# 物理代写|理论力学作业代写Theoretical Mechanics代考|PHYS3020

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Simple Pendulum

As an additional simple problem of dynamics we will now discuss the simple (thread) pendulum (Fig. 2.20), which sometimes is also called mathematical pendulum because it represents a somewhat mathematical abstraction. One considers the motion of a mass point which is fixed by a massless thread. The latter has a constant length $l$ so that the mass point performs a planar motion on a circular arc with radius $l$. The gravitational force acts on the mass point:
$$\mathbf{F}=m_{\mathrm{s}} \mathbf{g} ; \quad \mathbf{g}=(g, 0,0) .$$

The simple pendulum is excellently suited to demonstrate the equivalence of the inertial and the gravitational mass. To show this, we will first distinguish again between these two masses.
The application of plane polar coordinates is the natural choice:
\begin{aligned} \mathbf{F} &=F_r \mathbf{e}r+F{\varphi} \mathbf{e}{\varphi} \ F_r &=m_h g \cos \varphi \ F{\varphi} &=-m_h g \sin \varphi \end{aligned}
The equation of motion written in detail by use of $(2.13)$ reads as:
$$m_{i n}\left[\left(\ddot{r}-r \dot{\varphi}^2\right) \mathbf{e}r+(r \ddot{\varphi}+2 \dot{r} \dot{\varphi}) \mathbf{e}{\varphi}\right]=\left(F_r+F_{\mathrm{F}}\right) \mathbf{e}r+F{\varphi} \mathbf{e}{\varphi} .$$ $F{\mathrm{F}}$ is called the
It is about a so-called ‘constraining force’ which realizes certain ‘constraints’. Here the constraint is the constant distance of the mass point from the center of rotation:
$$r=l=\text { const } ; \quad \dot{r}=\ddot{r}=0 .$$
$F_{\mathrm{F}}$ thus prevents the free fall of the mass point and takes care for a static problem in the radial direction:
$$F_{\mathrm{F}}=-m_{h r} g \cos \varphi-m_{i n} l \dot{\varphi}^2 .$$
So only the movement in $\mathbf{e}{\varphi}$-direction is of interest: $$m{\text {in }} l \ddot{\varphi}=-m_h g \sin \varphi \Longrightarrow \ddot{\varphi}+\frac{g}{l} \frac{m_h}{m_{\text {in }}} \sin \varphi=0 .$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Complex Numbers

For the solution of the oscillation equation (2.125) we looked for a function which essentially reproduces itself after twofold differentiation. That happens indeed to the trigonometric functions sine and cosine. But the exponential function, which for a variety of reasons is tractable mathematically easier, also possesses a similar property. However, the ansatz $e^{\alpha t}$ would have led to the conditional equation
$$e^{\alpha t}\left(\alpha^2+\frac{g}{l}\right)=0 ; \quad e^{\alpha t} \neq 0,$$
an equation being not solvable for real $\alpha$. The equation becomes, however, solvable if one allows for complex numbers which we did not yet introduce so far.

By application of complex numbers and functions many issues in Theoretical Physics turn out to be mathematically essentially simpler. it is needless to say that all measurable quantities, which we call ‘observables’, are in any case real so that we must be able uniquely to relate real and complex representations. That will be treated in this section.
(a) Imaginary Numbers
The new number type of the imaginary numbers is characterized by the fact that their squares are always negative real numbers.
Definition 2.3.1 ‘Unit of imaginary numbers’
$$i^2=-1 \Longleftrightarrow i=\sqrt{-1} .$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Simple Pendulum

$$\mathbf{F}=m_{\mathrm{s}} \mathbf{g} ; \quad \mathbf{g}=(g, 0,0) .$$

$$\mathbf{F}=F_r \mathbf{e} r+F \varphi \mathbf{e} \varphi F_r \quad=m_h g \cos \varphi F \varphi=-m_h g \sin \varphi$$

$$m_{i n}\left[\left(\ddot{r}-r \dot{\varphi}^2\right) \mathbf{e} r+(r \ddot{\varphi}+2 \dot{r} \dot{\varphi}) \mathbf{e} \varphi\right]=\left(F_r+F_{\mathrm{F}}\right) \mathbf{e} r+F \varphi \mathbf{e} \varphi .$$
$F F$ 被称为
“线张力”

$$r=l=\text { const } ; \quad \dot{r}=\ddot{r}=0 .$$
$F_{\mathrm{F}}$ 从而防止质点自由落体并解决径向上的静态问题:
$$F_{\mathrm{F}}=-m_{h r} g \cos \varphi-m_{\text {in }} l \dot{\varphi}^2 .$$

$$\min l \ddot{\varphi}=-m_h g \sin \varphi \Longrightarrow \ddot{\varphi}+\frac{g}{l} \frac{m_h}{m_{\text {in }}} \sin \varphi=0 .$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Complex Numbers

$$e^{\alpha t}\left(\alpha^2+\frac{g}{l}\right)=0 ; \quad e^{\alpha t} \neq 0,$$

(a) 虚数 虚数

$$i^2=-1 \Longleftrightarrow i=\sqrt{-1} .$$

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