Resultant (statics)

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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 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

${\displaystyle G=\sum _i{F}_i}$

${\displaystyle s\times G=\sum _i{r}_i\times F_i.}$

If ${\displaystyle \sum _i{r}_i\times F_i=0}$, there is no net moment and the conditions are satisfied by taking ${\displaystyle G=\sum _i{F}_i}$ and s=0.

If ${\displaystyle \sum _i{r}_i\times F_i\ne 0}$, the second condition is soluble only if ${\displaystyle \sum _i{F}_i}$ is perpendicular to ${\displaystyle \sum _i{r}_i\times F_i}$. 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

${\displaystyle \left(\sum _i{F}_i\right)\cdot (\sum _i{r}_i\times F_i)=0.}$

References[edit]

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