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# 数学代写|数学建模代写Mathematical Modeling代考|The Universality of Mathematical Models

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## 数学代写|数学建模代写Mathematical Modeling代考|Fluid in a U-shaped flask.

Fluid in a U-shaped flask. The fluid fills part of a U-shaped flask; it represents a bent pipe of a radius $r_0$ (Fig. 15). The mass of the fluid is $M_0$, its density is $\rho_0$. The walls of the flask are ideally smooth, the surface tension is neglected, atmospheric pressure $P_0$ and acceleration of gravity $g$ are constant.

At the equilibrium the fluid, is obviously motionless, its height at either side of the U-bend is identical. If it is removed from the equilibrium, the motion will start with a character we will below establish with the help of energy conservation law, as long as our assumption that no energy loss exists in the system is correct.

We will calculate the potential energy of a system through work, which is necessary to shift it from the equilibrium state (where $h_1=h_2$ ) to a position represented in Fig. 15. It is
$$E_p=-\int_{\bar{h}}^{h_2} P d h_2=-\int_{\bar{h}}^{h_2} \rho_0 s_0\left(h_1-h\right) g d h, \quad \bar{h}=\frac{h_1+h_2}{2}, \quad s_0=\pi r_0^2,$$

where $P$ is the weight of the part of the fluid to the left side of the bend; its level exceeds the magnitude $h_2$. The work of the forces of atmospheric pressure is equal to zero, in so far as for different bends the corresponding displacements have different directions.

The unknown quantities $h_1(t)$ and $h_2(t)$ are connected by an obvious relation $h_1(t)+h_2(t)=$ const $>0$, expressing a constance of total length of the pillar of the fluid with constant cross-section. Substituting the last equality into the expression for $E_p$, we obtain after integration
$$E_p=-\rho_0 s_0 g\left[-h_2^2(t)+C h_2(t)+C_1\right]$$

## 数学代写|数学建模代写Mathematical Modeling代考|An oscillatory electrical circuit

An oscillatory electrical circuit. This device represents a capacitor, connected with wires to an inductive coil. At the moment $t=0$ the circuit is closed and the charge of the plates of the capacitor begins to propagate over the circuit (Fig. 16).

The resistance of wires is considered equal to zero, the capacity of the capacitor is $C$, induction of the coil is $L$. For changing in time quantity $q(t)$, where $q(t)$ is the charge on the plates of the capacitor, is necessary to obtain the appropriate equation. Obviously, the current $i(t)$ and the voltage $v(t)$ are also functions of time.

By the physical content of the quantity $C$ at any moment of time we have the equality $v(t)=q(t) C$ (the capacity is equal to the magnitude of the charge on the plate of the capacitor required for the increase of a potential difference by unity).

In so far as the electrical resistance in the circuit is absent, no loss of voltage in the wires exists, and the difference of potentials $v(t)$ of the capacitor, is immediately passed to the coil. At variable current in a coil an electromotive force of self-induction appears, equal to $\varepsilon=-L d i / d t$. The Ohm law for the circuit in the absence of a resistance is as follows:
$$v(t)=-\varepsilon(t)$$
or
$$q(t) C=-\varepsilon(t)=L d i / d t$$
So far as by definition $i=-d q / d t$ (at a change of the charge on the capacitor the current appears in a circuit), from the last relation we obtain
$$L \frac{d^2 q}{d t^2}=-C q$$
describing the oscillations of $q(t)$ (and consequently of $i(t), v(t))$ in the simplest electrical circuit identical to (1) of section 4. In the system “capacityinduction” the oscillations occur in the same manner and in the system “ball-spring” and analogously the models become more complicated when additional processes are taken into account – see exercise 2.

## 数学代写|数学建模代写Mathematical Modeling代考|Fluid in a U-shaped flask.

$\mathrm{U}$ 形烧瓶中的液体。液体充满了 $\mathrm{U}$ 形烧瓶的一部分；它代表一个半径的弯管 $r_0$ (图 15)。流体的质量是 $M_0$ ， 它的密度是 $\rho_0$. 烧并敬理想情况下是光滑的，忽略表面张力，大气压 $P_0$ 和重力加速度 $g$ 是恒定的。

$$E_p=-\int_{\bar{h}}^{h_2} P d h_2=-\int_{\bar{h}}^{h_2} \rho_0 s_0\left(h_1-h\right) g d h, \quad \bar{h}=\frac{h_1+h_2}{2}, \quad s_0=\pi r_0^2,$$

$$E_p=-\rho_0 s_0 g\left[-h_2^2(t)+C h_2(t)+C_1\right]$$

## 数学代写|数学建模代写Mathematical Modeling代考|An oscillatory electrical circuit

$$v(t)=-\varepsilon(t)$$

$$q(t) C=-\varepsilon(t)=L d i / d t$$

$$L \frac{d^2 q}{d t^2}=-C q$$

## 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。