PlanetPhysics/Concurrent Forces in Equilibrium in Three Dimensions

Any set of concurrent forces (happening at the same time) whose vector sum or resultant vanishes is said to be in equilibrium because any particle subjected to the action of such a set of forces at their point of concurrence would not have its velocity altered and would therfore remain static. If $$n$$ forces are in equilibrium, we may write the vector equation

$$ \vec{F}_1 + \vec{F}_2 + \cdots + \vec{F}_n = \sum_{s=1}^{s=n} \vec{F}_s = \sum \vec{F} = 0 $$

Note that this vector equation does not imply the scalar equation $$\sum F = 0$$ except in the very special case of collinear forces. Using double subscripts to represent the $$x$$, $$y$$, and $$z$$ components as well as to identify the forces, we note, however, that this vector equation is equivalent to the following set of three scalar equation:

$$ F_{x1} + F_{x2} + \cdots + F_{xn} = \sum F_x = 0; \,\,\, \sum F_y = 0; \,\,\, \sum F_z = 0 $$

Here $$F_{x1}$$ represents the $$x$$ component (scalar) of $$\vec{F}_1$$, etc. Any one of a set of forces in equilibrium is the equilibrant ( the negative of the resultant) of all the others.

In terms of the direction cosines of the direction angles of the individual forces designated we may also write the three scalar Eqs. (2) in the form:

$$ F_1 \cos \theta_{x1} +F_2 \cos \theta_{x2} + \cdots + F_n \cos \theta_{xn} = l_1 F_1 +l_2F_2 + \cdots + l_n F_n = \sum l F = 0 $$

$$ \sum m F = 0;  \,\,\,\,\,\, \sum n F = 0 $$

Here $$\theta_{x1}$$ is the angle between $$\vec{F}_1$$ and the positive $$x$$ direction, while $$l_1 = \cos \theta_{x1}$$, etc.

With regard to the amount of information required in order to solve a problem on concurrent forces, or statics of a particle, it is important to note that the single vector of Eq. (1) or the equivalent three scalar algebraic Eqs. (2) provide the necessary and sufficient conditions for equilibrium. In the study of algebra we learned that three independent, simultaneous, linear equations such as those constituting Eqs. (2) are just sufficient to determine three unknown quantities. For a general system of concurrent forces, then, we could use Eqs. (2) to solve for as many as three unkown components, provided all the others were given. Instead of being required specifically to solve for components, we might be given everything but the magnitude and direction of one of the forces. This would be a problem with three unknowns.

The three conditions of Eqs. (2) for the equilibrium of a particle or a system of concurrent forces may be associated with the fact that three coordinates are required to specif the position of a particle in three dimensional space. Because each of these may be varied independently of the others, and three equations are required to describe its motion fully, we say that a particle whose motion is not constrained in any way has three degrees of freedom of motion. We shall find in general for any body that the number of conditions of equilibrium is equal to the number of degrees of freedom of its motion which could result from lack of equilibrium.


 * derivative of the Public domain work of [Broxon].