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## 物理代写|理论力学代写Theoretical Mechanics代考|Motion in the Homogeneous Gravitational Field

According to the above-given program we have to at first formulate the equation of motion. Using (2.48) together with (2.43) and exploiting the equality of inertial and heavy mass we can write:
$$\ddot{\mathbf{r}}=\mathbf{g} ; \quad \mathbf{g}=(0,0,-g) .$$
The mass is eliminated; in the gravitational field all bodies are therefrom equally accelerated. It results in a
uniformly accelerated motion
as we have discussed it already in Sect. 2.1.2. We can directly take the former results $(2.30)$ and $(2.31)$ :
\begin{aligned} &\mathbf{v}(t)=\mathbf{v}\left(t_0\right)+\mathbf{g} \cdot\left(t-t_0\right) \ &\mathbf{r}(t)=\mathbf{r}\left(t_0\right)+\mathbf{v}\left(t_0\right)\left(t-t_0\right)+\frac{1}{2} \mathbf{g} \cdot\left(t-t_0\right)^2 . \end{aligned}

## 物理代写|理论力学代写Theoretical Mechanics代考|Linear Differential Equations

We refer to
$$x^{(n)}(t)=\frac{d^n}{d t^n} x(t)$$
as the $n$-th derivative of the function $x(t)$. A relation which contains one or more derivatives of a given function, where the $n$-th derivative appears as the highest,
$$f\left(x^{(n)}, x^{(n-1)}, \ldots, \dot{x}, x, t\right)=0,$$

is called a differential equation of $n$-th order. The goal is to derive the solution function $x(t)$ from such a relation. The basic dynamical equation of Classical Mechanics (2.43) written in Cartesian coordinates, e.g., has just this shape:
$$m \ddot{x}_i-F_i\left(\dot{x}_1, \dot{x}_2, \dot{x}_3, x_1, x_2, x_3, t\right)=0, \quad i=1,2,3 .$$
This is a coupled system of three differential equations of second order for the three functions $x_1(t), x_2(t), x_3(t)$.

Let us first focus, however, on a general relation of the type (2.95). The central statement is formulated in the following

Theorem 2.3.1 The general solution of a differential equation of $n$-th order (2.95) is an ensemble of solutions
$$x=x\left(t \mid \gamma_1, \gamma_2, \ldots, \gamma_n\right),$$
which depends on $n$ independent parameters $\gamma_1, \gamma_2, \ldots, \gamma_n$. Every set of $\gamma_i$ ‘s which are fixed in advance then leads to a special (particular) solution.

## 物理代写|理论力学代写Theoretical Mechanics代考|Motion in the Homogeneous Gravitational Field

$$\ddot{\mathbf{r}}=\mathbf{g} ; \quad \mathbf{g}=(0,0,-g) .$$

2.1.2. 我们可以直接取前一个结果 $(2.30)$ 和 $(2.31)$ :
$$\mathbf{v}(t)=\mathbf{v}\left(t_0\right)+\mathbf{g} \cdot\left(t-t_0\right) \quad \mathbf{r}(t)=\mathbf{r}\left(t_0\right)+\mathbf{v}\left(t_0\right)\left(t-t_0\right)+\frac{1}{2} \mathbf{g} \cdot\left(t-t_0\right)^2 .$$

## 物理代写|理论力学代写Theoretical Mechanics代考|Linear Differential Equations

$$x^{(n)}(t)=\frac{d^n}{d t^n} x(t)$$

$$f\left(x^{(n)}, x^{(n-1)}, \ldots, \dot{x}, x, t\right)=0,$$

$$m \ddot{x}_i-F_i\left(\dot{x}_1, \dot{x}_2, \dot{x}_3, x_1, x_2, x_3, t\right)=0, \quad i=1,2,3 .$$

$$x=x\left(t \mid \gamma_1, \gamma_2, \ldots, \gamma_n\right),$$

## 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。

Posted on Categories:Theoretical mechanics, 物理代写, 理论力学

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## 物理代写|理论力学代写Theoretical Mechanics代考|Forces

At the beginning of this chapter we recognized as the elementary task of each physical theory, in particular the Classical Mechanics, to derive conclusions from preformulated postulates (basis definitions, axioms). The axioms and the fundamental definition of mass are now available. The law of motion (2.42) and (2.43), respectively, have become the principal dynamical equation of Classical Mechanics. This equation is to be solved. As a rule, mathematically that means, for a given force $\mathbf{F}$, one has to solve a differential equation of second order.

More precise than the term force in this connection is, strictly speaking, the concept of the
‘Force Field’
$$\mathbf{F}=\mathbf{F}(\mathbf{r}, \dot{\mathbf{r}}, t) .$$

## 物理代写|理论力学代写Theoretical Mechanics代考|Theorem of Kinetic Energy

To each space point a force that acts on the mass point is assigned, which in general can even be time dependent and, additionally, may depend on the particle velocity. Dependence on acceleration $\ddot{\mathbf{r}}$, however, will not appear.

All the matter is built by elementary constituents (molecules, atoms, nucleons, electrons, …). Therefore, in the last analysis each force can be traced back to the interactions between these elementary constituents. To do this in all detail, however, is beyond the framework of Classical Mechanics which only asks for the consequences and not for the elementary causes of the forces. Normally one restricts oneself to mathematically as simple as possible and empirically reasoned model representations
Some frequently used examples are listed in the following:
(a) Weight, Gravitational Force
Each body is ‘heavy’. $1 \mathrm{~m}^3$ of iron is ‘heavier’ than $1 \mathrm{~cm}^3$ of iron. By this everyday experience a new material quantity is documented which is denoted as
gravitational (heavy) mass $m_h$.
It manifests itself in the ‘gravitational force’
$$\mathbf{F}_g=m_h \mathbf{g},$$
which acts on a stationary (motionless) mass point in the gravitational field of the earth. $g$ close to the earth’s surface is a nearly constant vector always pointing downwards in direction to the earth’s center. If we define this direction as the negative $x_3$ direction of a Cartesian coordinate system then we write
$$\mathbf{g}=-(0,0, g) ; \quad g=9.81 \mathrm{~m} \mathrm{~s}^{-2} \text { ‘gravity acceleration’. }$$

## 物理代写|理论力学代写Theoretical Mechanics代考|Forces

$$\mathbf{F}=\mathbf{F}(\mathbf{r}, \dot{\mathbf{r}}, t)$$

## 物理代写|理论力学代写Theoretical Mechanics代考|Theorem of Kinetic Energy

(a) 重量，重力

$$\mathbf{F}_g=m_h \mathbf{g}$$

$$\mathbf{g}=-(0,0, g) ; \quad g=9.81 \mathrm{~m} \mathrm{~s}^{-2} \text { ‘gravity acceleration’. }$$

## 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。

Posted on Categories:Theoretical mechanics, 物理代写, 理论力学

## avatest™帮您通过考试

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## 物理代写|理论力学代写Theoretical Mechanics代考|Equilibrium Equations for One Rigid Body System

Let us continue to discuss the equilibrium conditions of the general coplanar force group. As mentioned in Chap. 4, the equilibrium conditions are that the principal vector and principal moment are both zeroes:
\begin{aligned} &\boldsymbol{R}=0 \ &M=0 \end{aligned}
In the Cartesian coordinate system, the above formulas are expressed as
\begin{aligned} &\Sigma X=0, \ &\Sigma Y=0, \ &\Sigma M_{A}=0 . \end{aligned}
This group of formulas includes two equations on force, and one equation on moment, so it is normally named as one moment format. In fact, there are also some other formats on the equilibrium conditions. For example, the two-moment format can be written as
\begin{aligned} &\Sigma X=0 \ &\Sigma M_{A}=0 \ &\Sigma M_{B}=0 \end{aligned}
The three-moment format can be expressed as
\begin{aligned} &\Sigma M_{A}=0 \ &\Sigma M_{B}=0 \ &\Sigma M_{C}=0 \end{aligned}

## 物理代写|理论力学代写Theoretical Mechanics代考|Rigid Multi-body System

The previous discussions are only confined to one rigid body. However, in most cases, we will face a system, which is assembled by several rigid bodies via constraints. This system is named as “rigid multi-body system”. There are two types of forces for this kind of system. One is the external force, which comes from the other objects outside the system, and the other is internal force, which is the interaction force among the rigid bodies in the system. For the rigid multi-body system, we have the following rules:
(1) When the whole system is in equilibrium, every rigid body in the system must be in equilibrium.
(2) We can select a portion of the system as an object, or can choose the whole system to investigate.
(3) When we consider the total system, the internal forces among the subsystems do not appear; and when we analyze one subsystem, it endures the action forces from the other subsystems.
(4) The action and reaction forces always appear together, with the opposite directions.
(5) For a system including $n$ rigid bodies, the number of the individual objects we can choose is $n$.

It can be seen that, for a rigid multi-body system including $n$ rigid bodies, we can write down $3 n$ equilibrium equations totally. If the number of the unknowns is equal to that of the equilibrium equations, all of these unknowns can be solved, and the case is called determinate problem. And vice versa, if the number of the unknowns is bigger than that of the equilibrium equations, not all of these unknowns can be solved, and the case is called indeterminate problem. How to solve the indeterminate problem needs considering the deformation of the object, which is out of the range of Statics.

## 物理代写|理论力学代写Theoretical Mechanics代考|Equilibrium Equations for One Rigid Body System

$$\boldsymbol{R}=0 \quad M=0$$

$$\Sigma X=0, \quad \Sigma Y=0, \Sigma M_{A}=0 .$$

$$\Sigma X=0 \quad \Sigma M_{A}=0 \Sigma M_{B}=0$$

$$\Sigma M_{A}=0 \quad \Sigma M_{B}=0 \Sigma M_{C}=0$$

## 物理代写|理论力学代写Theoretical Mechanics代考|Rigid Multi-body System

(1)当整个㒶统处于平衡状态时，系统中的每个刚体都必须处于平衡状态。
(2)我们可以选择系统的一部分作为对象，也可以选择整个系统进行调育。
（3）当我们考慮整个系统时，子系统之间的内力不出现；当我们分析一个子系统时，它会承受来自其他子系统的作用力。
(4) 作用力和反作用力总是同时出现，方向相反。
(5) 对于一个系统，包括 $n$ 刚体，我们可以选择的单个物体的数量是 $n$.

## 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。