vault backup: 2025-06-11 08:19:47
This commit is contained in:
parent
35433c8acd
commit
368da9a22e
@ -2626,7 +2626,7 @@ $$
|
||||
\begin{array}{r l}&{\underline{{p}}_{0}=r\,\bar{e}_{1},}\\ &{\quad\underline{{v}}=\dot{r}\,\bar{e}_{1}+r\dot{\phi}\,\bar{e}_{2}+r\dot{\theta}\sin\phi\,\bar{e}_{3},}\\ &{\underline{{a}}=\left(\ddot{r}-r\dot{\phi}^{2}-r\dot{\theta}^{2}\sin^{2}\phi\right)\,\bar{e}_{1}+\left(r\ddot{\phi}+2\dot{r}\dot{\phi}-r\dot{\theta}^{2}\sin\phi\cos\phi\right)\,\bar{e}_{2}}\\ &{\quad\,+\left(r\ddot{\theta}\sin\phi+2\dot{r}\dot{\theta}\sin\phi+2r\dot{\phi}\dot{\theta}\cos\phi\right)\,\bar{e}_{3}.}\end{array}
|
||||
$$
|
||||
|
||||
# Basic principles
|
||||
# 3 Basic principles
|
||||
|
||||
This chapter reviews the basic principles of dynamics. Newton’s laws are the foundation of mechanics and dynamics and deal with the behavior of particles subjected to forces. Section 3.1 presents Newton’s three laws and the principle of work and energy. Section 3.2 introduces the concept of conservative forces that play a fundamental role in dynamics. The principle of conservation of energy is discussed in section 3.2.1.
|
||||
|
||||
@ -2640,11 +2640,11 @@ Newton’s law only apply to a single particle; section 3.4 introduces Euler’s
|
||||
|
||||
牛顿定律仅适用于单个粒子；第3.4节介绍了欧拉的第一和第二定律，这些定律适用于非常普遍的粒子系统。
|
||||
|
||||
# 3.1 Newtonian mechanics for a particle
|
||||
## 3.1 Newtonian mechanics for a particle
|
||||
|
||||
Newton’s laws deal with the motion of a particle, i.e., a body of mass $m$ that presents no physical dimension. This abstraction can be visualized by considering a body of mass $m$ and finite dimensions. Next, the dimensions of the body are allowed to shrink, while the mass remains constant; at the limit, a particle of mass $m$ is obtained that occupies a single point in space. As the particle moves, the locus of all positions it occupies in time describes a curve in three-dimensional space called the path of the particle.
|
||||
牛顿定律处理的是粒子的运动，即一个质量为 $m$ 且没有物理尺寸的物体。 这种抽象概念可以通过考虑一个质量为 $m$ 且具有有限尺寸的物体来可视化。 接下来，允许物体的尺寸缩小，而质量保持不变；在极限情况下，得到一个质量为 $m$ 且占据空间中单个点的粒子。 粒子运动时，它在时间中占据的所有位置的轨迹描述了空间中一条三维曲线，称为粒子的路径。
|
||||
# 3.1.1 Kinematics of a particle 质点运动学
|
||||
### 3.1.1 Kinematics of a particle 质点运动学
|
||||
|
||||
The position vector of particle $\mathbf{P}$ with respect to an inertial frame will be denoted as $\underline{{x}}_{P/O}$ , meaning “position vector of particle $\mathbf{P}$ with respect to point O,” which is the origin of the inertial frame. Newton’s laws assume the existence of an inertial frame, that is, a frame that is stationary with respect to the distant stars. In many practical applications, a frame attached to the earth may be used as an inertial frame. For instance, when studying the dynamics of a jet engine on a test bench, a frame of reference attached to the test bench is appropriate. If the same engine is mounted on an aircraft wing, a frame attached to the wing would not be inertial, because the aircraft is itself moving; for such a problem, a frame attached to the surface of the earth could be considered to be inertial. Finally, when studying the motion of satellites, it becomes necessary to select an inertial frame attached to the sun.
|
||||
|
||||
@ -2670,11 +2670,11 @@ $$
|
||||
\underline{{a}}=\frac{\mathrm{d}\underline{{v}}}{\mathrm{d}t}=\frac{\mathrm{d}^{2}\underline{{x}}_{P/O}}{\mathrm{d}t^{2}}.
|
||||
$$
|
||||
|
||||
# 3.1.2 Newton’s laws
|
||||
### 3.1.2 Newton’s laws
|
||||
|
||||
This section presents Newton’s three laws and Newton’s law of gravitation. These laws provide the foundation of dynamics and mechanics.
|
||||
本节介绍牛顿三定律和牛顿万有引力定律。这些定律是动力学和力学的基础。
|
||||
# Newton’s first law
|
||||
#### Newton’s first law
|
||||
|
||||
Newton’s first law of motion states that every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. The expression “state of uniform motion” means that the object moves at a constant velocity. If several forces are applied to the object, the “external force” is, in fact, the resultant, i.e., the vector sum, of all externally applied forces. Finally, the “object” mentioned in the law is to be understood as a particle, as defined in the previous section.
|
||||
|
||||
@ -2696,7 +2696,7 @@ For statics problems, is is customary to focus on particles at rest rather than
|
||||
|
||||
对于静力学问题，通常关注的是静止的粒子，而不是以恒定速度运动的粒子。 在这个框架下，牛顿第一定律变为：一个粒子静止，当且仅当作用于其上的外力之和为零。 这个陈述提供了静力学平衡的定义，并且是静力学和结构力学的基础。
|
||||
|
||||
# Newton’s second law
|
||||
#### Newton’s second law
|
||||
|
||||
Newton’s second law states that if a force is acting on a particle, its acceleration is proportional to this force; the constant of proportionality is the mass of the particle. Here again, the force acting on the particle is the vector sum of all externally applied forces. Both externally applied force and resulting acceleration must be understood as vector quantities, and furthermore, the acceleration vector is the inertial acceleration vector as defined by eq. (3.3). Newton’s second law then states
|
||||
|
||||
@ -2725,20 +2725,21 @@ $$
|
||||
|
||||
显然，牛顿第一定律是由第二定律蕴含的。牛顿第二定律提供了粒子的运动方程；它将粒子的运动与外部施加的力联系起来。
|
||||
|
||||
# Newton’s third law
|
||||
#### Newton’s third law
|
||||
|
||||
Newton’s third law is also of fundamental importance to dynamics. It states: if particle A exerts a force on particle $\textbf{\emph{B}}$ , particle $\textbf{\emph{B}}$ simultaneously exerts on particle A a force of identical magnitude and opposite direction. It is also postulated that these two forces share a common line of action. In a more compact manner, Newton’s third law states that
|
||||
|
||||
**Law 3 (Newton’s third law)** Two interacting particles exert on each other forces of equal magnitude, opposite directions, and sharing a common line of action.
|
||||
|
||||
Newton’s third law is most useful when dealing with systems of particles: it enables the appropriate modeling of the interaction forces among the particles. It also allows “isolating” or “disconnecting” a particle from its surroundings and replacing the connection by a set of forces of equal magnitudes, opposite directions, and sharing a common line of action. This technique is the basis for drawing free body diagrams of a particle or system of particles.
|
||||
|
||||
牛顿第三定律在动力学中也具有根本的重要性。它指出：如果粒子 A 对粒子 $\textbf{\emph{B}}$ 施加一个力，粒子 $\textbf{\emph{B}}$ 同时对粒子 A 施加一个大小相同、方向相反的力。 此外，还假定这两个力共线。 更简洁地表述，牛顿第三定律指出：
|
||||
|
||||
**定律 3 (牛顿第三定律)** 相互作用的两个粒子对彼此施加大小相等、方向相反、共线的力。
|
||||
|
||||
牛顿第三定律在处理由粒子组成的系统时最为有用：它能够对粒子间的相互作用力进行适当的建模。 它还允许“隔离”或“断开”一个粒子与其周围环境的连接，并用一组大小相等、方向相反、共线的力来代替这种连接。 这种技术是绘制粒子或粒子系统的受力图的基础。
|
||||
|
||||
# Newton’s law of gravitation
|
||||
#### Newton’s law of gravitation
|
||||
|
||||
Newton’s law of gravitation also plays an important role in dynamics. It states that
|
||||
|
||||
@ -2752,12 +2753,24 @@ $$
|
||||
|
||||
where $F$ is the magnitude of the attractive force, $m_{1}$ and $m_{2}$ the masses of the two particles, $r$ their relative distance, and $G$ the constant of proportionality know as the universal constant of gravitation.
|
||||
|
||||
牛顿万有引力定律在动力学中也扮演着重要的角色。它阐述如下：
|
||||
|
||||
法 4 (牛顿万有引力定律) 两个粒子之间相互吸引，其吸引力与它们的质量成正比，与它们相对距离的平方成反比。作用线的方向连接这两个粒子。
|
||||
|
||||
这暗示着：
|
||||
|
||||
$$
|
||||
F=G{\frac{m_{1}m_{2}}{r^{2}}},
|
||||
$$
|
||||
|
||||
其中，$F$ 是吸引力的量值，$m_{1}$ 和 $m_{2}$ 是两个粒子的质量，$r$ 是它们的相对距离，$G$ 是比例常数，又称万有引力常量。
|
||||
|
||||
Fig. 3.1. Gravitation force acting between two particles.
|
||||
|
||||
Figure 3.1 shows the force, $\underline{{F}}_{12}$ , that the second particle exerts on the first, and the force, $\underline{{F}}_{21}$ , that the first exerts on the second. Forces $\underline{{F}}_{12}$ and $\underline{{F}}_{21}$ have the same magnitude ${\textbf{\textit{F}}}=$ $\|\underline{{F}}_{12}\|\ =\ \|\underline{{F}}_{21}\|$ , opposite directions $\underline{{F}}_{12}+$ $\underline{{F}}_{21}\;=\;0$ , and share a common line of action that joins the two particles. Clearly, these two forces present an important example of Newton’s third law.
|
||||
|
||||
# 3.1.3 Systems of units
|
||||
图 3.1 显示了第二个粒子对第一个粒子施加的力，$\underline{{F}}_{12}$，以及第一个粒子对第二个粒子施加的力，$\underline{{F}}_{21}$。力 $\underline{{F}}_{12}$ 和 $\underline{{F}}_{21}$ 具有相同的 magnitude ${\textbf{\textit{F}}}=$ $\|\underline{{F}}_{12}\|\ =\ \|\underline{{F}}_{21}\|$，相反的方向 $\underline{{F}}_{12}+$ $\underline{{F}}_{21}\;=\;0$，并且共享一条连接两个粒子的作用线。显然，这两个力呈现了牛顿第三定律的一个重要例子。
|
||||
### 3.1.3 Systems of units
|
||||
|
||||
The quantities involved in Newton’s three laws are length, mass, time, and force, denoted $L,\,M,\,T$ , and $F$ , respectively. In view of Newton’s second law, eq. (3.4), these three quantities are not independent, rather $F=M L/T^{2}$ .
|
||||
|
||||
@ -2765,6 +2778,11 @@ This text uses the $S I$ system of units exclusively. In this system of units, t
|
||||
|
||||
In this set of units, the universal constant of gravitation is
|
||||
|
||||
涉及牛顿三大定律的量包括长度、质量、时间和力，分别用 $L,\,M,\,T$ 和 $F$ 表示。根据牛顿第二定律，公式 (3.4) 表明，这三个量并非独立，而是 $F=M L/T^{2}$。
|
||||
|
||||
本文本专 exclusively 使用 $S I$ 单位制。在这个单位制中，三个基本单位是长度、质量和时间，分别以米（$\mathrm{^{\circ}m}$）、千克（“kg”）和秒（“s”）为单位进行测量。力则是一个以牛顿（“N”）为单位测量的导出单位。一个 $1\,\mathrm{N}$ 的力可以使 $1\,\mathrm{kg}$ 的质量获得 $1\;\mathrm{m}/\mathrm{s}^{2}$ 的加速度。以质量为基本单位的单位制被称为绝对单位制：$S I$ 单位制是一种绝对单位制。
|
||||
|
||||
在这个单位制中，万有引力常数是
|
||||
$$
|
||||
G=6.6732\;10^{-11}\;\mathrm{m}^{3}/(\mathrm{kg}\cdot\mathrm{s}^{2}).
|
||||
$$
|
||||
@ -2773,6 +2791,9 @@ In view of the small value of this constant, the attractive force acting between
|
||||
|
||||
The weight, $w$ , of a particle at the surface of the earth is defined as the gravitational force applied by the earth to the particle,
|
||||
|
||||
鉴于该常数值很小，小物体的吸引力非常微弱。然而，如果其中一个粒子具有很大的质量，则粒子间的吸引力会很大。
|
||||
|
||||
地球表面粒子的重量，表示为 $w$ ，定义为地球施加于该粒子的引力，
|
||||
$$
|
||||
w={\frac{G M}{r_{e}^{2}}}\;m,
|
||||
$$
|
||||
@ -2783,11 +2804,22 @@ Because the earth is not a perfect sphere and its mass distribution is not unifo
|
||||
|
||||
In the US customary system of units, the three basic unit are length, time, and force, measured in feet, denoted “ft,” seconds, denoted “s,” and pounds, denoted “lbs,” respectively. In this system, mass is then a derived unit measured in slugs, denoted “slug.” A mass of 1 slug weighs $1\,1\mathrm{b}$ when subjected to a gravitational acceleration of $1\:\mathrm{ft/s^{2}}$ . Systems of units where force is a basic unit are said to be gravitational: the US customary system is a gravitational system. In the US customary system, $g=32.17\,\mathrm{ft/s^{2}}$ , and the mass of a particle at the surface of the earth is then found as $m=w/g$ . It should be noted that in the US customary system, length is sometimes measured in inches rather than feet; in this case, $g=386\:\mathrm{in/s^{2}}$ .
|
||||
|
||||
# 3.1.4 The principle of work and energy
|
||||
其中 $M=5.976~10^{24}\,\mathrm{kg}$ 是地球的质量，$r_{e}=6,378\,\mathrm{km}$ 是其半径，而 $m$ 是位于地球表面的粒子的质量。利用这些常数，可以得出粒子的重量为 $w=9.803\;m=g m$ ，其中 $g=9.803\:\mathrm{m/s^{2}}$ 是地球表面的重力加速度常数。
|
||||
|
||||
由于地球不是完美的球体，且其质量分布不均匀，因此应预计从一点到另一点的重力加速度常数会有小幅度变化。对于大多数动力学问题，$g=9.81\:\mathrm{m/s^{2}}$ 将是重力加速度常数的一个足够精确的值。在地球表面，一个 $80\,\mathrm{kg}$ 的人的重量为 $w=9.81\times80=785\,\mathrm{\,N}$ 。
|
||||
|
||||
在英制单位制中，三个基本单位是长度、时间和力，分别以英尺（“ft”）、秒（“s”）和磅（“lbs”）为单位。在这个系统中，质量是派生单位，以斯莱格（“slug”）为单位。质量为 1 斯莱格的物体在重力加速度为 $1\:\mathrm{ft/s^{2}}$ 时重 $1\,1\mathrm{b}$。以力为基本单位的单位系统被称为重力系统：英制单位制是一个重力系统。在英制单位制中，$g=32.17\,\mathrm{ft/s^{2}}$ ，因此在地球表面，粒子的质量可以表示为 $m=w/g$ 。需要注意的是，在英制单位制中，长度有时以英寸而不是英尺为单位；在这种情况下，$g=386\:\mathrm{in/s^{2}}$ 。
|
||||
|
||||
|
||||
### 3.1.4 The principle of work and energy 能量守恒原理
|
||||
|
||||
Figure 3.2 depicts a particle of mass $m$ whose position is described by position vector ${\underline{{r}}}(t)$ with respect to an inertial frame, $\begin{array}{r l}{\mathcal{F}^{I}}&{{}=}\end{array}$ $[\mathbf{O},\mathcal{T}=(\bar{\iota}_{1},\bar{\iota}_{2},\bar{\iota}_{3})]$ . While moving along its path, the particle is acted upon by forces, the resultant of which is $\underline{{F}}(t)$ . These forces are called externally applied forces, or impressed forces.
|
||||
|
||||
The differential work, dW, the resultant force performs on the particle as it moves by an differential distance, $\mathrm{d}\underline{{r}},$ is defined as the scalar product of the force vector by the differential displacement vector of its point of application
|
||||
图 3.2 描绘了一个质量为 $m$ 的粒子，其位置由相对于惯性参考系的位移矢量 ${\underline{{r}}}(t)$ 描述，$\begin{array}{r l}{\mathcal{F}^{1}}&{{}=}\end{array}$ $[\mathbf{O},\mathcal{T}=(\bar{\iota}_{1},\bar{\iota}_{2},\bar{\iota}_{3})]$ 。在运动轨迹上，粒子受到力的作用，其合力为 $\underline{{F}}(t)$ 。这些力被称为外部施加力，或冲击力。
|
||||
|
||||
合力所做的微分功，dW，当粒子在微分位移 $\mathrm{d}\underline{{r}}$ 移动时，定义为力矢量与应用点微分位移矢量的标量积。
|
||||
|
||||
|
||||
|
||||
Fig. 3.2. Force acting on a particle.
|
||||
@ -2800,6 +2832,9 @@ In view of the definition of the scalar product, this differential work can be w
|
||||
|
||||
Introducing Newton’s second law, eq. (3.4), into the definition of the differential work leads to
|
||||
|
||||
鉴于标量积的定义，这个微分功可以写成 $\mathrm{d}W\,=\,\|\underline{{F}}\|\|\mathrm{d}\underline{{r}}\|\cos\theta$ ，其中 $\theta$ 是力矢量和微分位移矢量之间的夹角，见图 3.2。如果力垂直于微分位移，微分功为零，尽管力的大小是有限的。符号 $\mathrm{d}W$ 用于表示微分功，但它并不暗示存在一个功函数 $W$，使得 $\mathrm{d}(W)$ 是微分功。
|
||||
|
||||
将牛顿第二定律，公式 (3.4)，引入微分功的定义，得到
|
||||
$$
|
||||
\mathrm{d}{\boldsymbol{W}}={\underline{{F}}}^{T}\mathrm{d}{\underline{{r}}}=m{\underline{{a}}}^{T}\mathrm{d}{\underline{{r}}}=m{\frac{\mathrm{d}{\underline{{v}}}^{T}}{\mathrm{d}t}}{\frac{\mathrm{d}{\underline{{r}}}}{\mathrm{d}t}}{\mathrm{~d}t}=m{\frac{\mathrm{d}{\underline{{v}}}^{T}}{\mathrm{d}t}}{\underline{{v}}}\,\mathrm{d}t=m\,{\underline{{v}}}^{T}\mathrm{d}{\underline{{v}}}.
|
||||
$$
|
||||
@ -2817,6 +2852,7 @@ $$
|
||||
$$
|
||||
|
||||
Consider now two arbitrary instants during the motion of the particle, say times $t_{i}$ and $t_{f}$ , as illustrated in fig. 3.2. The work done by the force over this period is denoted $W_{t_{i}\to t_{f}}$ and can be evaluated as follows
|
||||
现在考虑粒子运动过程中的两个任意时刻，例如图 3.2 所示的 $t_{i}$ 和 $t_{f}$。这段时间内力所做的功，记为 $W_{t_{i}\to t_{f}}$，计算方法如下：
|
||||
|
||||
$$
|
||||
W_{t_{i}\rightarrow t_{f}}=\int_{t_{i}}^{t_{f}}\underline{{F}}^{T}\mathrm{d}\underline{{r}}=\int_{t_{i}}^{t_{f}}\mathrm{d}(K)=K(t_{f})-K(t_{i})=K_{f}-K_{i}=\varDelta K.
|
||||
@ -2825,10 +2861,13 @@ $$
|
||||
This result is known as the principle of work and energy.
|
||||
|
||||
Principle 1 (Principle of work and energy for a particle) The work done by the external forces acting on a particle equals the change in the particle’s kinetic energy.
|
||||
这个结果被称为能量守恒原理。
|
||||
|
||||
# 3.2 Conservative forces
|
||||
原理 1 (能量守恒原理，适用于单个质点) 作用于质点的外力所做的功等于质点动能的变化。
|
||||
## 3.2 Conservative forces
|
||||
|
||||
Figure 3.2 depicts a particle of mass $m$ whose position is described by position vector ${\underline{{r}}}(t)$ with respect to an inertial frame, $\mathcal{F}^{I}=[\mathbf{O},\mathcal{T}=(\bar{\iota}_{1},\bar{\iota}_{2},\bar{\iota}_{3})]$ . Conservative forces are a class of forces that depend only upon the position of the particles on which they act, $\underline{{F}}=\underline{{F}}(\underline{{r}})$ . Although these forces may vary with time as the particle moves, they do not depend explicitly on time or velocity. Figure 3.3 shows two arbitrary paths, denoted ACB and ADB, along which the particle moves in space from point A to point $\mathbf{B}$ .
|
||||
图 3.2 描绘了一个质量为 $m$ 的粒子，其位置由相对于惯性系 $\mathcal{F}^{I}=[\mathbf{O},\mathcal{T}=(\bar{\iota}_{1},\bar{\iota}_{2},\bar{\iota}_{3})]$ 的位置矢量 ${\underline{{r}}}(t)$ 描述。保守力是一类仅取决于它们作用的粒子位置的力，$\underline{{F}}=\underline{{F}}(\underline{{r}})$ 。 尽管这些力在粒子运动时可能随时间变化，但它们不显式地依赖于时间或速度。图 3.3 显示了两个任意路径，分别标记为 ACB 和 ADB，粒子沿这些路径在空间中从点 A 移动到点 $\mathbf{B}$ 。
|
||||
|
||||
# Definition
|
||||
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user