Force Resultant

R = (\sum F_{x})^{2} + (\sum F_{y})^{2} θ = tan^{- 1} (\sum F_{y} / \sum F_{x})

The resultant of a concurrent 2D force system is the vector sum. Resolve each force into components, add, then recover magnitude and direction. The sense of θ follows the signs of ΣFx and ΣFy.

$R$ (N): Resultant magnitude
$F_{x}, F_{y}$ (N): Rectangular components
$θ$: Direction from +x

Step-by-step solved example

Forces 40 N at 0° and 30 N at 90° act at a point. Find R and θ.

Concurrent forces add as a parallelogram / components.

1. Components
ΣFx = 40, ΣFy = 30.
2. Magnitude
R = √(40²+30²) = 50 N.
$R = 50 t e x t N$
3. Angle
θ = tan⁻¹(30/40) = 36.9°.
$\theta=36.9^\\circ$

Answer: R = 50 N at 36.9°

10 practice questions

0/10 correct

1.3-4-5 components give R =

2.ΣFx = 0 and ΣFy = 20 N → R =

3.Two equal opposite collinear forces have R =

4.θ = 180° means the resultant points

5.30 N east and 40 N north. R =

6.A 2D concurrent system is in equilibrium if

7.Closing the force polygon gives

8.ΣFx = −6, ΣFy = 8. R =

9.Resolve 100 N at 30° to x: Fx =

10.The resultant of concurrent forces always passes through