Riemann surface
matlab
无需担心！我们的理论力学代写专家团队将专业地解决您在理论力学学习中遇到的各种挑战。我们拥有广泛的专业知识和丰富的经验，可以协助您完成高水平的作业和论文，确保您在学习道路上顺利前行！
以下是一些我们可以帮助您解决的问题：
力学基础概念：涵盖质点运动、牛顿定律、功与能等各种常用理论力学概念的定义、性质和分类。
运动方程求解：研究和应用于理论力学中的运动方程求解，包括微分方程的求解方法和数值模拟等。
刚体力学：常见的刚体力学理论和刚体运动分析方法，如欧拉角、转动惯量等。
振动与波动：研究振动与波动现象的力学模型和解析方法，如简谐振动、波动方程等。
相对论力学：介绍相对论力学的基本原理和应用，如相对论运动动力学、洛伦兹变换等。
非线性力学：研究非线性力学问题的建模和分析方法，如混沌现象、非线性振动等。
力学系统建模：应用力学理论对实际系统进行建模和分析，如机械系统、流体力学系统等。
无论您面临的理论力学问题是什么，我们都会竭尽全力提供专业的帮助，确保您的学习之旅顺利无阻！
Question 2.3
The active gravitational mass $\left(m^A\right)$ of a particle is an attribute that enables it to establish a gravitational field in space, whereas the passive gravitational mass $\left(m^{\mathrm{P}}\right)$ is an atribute that enables the particle to respond to this field.
(a) Write Newton’s law of universal gravitation in terms of the relevant active and passive gravitational masses.
(b) Show that the third law of motion makes it unnecessary to distinguish between active and passive gravitational mass.
Solution
(a) The gravitational force $\mathbf{F}{12}$ that particle 1 exerts on particle 2 is proportional to the product of the active gravitational mass $m_1^A$ of particle 1 and the passive gravitational mass $m_2^{\mathrm{P}}$ of particle 2. Thus, the inverse-square law of gravitation is $$ \mathbf{F}{12}=-G \frac{m_1^{\mathrm{A}} m_2^{\mathrm{P}}}{r^2} \hat{\mathbf{r}}
$$
where $G$ is the universal constant of gravitation, $r$ is the distance between the particles and $\hat{\mathbf{r}}$ is a unit vector directed from particle 1 to particle 2. By the same token, the force $\mathbf{F}{21}$ which particle 2 exerts on particle 1 is $$ \mathbf{F}{21}=G \frac{m_2^{\mathrm{A}} m_1^{\mathrm{P}}}{r^2} \hat{\mathbf{r}} .
$$
(b) According to Newton’s third law, $\mathbf{F}{12}=-\mathbf{F}{21}$. It therefore follows from (1) and (2) that
$$
\frac{m_2^{\mathrm{A}}}{m_2^{\mathrm{P}}}=\frac{m_1^{\mathrm{A}}}{m_1^{\mathrm{P}}} .
$$
We conclude from (3) that the ratio of the active to the passive gravitational mass of a particle is a universal constant. Furthermore, this constant can be
14 Solved Problems in Classical Mechanics
incorporated in the universal constant $G$, which is already present in (1) and (2). That is, we can set $m^{\mathrm{P}}=m^{\mathrm{A}}$. There is no need to distinguish between active and passive gravitational masses; it is sufficient to work with just gravitational mass $m^{\mathrm{G}}$ and to write (1) as
$$
\mathbf{F}{12}=-G \frac{m_1^G m_2^G}{r^2} \hat{\mathbf{r}} $$ Comment Evidently, the same reasoning applies to the notions of active and passive electric charge. Thus, if one were to write Coulomb’s law for the electrostatic force between two charges in vacuum as $$ \mathbf{F}{12}=k \frac{q_1^{\mathrm{A}} q_2^{\mathrm{P}}}{r^2} \hat{\mathbf{r}}
$$
where $k$ is a universal constant, a discussion similar to the above would lead to
$$
\frac{q_2^{\mathrm{A}}}{q_2^{\mathrm{P}}}=\frac{q_1^{\mathrm{A}}}{q_1^{\mathrm{P}}}
$$
Consequently, the ratio of active to passive charge is a universal constant that can be included in $k$ in (5); it is sufficient to consider just electric charge $q$.
Question 2.4
The inertial mass of a particle is, by definition, the mass that appears in Newton’s second law. Consider free fall of a particle with gravitational mass $m^{\mathrm{G}}$ and inertial mass $m^{\mathrm{I}}$ near the surface of a homogeneous planet having gravitational mass $M^{\mathrm{G}}$ and radius $R$. Express the gravitational acceleration $a$ of the particle in terms of these quantities. (Neglect any frictional forces.)
Solution
The equation of motion is
$$
m^{\mathrm{r}} a=F
$$
where $F$ is the gravitational force exerted by the planet
$$
F=G \frac{M^{\mathrm{G}} m^{\mathrm{G}}}{R^2}
$$
(see Question 11.17). Thus
$$
a=\frac{m^{\mathrm{G}}}{m^1} \frac{G M^{\mathrm{G}}}{R^2}
$$
E-mail: help-assignment@gmail.com 微信:shuxuejun
help-assignment™是一个服务全球中国留学生的专业代写公司
专注提供稳定可靠的北美、澳洲、英国代写服务
专注于数学，统计，金融，经济，计算机科学，物理的作业代写服务