﻿ 非惯性系下质点的运动规律研究

# 非惯性系下质点的运动规律研究Study on the Motion Law of Particles in Non-Inertial System

Abstract: In mechanics textbook, according to Newton’s law motion, only the mathematical expression of particle motion theorem and its corresponding conservation law in inertial system and “special Non-Inertial system” (center of mass system) are deduced. In order to study the motion law of particle in “general Non-Inertial system”, based on Newton’s law of motion, this paper deduces the momentum theorem, kinetic energy theorem, angular momentum theorem of particles in “general Non-Inertial system” and their corresponding conservation laws.

1. 引言

2. 惯性系和非惯性系下质点的受力关系

$r={r}_{0}+{r}^{\prime }$ (1)

$\frac{\text{d}r}{\text{d}t}=\frac{\text{d}{r}_{0}}{\text{d}t}+\frac{\text{d}{r}^{\prime }}{\text{d}t}$ (2)

$v=\frac{\text{d}r}{\text{d}t}=\frac{\text{d}{r}_{x}}{\text{d}t}i+\frac{\text{d}{r}_{y}}{\text{d}t}j+\frac{\text{d}{r}_{z}}{\text{d}t}k$

${v}_{0}=\frac{\text{d}{r}_{0}}{\text{d}t}=\frac{\text{d}{r}_{0x}}{\text{d}t}i+\frac{\text{d}{r}_{0y}}{\text{d}t}j+\frac{\text{d}{r}_{0z}}{\text{d}t}k$

$\begin{array}{c}{v}^{\prime }=\frac{\text{d}{r}^{\prime }}{\text{d}t}=\frac{\text{d}{{r}^{\prime }}_{x}}{\text{d}t}{i}^{\prime }+\frac{\text{d}{{r}^{\prime }}_{y}}{\text{d}t}{j}^{\prime }+\frac{\text{d}{{r}^{\prime }}_{z}}{\text{d}t}{k}^{\prime }+{{r}^{\prime }}_{x}\frac{\text{d}{i}^{\prime }}{\text{d}t}+{{r}^{\prime }}_{y}\frac{\text{d}{j}^{\prime }}{\text{d}t}+{{r}^{\prime }}_{z}\frac{\text{d}{k}^{\prime }}{\text{d}t}\\ =\frac{{\text{d}}^{*}{r}^{\prime }}{\text{d}t}+\omega ×{r}^{\prime }={v}_{r}+\omega ×{r}^{\prime }\end{array}$

$v={v}_{0}+{v}_{r}+\omega ×{r}^{\prime }$ (3)

(3)式左右两边求导，得：

$\frac{\text{d}v}{\text{d}t}=\frac{\text{d}{v}_{0}}{\text{d}t}+\frac{\text{d}{v}_{r}}{\text{d}t}+\frac{\text{d}\left(\omega ×{r}^{\prime }\right)}{\text{d}t}$ (4)

$\begin{array}{l}a=\frac{\text{d}v}{\text{d}t}=\frac{{\text{d}}^{2}{r}_{x}}{\text{d}{t}^{2}}i+\frac{{\text{d}}^{2}{r}_{y}}{\text{d}{t}^{2}}j+\frac{{\text{d}}^{2}{r}_{z}}{\text{d}{t}^{2}}k\\ {a}_{0}=\frac{\text{d}{v}_{0}}{\text{d}t}=\frac{{\text{d}}^{2}{r}_{0x}}{\text{d}{t}^{2}}i+\frac{{\text{d}}^{2}{r}_{0y}}{\text{d}{t}^{2}}j+\frac{{\text{d}}^{2}{r}_{0z}}{\text{d}{t}^{2}}k\end{array}$ (5)

$\begin{array}{c}{a}^{\prime }=\frac{\text{d}{v}^{\prime }}{\text{d}t}=\frac{\text{d}{v}_{r}}{\text{d}t}+\frac{\text{d}\omega ×{r}^{\prime }}{\text{d}t}\\ =\frac{\text{d}}{\text{d}}\left(\frac{\text{d}{{r}^{\prime }}_{x}}{\text{d}t}i+\frac{\text{d}{{r}^{\prime }}_{y}}{\text{d}t}j+\frac{\text{d}{{r}^{\prime }}_{z}}{\text{d}t}k\right)+\frac{\text{d}\omega }{\text{d}t}×{r}^{\prime }+\omega ×\frac{\text{d}{r}^{\prime }}{\text{d}t}\\ ={a}_{r}+\omega ×{v}_{r}+\stackrel{˙}{\omega }×{r}^{\prime }+\omega ×\left(\omega ×{r}^{\prime }\right)+\omega ×{v}_{r}\\ ={a}_{r}+\stackrel{˙}{\omega }×{r}^{\prime }+\omega ×\left(\omega ×{r}^{\prime }\right)+2\omega ×{v}_{r}\end{array}$ (6)

$a={a}_{r}+{a}_{0}+\stackrel{˙}{\omega }×{r}^{\prime }+\omega ×\left(\omega ×{r}^{\prime }\right)+2\omega ×{v}_{r}$ (7)

$a={a}_{r}+{a}_{0}+{a}_{c}$ (8)

${a}_{c}=\stackrel{˙}{\omega }×{r}^{\prime }+\omega ×\left(\omega ×{r}^{\prime }\right)+2\omega ×{v}_{r}$

$ma=m{a}_{r}+m{a}_{0}+m\stackrel{˙}{\omega }×{r}^{\prime }+m\omega ×\left(\omega ×{r}^{\prime }\right)+2m\omega ×{v}_{r}$ (9)

$ma=m{a}_{r}+m{a}_{0}+m{a}_{c}$ (10)

${F}^{*}=F+{F}_{t}+{F}_{c}$ (11)

3. 非惯性系下质点运动定理的数学表达式

${F}^{*}\text{d}t=F\text{d}t+{F}_{t}\text{d}t+{F}_{c}\text{d}t$ (12)

${F}^{*}\text{d}t=m\frac{{\text{d}}^{*}{v}_{r}}{\text{d}t}\text{d}t=\text{d}\left(m{v}_{r}\right)$

${\int }_{{v}_{1r}}^{{v}_{2r}}\text{d}\left(m{v}_{r}\right)={\int }_{{t}_{1}}^{{t}_{2}}F\text{d}t+{\int }_{{t}_{1}}^{{t}_{2}}{F}_{t}\text{d}t+{\int }_{{t}_{1}}^{{t}_{2}}{F}_{c}\text{d}t$ (13)

${\int }_{{v}_{1r}}^{{v}_{2r}}\text{d}\left(m{v}_{r}\right)={{p}^{\prime }}_{2}-{{p}^{\prime }}_{1}$$I={\int }_{{t}_{1}}^{{t}_{2}}F\text{d}t$${I}_{t}={\int }_{{t}_{1}}^{{t}_{2}}{F}_{t}\text{d}t$${I}_{c}={\int }_{{t}_{1}}^{{t}_{2}}{F}_{c}\text{d}t$

${{p}^{\prime }}_{2}-{{p}^{\prime }}_{1}=I+{I}_{t}+{I}_{c}$ (14)

(14)式表明：质点在“一般非惯性系”下的相对动量增量等于作用于质点上的真实力 $F$ 和惯性力 ${F}_{t}$${F}_{c}$ 在相同时间段内的冲量和，称为“一般非惯性系”下质点的动量定理。

${F}^{*}\cdot \text{d}{r}^{\prime }=F\cdot \text{d}{r}^{\prime }+{F}_{t}\cdot \text{d}{r}^{\prime }+{F}_{c}\cdot \text{d}{r}^{\prime }$ (15)

${F}^{*}\cdot \text{d}{r}^{\prime }=m\frac{{\text{d}}^{*}{v}_{r}}{\text{d}t}\cdot \left[\left({v}_{r}+\omega ×{r}^{\prime }\right)\text{d}t\right]=\text{d}\left(\frac{1}{2}m{v}_{r}^{2}\right)+m\left(\omega ×{r}^{\prime }\right)\cdot {\text{d}}^{\text{*}}{v}_{r}$

$F\cdot \text{d}{r}^{\prime }=ma\cdot \text{d}{r}^{\prime }$

${F}_{t}\cdot \text{d}{r}^{\prime }=-\left[m{a}_{0}+m\stackrel{˙}{\omega }×{r}^{\prime }+m\omega ×\left(\omega ×{r}^{\prime }\right)\right]\cdot \text{d}{r}^{\prime }$

$\text{d}\left(\frac{1}{2}m{v}_{r}^{2}\right)=ma\cdot \text{d}{r}^{\prime }-\left[m{a}_{0}+m\stackrel{˙}{\omega }×{r}^{\prime }+m\omega ×\left(\omega ×{r}^{\prime }\right)\right]\cdot \text{d}{r}^{\prime }=F\cdot \text{d}{r}^{\prime }+{F}_{t}\cdot \text{d}{r}^{\prime }$ (16)

$\frac{1}{2}m{v}_{r2}^{2}-\frac{\text{1}}{\text{2}}m{v}_{r1}^{2}=A+{A}_{t}$ (17)

(17)式表明：质点在非惯性系的相对动能的变量等于作用于该质点的真实力和牵连惯性力在相对运动上的路程所做功之和。该结论为“一般非惯性系”下质点的动能定理。而在上讨论过科里奥利力的方向总是与质点运动的位矢垂直，所以在讨论质点在非惯性系的相对动能不用考虑科里奥利力所做的功，实际上，科里奥利力会改变质点的运动方向。

${r}^{\prime }×{F}^{\text{*}}={r}^{\prime }×\left(F+{F}_{t}+{F}_{c}\right)$ (18)

(18)式左侧可进一步表示为：

${r}^{\prime }×{F}^{*}=\frac{{\text{d}}^{\text{*}}\left({r}^{\prime }×{p}^{\prime }\right)}{\text{d}t}=\frac{{\text{d}}^{*}{r}^{\prime }}{\text{d}t}×m{v}_{r}+{r}^{\prime }×m\frac{{\text{d}}^{*}{v}_{r}}{\text{d}t}={r}^{\prime }×\frac{{\text{d}}^{*}p}{\text{d}t}=\frac{{\text{d}}^{*}}{\text{d}t}{L}^{\prime }$

(18)式可简写成：

$\frac{{\text{d}}^{*}}{\text{d}t}{L}^{\prime }=M+{{M}^{\prime }}_{t}+{{M}^{\prime }}_{c}$ (19)

(19)式表明：在“一般非惯性系”下，质点对参考点的角动量对时间的变化率等于作用于质点的真实力 $F$ 、惯性力 ${F}_{t}$ 和科里奥利力 ${F}_{c}$ 对该参考点的位矢的力矩，该定理称为“一般非惯性系”下质点的角动量定理。

${{F}^{\prime }}_{保}\cdot \text{d}{r}^{\prime }=-\text{d}{U}^{\prime }$ (20)

${F}^{\text{*}}\cdot \text{d}{r}^{\prime }=\text{d}\left(\frac{1}{2}m{v}_{r}^{2}\right)$

$F\cdot \text{d}{r}^{\prime }=-\text{d}U=-ma\cdot \text{d}{r}^{\prime }$

${F}_{t}\cdot \text{d}{r}^{\prime }=-\text{d}{{U}^{\prime }}_{t}=\left[m{a}_{0}+m\stackrel{˙}{\omega }×{r}^{\prime }+m\omega ×\left(\omega ×{r}^{\prime }\right)\right]\cdot \text{d}{r}^{\prime }$

${F}_{c}\cdot \text{d}{r}^{\prime }=-2m\omega ×{v}_{r}\cdot \text{d}{r}^{\prime }=0$

$\text{d}\left(\frac{1}{2}m{v}_{r}^{2}\right)=-\text{d}U-\text{d}{{U}^{\prime }}_{t}$ (21)

(21)式可写成：

$\text{d}\left({T}^{\prime }\right)=-\text{d}U-\text{d}{{U}^{\prime }}_{t}$ (22)

${T}^{\prime }+U+{{U}^{\prime }}_{t}=c$ (23)

(23)式表明：若“一般非惯性系”下的质点所受的真实力 $F$ 和惯性力 ${F}_{t}$ 都是保守力的情况下，质点相对一般非惯性的相对动能和保守力 $F$ 对应的势能与保守惯性力 ${F}_{t}$ 所对应的惯性势能之和是一个定值。该表达式称为“一般非惯性系”下的机械能守恒定理。

4. 应用

$\frac{1}{2}m{v}_{1}^{2}-\frac{\text{1}}{\text{2}}m{v}^{2}=A+{A}_{t}$ (24)

5. 结论

NOTES

*通讯作者。

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