## Question:

A biomechanical model of the lumbar region of the human trunk is shown. The forces acting in the four muscle groups consist of ${ F }_{ R }$ for the rectus, ${ F }_{ O }$ for the oblique, ${ F }_{ L }$ for the lumbar latissimus dorsi, and ${ F }_{ E }$ for the erector spinae. These loadings are symmetric with respect to the $y - z$ plane. Replace this system of parallel forces by an equivalent force and couple moment acting at the spine, point $O$. Express the results in Cartesian vector form.

Given:

${ F }_{ R }$ = $35$ $N\quad a$ = $75$ $mm$

${ F }_{ O }$ = $45$ $N\quad b$ = $45$ $mm$

${ F }_{ L }$ = $23$ $N\quad c$ = $15$ $mm$

${ F }_{ E }$ = $32$ $N\quad d$ = $50$ $mm$

$e$ = $40$ $mm\quad f$ = $30$ $mm$

## Step-by-step

${ F }_{ Res }$ =$\sum { { F }_{ i }; } \quad { F }_{ Res }$=$2({ F }_{ R }+{ F }_{ O }+{ F }_{ L }+{ F }_{ E })$$\quad \quad$ ${ F }_{ Res }$= $270N$

${ M }_{ RO }$=$\sum { { M }_{ Ox } } ;\quad$ ${ M }_{ RO }$=$-2{ F }_{ R }a+2{ F }_{ E }c+2{ F }_{ L }b\quad \quad$ ${ M }_{ RO }$=$-2.22N.m$