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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \sum_i F_i \,}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \times G = \sum_i r_i \times F_i .}

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_i r_i \times F_i = 0} , there is no net moment and the conditions are satisfied by taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \sum_i F_i \,} and s=0.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_i r_i \times F_i \neq 0} , the second condition is soluble only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_i F_i \,} is perpendicular to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.