Resultant (statics)

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This editable Main Article is under development and subject to a disclaimer.

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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 Fi act at points ri. 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=∑iFi
s×G=∑iri×Fi.

If ∑iri×Fi=0, there is no net moment and the conditions are satisfied by taking G=∑iFi and s=0.

If ∑iri×Fi≠0, the second condition is soluble only if ∑iFi is perpendicular to ∑iri×Fi. 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

(∑iFi)⋅(∑iri×Fi)=0.

## References[edit]

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