Equation of equilibrium and motion

we introduce a Cartesian system of coordinates with an origin 0 at the center of mass, Fig. 3.11b. The x and y axes are directed to north and east, respectively, and the z-axis is along the vertical. By definition, the moment of the force F(x, y, z) with components F, Fy, and F with respect to point 0 is 

In the case when masses can be treated as elementary and it is possible to neglect the bar s mass, the volume integral disappears and we have 

At equilibrium the moment tg is balanced by the moment of the elastic force of the thread 

Here k is the coefficient of resistance and I the moment of inertia of the system. 

The index 0 means that derivatives are taken at the origin where x — y — z — 0. Inasmuch as the thread is oriented along the vertical (plumb line), the gravitational field at the origin has only a vertical component, that is. 

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