# 数学代写|数学建模代写Mathematical Modeling代考|Simole Harmonic Motion

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 数学代写|数学建模代写Mathematical Modeling代考|Simole Harmonic Motion

Here a particle moves in a straight line in such a manner that its acceleration is always proportional to its distance from the origin and is always directed toward the origin, so that
$$v \frac{d v}{d x}=-\mu x$$
integrating
$$v^2=\mu\left(a^2-x^2\right)$$
where the particle is initially at rest at $x=a$. Equation (44) gives
$$\frac{d x}{d t}=-\sqrt{\mu} \sqrt{a^2-x^2}$$
We take the negative sign since velocity increases as $x$ decreases (Figure 2.6).
Integrating again and using the condition that at $t=0, x=a$
$$x(t)=a \cos \sqrt{\mu} t$$
so that
$$v(t)=-a \sqrt{\mu} \sin \sqrt{\mu} t$$
Thus in simple harmonic motion, both displacement and velocity are periodic functions with period $2 \pi / \sqrt{\mu}$.
The particle starts from $A$ with zero velocity and moves toward 0 with increasing velocity and reaches 0 at time $\pi / 2 \sqrt{\mu}$ with velocity $\sqrt{\mu} a$. It continues to move in the same direction, but now with decreasing velocity till it reaches $A^{\prime}\left(0 A^{\prime}=a\right)$ where its velocity is again zero. It then begins moving toward 0 with increasing velocity and reaches 0 with velocity $\sqrt{\mu} a$ and again comes to rest at $A$ after a total time period $2 \pi / \sqrt{\mu}$. The periodic motion then repeats itself.
As one example of SHM, consider a particle of mass $m$ attached to one end of a perfectly elastic string, the other end of which is attached to a fixed point 0 (Figure 2.7). The particle moves under gravity in a vacuum.

## 数学代写|数学建模代写Mathematical Modeling代考|Motion Under Gravity in a Resisting Medium

A particle falls under gravity in a medium in which the resistance is proportional to the velocity. The equation of motion is
or
$$\begin{gathered} m \frac{d v}{d i}=m g-m k v \ \frac{d v}{V-v}=k d t ; V=\frac{g}{k} \end{gathered}$$
Integrating
$$V-v=V e^{-k t}$$
if the particle starts from rest with zero velocity, Equation (50) gives
$$v=V\left(1-e^{-k t}\right),$$
so that the velocity goes on increasing and approaches the limiting velocity $g / k$ as $t \rightarrow \infty$. Replacing $v$ by $\mathrm{d} x / d t$, we get
$$\frac{d x}{d t}=V\left(1-e^{-k t}\right)$$
Integrating and using $x=0$ when $t=0$, we get
$$x=V t+\frac{V e^{-k t}}{k}-\frac{V}{k}$$

## 数学代写|数学建模代写Mathematical Modeling代考|Simole Harmonic Motion

$$v \frac{d v}{d x}=-\mu x$$

$$v^2=\mu\left(a^2-x^2\right)$$

$$\frac{d x}{d t}=-\sqrt{\mu} \sqrt{a^2-x^2}$$

$$x(t)=a \cos \sqrt{\mu} t$$

$$v(t)=-a \sqrt{\mu} \sin \sqrt{\mu} t$$

## 数学代写|数学建模代写Mathematical Modeling代考|Motion Under Gravity in a Resisting Medium

$$\begin{gathered} m \frac{d v}{d i}=m g-m k v \ \frac{d v}{V-v}=k d t ; V=\frac{g}{k} \end{gathered}$$

$$V-v=V e^{-k t}$$

$$v=V\left(1-e^{-k t}\right),$$
，使速度继续增加，接近极限速度$g / k$为$t \rightarrow \infty$。将$v$替换为$\mathrm{d} x / d t$，我们得到
$$\frac{d x}{d t}=V\left(1-e^{-k t}\right)$$

$$x=V t+\frac{V e^{-k t}}{k}-\frac{V}{k}$$

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。