For the loading condition shown calculate the erector spinae

For the loading condition shown, calculate the erector spinae muscle force F_M and the compressive and shear components of joint reaction force (F_JC and F_JS) at the L5 vertebrae (red square)

Solution

Equating torques about the point O:

Fm*0.05 = 450*0.25 + 200*0.4

So, Fm = 3850 N <-------answer

Equating forces in vertical:

450 + 200 + Fm*cos(35) = Fjc*cos(30) + Fjs*sin(35)

So, 450 + 200 + 3850*cos(35 deg) = Fjc*cos(35 deg) + Fjs*sin(35 deg) ------- (1)

Now, equating forces in horizontal:

3850*sin(35 deg) + Fjs*cos(35 deg) = Fjc*sin(35 deg) -------- (2)

Solving both equations simultaneously we get :

Fjc = 4382.4 N <-----answer

Fjs = 372.8 N <-------answer