Force Systems in Space

Forces in Space (3D Forces)
Magnitude of a force F in space

$F = \sqrt{{F_x}^2 + {F_y}^2 + {F_z}^2}$

Components of a force in space

$F_x = F \cos \theta_x$

$F_y = F \cos \theta_y$

$F_z = F \cos \theta_z$

Direction cosines

$\cos \theta_x = \dfrac{F_x}{F}$

$\cos \theta_y = \dfrac{F_y}{F}$

$\cos \theta_z = \dfrac{F_z}{F}$

Proportion of components

$\dfrac{F_x}{x} = \dfrac{F_y}{y} = \dfrac{F_z}{z} = \dfrac{F}{d}$

Moment of a force about an axis

$M_x = zF_y \pm yF_z$

$M_y = zF_x \pm xF_z$

$M_z = yF_x \pm xF_y$


Resultant of Concurrent Force Systems in Space
Components of the resultant

$R_x = \Sigma F_x$

$R_y = \Sigma F_y$

$R_z = \Sigma F_z$

Magnitude of the resultant

$R = \sqrt{{R_x}^2 + {R_y}^2 + {R_z}^2}$


Equilibrium of Concurrent Space Forces
The resultant of all forces is zero

$\Sigma F_x = 0$

$\Sigma F_y = 0$

$\Sigma F_z = 0$

The sum of moment is zero

$\Sigma M_x = 0$

$\Sigma M_y = 0$

$\Sigma M_z = 0$