第八次作业

作业

---

name:姚逸飞
student number：2013301020096

homework

• $\Bbb homework\; L1:$ 3.4
• $\Bbb homework\; L1:$ 3.5

Abstract

This passage is mainly discuss about the Oscillatory Motion. We have already learned the simple pendulum from senior high school.Now it's a perfect choice to discuss about the period of the simple pendulum.
This passage will discuss about two different method :Euler method and Euler-Cromer method.And we can purely see the strength of Euler-Cromer method.

Passage

The picture shows a simple pendulum.From the Newton's second law ,we can achieve the equation as follow

$$\frac{{\rm d^2}\theta}{{\rm d}t}=-\frac{g}{l}\theta$$

Here $l$ is the length of string,$\theta$ is the oscillation of angle ,$g$ is the gravity.
So we can give the answer of the equation as follows:

$$\theta=\theta_0{\rm cos}(\sqrt{\frac{g}{l}}t)$$
It's a simple answer ,and we'll use this to compare the Euler Method:
$$\begin{cases} \theta_{i+1}=\theta_i+\omega_i\Delta t\\ \omega_{i+1}=\omega_{i}-\frac{g}{l}\theta_{i}\Delta t \end{cases}$$
and Eulwe-Cromer Method:
$$\begin{cases} \theta_{i+1}=\theta_{i}+\omega_{i+1}\Delta t\\ \omega_{i+1}=\omega_{i}-\frac{g}{l}\theta_{i}\Delta t \end{cases}$$
As we can see from the two method ,there only a little difference bwtween them. In many cases ,the results of two method are very close.But in the Oscillatory Motion,we can see that the results are prominently defferent.
The different pictures are shown as follows:

$$E=\frac{1}{2}ml^2\omega^2+mgl{\rm cos^2}\theta$$
We can see from the picture that the energy of Euler method is diffuse but that of Euler_cromer method is not.

The program is as follow: Here
When we consider the driven force ,we can get the different pictures:

conlusion

Although this homework is very easy ,I begin to learn how to write programe with the class,it's a huge step for me .

Reference

• Thanks for the Cmd Markdown: https://www.zybuluo.com/codeep/note/163962#9
• Thanks for Chen Yangyao 's example