果。)