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Statics

The enormous technology of Mechanical Engineering can be in some naïve sense be reduced to the two equations

$$\dot{\mbox{\boldmath $\vec{p}$\unboldmath }} = \mbox{\boldm . . . . . . oldmath }}_O = \mbox{\boldmath $\vec{\tau}$\unboldmath }_O .$$

Whole courses are taught on what amounts to these two equations and the various tricks for solving them in different types of situations. Fortunately, this isn't one of them! Just to give a flavour, however, I will mention the basic problem-solving technique of Statics, the science of things that are sitting still!^11.18 That means   ${\displaystyle \dot{\mbox{\boldmath$\vec{p}$\unboldmath }} = 0}$  and   ${\displaystyle \dot{\mbox{\boldmath$\vec{L}$\unboldmath }}_O = 0}$  so that the relevant equations are now

$$\sum \; \mbox{\boldmath $\vec{F}$\unboldmath } \; = \; 0 \q . . . . . . \sum \; \mbox{\boldmath $\vec{\tau}$\unboldmath }_O \; = \; 0$$

where the  $\sum$  [summation] symbols emphasize that there is never just one force or one torque acting on a rigid body in equilibrium; if there were, it (the force or torque) would be unbalanced and acceleration would inevitably result!

To solve complex three-dimensional Statics problems it is often useful to back away from our nice tidy vector formalism and explicitly write out the ``equations of equilibrium'' in terms of the components of the forces along the   $\hat{x}, \hat{y}$ and $\hat{z}$  directions as well as the torques about the  x, y and z  axes [which meet at the origin  O]:

 

$$\sum F_x = \; 0 \qquad \qquad \sum \tau_x = \; 0$$ (11.22)

$$\sum F_y = \; 0 \qquad \qquad \sum \tau_y = \; 0$$ (11.23)

$$\sum F_z = \; 0 \qquad \qquad \sum \tau_z = \; 0$$ (11.24)

If you have some civil engineering to do, you can work it out with these equations. Or hire an Engineer. I suggest the latter.