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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

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