Resultant (statics)

In statics the resultant of a system of forces acting at various points on a rigid body or system of particles is a single force, acting at a single point, if one exists, which is equivalent to the given system.

Suppose that forces F_i act at points r_i. The resultant would be a single force G acting at a point s. The systems are equivalent if they have the same net force and the same net moment about any point.

These condition are equivalent to requiring that

$G=\sum _{i}F_{i}\,$

$s\times G=\sum _{i}r_{i}\times F_{i}.$

If $\sum _{i}r_{i}\times F_{i}=0$, there is no net moment and the conditions are satisfied by taking $G=\sum _{i}F_{i}\,$ and s=0.

If $\sum _{i}r_{i}\times F_{i}\neq 0$, the second condition is soluble only if $\sum _{i}F_{i}\,$ is perpendicular to $\sum _{i}r_{i}\times F_{i}$. Suppose that this necessary condition is satisfied. It is then the case that an appropriate s can be found.

We conclude that a necessary and sufficient condition for the system of forces to have a resultant is that

$\left(\sum _{i}F_{i}\right)\cdot \left(\sum _{i}r_{i}\times F_{i}\right)=0.\,$

References

• D.A. Quadling; A.R.D. Ramsay (1964). An Introduction to Advanced Mechanics. G. Bell and Sons, 102-103.