《理论力学 Theoretical Mechanics——动力学》（双语版）第十二章 动量定理 theorem of momentum

Theoretical Mechanics Chapter 12: theorem of momentum

1 Theoretical Mechanics

理论力学

2

Summary of the general theorems of dynamics Problem for one particle dynamics Establish the differential equations of motion for a particle and solve them Problem for the dynamics of a system of particles. In principle, we can write down 3n differential equations for a system of n particles Practical problems are 1. Combining and solving differential equations(performing the integral operation) is very difficult 2. In a great number of problems we only need to investigate the motion of the system of particles as a whole without the necessity to know the motion of every particle in the system. Starting with the present chapter we introduce other methods of solving dynamical problems. The general theorems of dynamics (including the theorem of momentum, the theorem of kinetic energy. the theorem of moment of momentum and some others theorems derived from them) will be introduced

3 Summary of the General Theorems of Dynamics Problem for one particle dynamics Establish the differential equations of motion for a particle and solve them. Problem for the dynamics of a system of particles. In principle, we can write down 3n differential equations for a system of n particles and then solve them. Practical problems are : 1.Combining and solving differential equations (performing the integral operation) is very difficult. 2.In a great number of problems we only need to investigate the motion of the system of particles as a whole without the necessity to know the motion of every particle in the system. Starting with the present chapter we introduce other methods of solving dynamical problems. The general theorems of dynamics (including the theorem of momentum, the theorem of kinetic energy, the theorem of moment of momentum and some others theorems derived from them) will be introduced

学 动力学普遍定理概述 对质点动力学问题:建立质点运动微分方程求解。 对质点系动力学问题:理论上讲,n个质点列出3n个微分方 程,联立求解它们即可 实际上的问题是:1、联立求解微分方程(尤其是积分问题)非 常困难。 2、大量的问题中,不需要了解每一个质 点的运动仅需要研究质点系整体的运 动情况。 从本章起,将要讲述解答动力学问题的其它方法,而首先要讨论 的是动力学普遍定理(包括动量定理、动量矩定理、动能定理及由此 推导出来的其它一些定理)

4 实际上的问题是： 1、联立求解微分方程(尤其是积分问题)非 常困难。 2、大量的问题中，不需要了解每一个质 点的运 动,仅需要研究质点系整体的运 动情况。 动力学普遍定理概述 对质点动力学问题： 建立质点运动微分方程求解。 对质点系动力学问题：理论上讲，n个质点列出3n个微分方 程， 联立求解它们即可。 从本章起, 将要讲述解答动力学问题的其它方法, 而首先要讨论 的是动力学普遍定理(包括动量定理、动量矩定理、动能定理及由此 推导出来的其它一些定理)

They show the dependence between two kinds of quantities in concise mathematical forms. One kind are the quantities related to the characteristics of motion(momentum, moment of momentum, kinetics energy etc), The second kind are the quantities related to the forces (impulse, moment of a force, work, etc. In this chapter we will investigate the theorem of momentum of a particle or a system of particles and establish the relation between the change of momentum and the impulse of a force. In addition we ill study another important form of the theorem of momentum, the theorem of motion of the center of mass

5 They show the dependence between two kinds of quantities in concise mathematical forms. One kind are the quantities related to the characteristics of motion (momentum, moment of momentum, kinetics energy etc), The second kind are the quantities related to the forces (impulse, moment of a force, work, etc.) In this chapter we will investigate the theorem of momentum of a particle or a system of particles and establish the relation between the change of momentum and the impulse of a force. In addition we will study another important form of the theorem of momentum, the theorem of motion of the center of mass

学 它们以简明的数学形式,表明两种量—一种是同运动 特征相关的量(动量、动量矩、动能等),一种是同力相关的量 (冲量、力矩、功等)—一之间的关系,从不同侧面对物体的 机械运动进行深入的研究。在一定条件下,用这些定理来解答 动力学问题非常方便简捷。 本章中研究质点和质点系的动量定理,建立了动量的改变 与力的冲量之间的关系,并研究质点系动量定理的另一重要形 式质心运动定理

6 它们以简明的数学形式， 表明两种量 —— 一种是同运动 特征相关的量(动量、动量矩、动能等)，一种是同力相关的量 (冲量、力 矩、功等) —— 之间的关系，从不同侧面对物体的 机械运动进行深入的研究。在一定条件下，用这些定理来解答 动力学问题非常方便简捷 。 本章中研究质点和质点系的动量定理，建立了动量的改变 与力的冲量之间的关系，并研究质点系动量定理的另一重要形 式——质心运动定理

Chapter 12: Theorem of momentum D812-1 The center of mass of a system of particles, external forces and internal forces 四§12-2 Momentum and impulse D$12-3 Theorem of momentum 四§12-4 Theorem of motion of the center of mass

7 §12–1 The center of mass of a system of particles, external forces and internal forces §12–2 Momentum and impulse §12–3 Theorem of momentum §12–4 Theorem of motion of the center of mass Chapter 12: Theorem of momentum

第十二章动量定理 §12-1质点系的质心,内力与外力 四§12-2动量与冲量 §12-3动量定理 四§12-4质心运动定理

8 §12–1 质点系的质心, 内力与外力 §12–2 动量与冲量 §12–3 动量定理 §12–4 质心运动定理 第十二章 动量定理

Dynamics 8 12-1 The center of mass of a system of particles, external forces and internal forces 1. The center of mass The center of mass of a system of particles is called center of mass. It is an important concept representing the distribution of mass in any system of particles The position of the center of mass c is (M=>m) ∑m OrMC示 ∑ From r=x i+y+=k, we got ∑mx ∑m ∑m M

9 1. The center of mass. The center of mass of a system of particles is called center of mass. It is an important concept representing the distribution of mass in any system of particles. §12-1 The center of mass of a system of particles, external forces and internal forces ( =  ) M mi  =  C = i i i i C Mr m r M m r r or From r x i y j z k ,we got c = c + c + c M m z z M m y y M m x x i i C i i C i i C    = , = , = The position of the center of mass c is

§12-1质点系的质心,内力与外力 一.质点系的质心 质点系的质量中心称为质心。是表征质点系质量分布情况的 个重要概念。 质心C点的位置:(M=∑m2) 元=一2或M=∑m万 设=x+y27+=k,则 ∑mx,y miyi ∑m1 M M 10

10 一.质点系的质心 质点系的质量中心称为质心。是表征质点系质量分布情况的 一个重要概念。 §12-1 质点系的质心，内力与外力 ( =  ) M mi   = C = i i i i C Mr m r M m r r 或 设rc = xc i + yc j + zc k ,则 M m z z M m y y M m x x i i C i i C i i C    = , = , = 质心 C 点的位置：