Equation of mass rotation

As before, suppose that at the beginning the spring extension compensates a weight and the lever spring system is in equilibrium. Then at some instant the mass is moved away from this position and an external force is removed. After this moment we start to observe a rotation of mass due to the resultant moment. As follows from Newton s second law, a motion of elementary mass is described by the equation 

Here I — ma is moment of inertia, a angular acceleration, and z the resultant moment. Note that we have neglected attenuation but in reality, of course, it is always present. This equation characterizes a motion for any angle a, but we consider only the vicinity of points of equilibrium. For this reason, the resultant moment in the linear approximation can be represented as 

In essence, we have expanded the moment in a power series in the vicinity of the point of equilibrium, a = a and assumed that within this range the derivative 5r/5a is constant. Taking into account the fact that at points of equilibrium r(a ) = 0 and introducing the notation 

Here /I is angular displacement of mass with respect to the point of equilibrium, 

with an increase of time, the mass m moves away from the point of equilibrium. 

---