Unit 1 General Equilibrium__ Conditions of A System

General Equilibrium Conditions of A System力系的一般平衡条件In this section, we shall consider the conditions that the forces and couples acting upon a body must satisfy in order to be in equilibrium.在这一部分，我们将研究为了使一个物体保持平衡，作用在其上的力和力偶所必须满足的条件。

According to Newton’s first law, the sum of the forces exerted on a body at rest must be zero. Notice, however, that this law says nothing about the moment, or rotational effects, of the forces. Clearly, the total moment must also be zero, else the body would rotate.根据牛顿第一定律，施加在一个静止物体上的力系的合力一定为零。

然而，请注意这个定律对力矩或力系的转动效应只字未提。

显然，合力矩也一定为零，否则物体将会转动。

The fundamental problem here is that Newton’s first law (and second law), as originally stated, applies only for very small bodies, or particles, with negligible dimensions and nonzero mass. However, it can be extended to bodies of finite size as follows.这里的基本问题是原先叙述的牛顿第一定律（和第二定律）只适用于非常小的物体，或者尺寸可以忽略的非零质量的粒子。

然而，它可以扩展到如下所述的有限尺寸的物体。

Consider a system consisting of two particles, and let 1f and 2f be the forces due to the interaction between them (Fig .1.1). These forces arecalled internal forces, since theyare due to interactions betweenbodies within the system.Assuming that the internal forcesobey Newton ’s third law, wehave 12f f =-.Suppose thatthere are also forces, such as1,2F F and 3F , exerted on theparticles due to interactions with bodies outside the system. Such forces are called external forces. Clearly, and the forces acting upon a particular particle must have the same point of application because a particle has negligible dimensions.考虑一个由两个质点组成的系统，假设1f 和2f 为它们之间的相互作用力(图.1.1)。

这些力称为内力，因为它们是由于系统内部的物体之间的相互作用而产生的。

假定内力服从牛顿第三定律，我们有12f f =-。

假如还有质点与系统外物体之间的相互作用力施加在质点上，如1,2F F 和3F ,这些力称为外力。

显然，作用在一个特定粒子上的力一定有相同的应用，因为粒子的尺寸可以忽略。

We shall say that the system is in equilibrium if each particle within it is in equilibrium. In this case, by Newton ’s first law, the sum of the forces acting upon each particle must be zero. For particle A we have如果系统内的每一个质点处于平衡，我们就可以说系统是平衡的。

在本例中，依据牛顿第一定律，作用在每个质点上的力的合力一定为零。

对质点A 我们有：∑=++=0121f F F F Aand for particle B而对质点B 有：∑032=+=F f F BThe total force acting upon the system is作用在系统上的力的总和为：123120A B F F F F F F f f =+=++++=∑∑∑Now let us consider the total moment of these forces about some point P. Referring to Fig.1.1, we have现在我们来研究这些力对于同一点P 的合力矩。

由图1.1，我们有：12()()P A B Mr F r F =⨯+⨯∑∑∑ But 0A B F F ==∑∑; so the total moment must also be zero, as stated previously.但0A B F F ==∑∑；所以，如先前所述，合力矩也一定为零。

Since the forces 1f and 2f have the same line of action, the moment condition can be rewritten as由于力1f 和2f 有相同的作用线，力矩的条件可以改写为1121223()0P M r F F f f r F =⨯++++⨯=∑But 12f f =-; so the conditions on the forces and moments reduce to 但12f f =-；所以力和力矩的条件简化为1210F FF F +=+=∑And和 111223()()()0P M r F r F r F =⨯+⨯+⨯=∑In other words, if the system is in equilibrium, the sum of the external forces acting upon it is zero and so is the sum of the moments of these forces about an arbitrary point. The internal forces need not be considered because their effects cancel out.换句话说，如果系统处于平衡，那么作用在其上的合外力一定为零，而且这些力对于任一点的合力矩也为零。

内力不需要考虑，因为它们的效应相互抵消了。

Although we shall not go through the details, it should not be too difficult to see that the preceding results hold for a system consisting of any number of particles acted upon by any number of external forces, provided the internal forces obey Newton ’s third law. In particular, these results apply to bodies of finite extent, since such bodies can be thought of as consisting of a large number very small pieces, or particles. Thus, we have the following general equilibrium conditions:尽管我们不去研究细节，也不难发现，倘若内力服从牛顿第三定律，前述结果适用于由许多有多个外力作用其上的质点组成的系统。

特别是，这些结果也适用于有限尺寸的物体，因为这样的物体可认为是由大量微体或质点组成的。

因此，我们得到下列一般平衡条件：If a system is in equilibrium, then如果系统处于平衡，那么0F =∑ and 0P M =∑ (1.1) whereF ∑ is the sum of the external forces acting upon the system and P M ∑ is the total moment of these forces about an arbitrary point, including the moment of any couples which may be acting.这里F ∑是作用在系统上的所有外力的总和，而P M ∑是这些力对任意点的合力矩，包括系统中可能作用有的力偶的矩。

Equations (1.1) are necessary conditions for equilibrium; i.e., if the system is in equilibrium, these equations must be satisfied. They are not, in general, sufficient conditions for equilibrium. This presents no difficulties, however, for we shall be dealing only with systems known to be in equilibrium. Equations (1.1) are both necessary and sufficient conditions for equilibrium of a rigid body. Proof that they are sufficient requires use of Newton’s second law and other knowledge beyond the level of this text.方程（1.1）是平衡的必要条件；也就是说，如果系统处于平衡，必须满足这些方程。