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Space Forces

December 8, 2016
Published By : Pratik Kataria
Categorised in:

Rectangular Components of a Force in Space

$F_x$ = F cos$θ_x$ , $F_y$ = cos$θ_y$ , $F_z$ = cos$θ_z$

$\ov{F}$ = $F_x$i + $F_y$j + $F_z$k

l = cos$θ_x$ = ${F_x}/{F}$ Similarly for others.

$l^2 + m^2 + n^2 = 1$

Unit Vector

$\ov{F}$ = F $\ov{e}$
where,
$\ov{e}$ = unit vector in the direction of force F = (cos$θ_x$)i + (cos$θ_y$)j + (cos$θ_z$)k

Unit Vector when Force is Specified by Two Points

$\ov{F}$ = F $[{(x_2 – x_1)i+(y_2 – y_1)j+(z_2 – z_1)k}/{√{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}}]$

Components of Force when Orientation of Planes are Given

$F_x$ = F sin$θ_y$ cosα

$F_y$ = F cos$θ_y$ and

$F_z$ = F sin$θ_y$ sinα

Resultant of Concurrent Forces in Space

$R_x$ = Σ $F_x$

$R_y$ = Σ $F_y$

$R_z$ = Σ $F_z$

Magnitude of Resultant, R = $√{(R_x)^2 + (R_y)^2 + (R_z)^2}$

and directions cos $θ_x$ = ${R_x}/R$ , cos $θ_y$ = ${R_y}/R$ , cos $θ_z$ = ${R_z}/R$

Moment of a Force about a Point

$\ov{M}_o$ = $\ov{r}$ * $\ov{F}$

where,
$\ov{r}$ = Position vector of point of application A

$\ov{F}$ = Force vector


$M_o$ = r * F = $|{\table i,j,k;r_x,r_y,r_z;F_x,F_y,F_z}|$

Consider a force passing through points P and Q.

Moment about origin $M_o$ = $|{\table i,j,k;x,y,z;F_x,F_y,F_z}|$

Moment of a Force about a Line not Passing through Origin

$M_{BL}$ = $|{\table λ_x,λ_y,λ_z;(x_1 – x_2),(y_1 – y_2),(z_1 – z_2);F_x,F_y,F_z}|$

Varignon’s Theorem (Law of Moments)

$\ov{M}_R$ = Σ $\ov{M}$ (about same point)